No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
Analysis and implementation of TR-BDF2
Abstract
Bank et al. (1985) developed a one-step method, TR-BDF2, for the simulation of circuits and semiconductor devices based on the trapezoidal rule and the backward differentiation formula of order 2 that provides some of the important advantages of BDF2 without the disadvantages of a memory. Its success and popularity in the context justify its study and further development for general-purpose codes. Here the method is shown to be strongly S-stable. It is shown to be optimal in a class of practical one-step methods. An efficient, globally C1 interpolation scheme is developed. The truncation error estimate of Bank et al. (1985) is not effective when the problem is very stiff. Coming to an understanding of this leads to a way of correcting the estimate and to a more effective implementation. These developments improve greatly the effectiveness of the method for very stiff problems.
... We will base our work on the strategy proposed in [19] for the θ -method and extended in [3] to the TR-BDF2 method as fundamental single rate solver. The TR-BDF2 method has been originally introduced in [2] and more thoroughly analyzed in [10]. It is a second order, one step, L-stable implicit method endowed with a number of interesting properties, as discussed in [10]. ...
... The TR-BDF2 method has been originally introduced in [2] and more thoroughly analyzed in [10]. It is a second order, one step, L-stable implicit method endowed with a number of interesting properties, as discussed in [10]. As in [3], in our approach the choice of the time step size at each step is based on the technique proposed in [6]. ...
... For γ = 2 − √ 2, the method is L-stable and also employs the same Jacobian matrix for the two stages. In [10] it has been interpreted as a Diagonally Implicit Runge Kutta (DIRK) method with two internal stages, proving the following properties: ...
This work focuses on the development of a self adjusting multirate strategy based on an implicit time discretization for the numerical solution of hyperbolic equations, that could benefit from different time steps in different areas of the spatial domain. We propose a novel mass conservative multirate approach, that can be generalized to various implicit time discretization methods. It is based on flux partitioning, so that flux exchanges between a cell and its neighbors are balanced. A number of numerical experiments on both non-linear scalar problems and systems of hyperbolic equations have been carried out to test the efficiency and accuracy of the proposed approach.
... In this paper, we propose an extension of the solver for single-phase incompressible Navier-Stokes equations with an artificial compressibility formulation presented in 16,17 , so as to overcome well know issues of projection methods. The time discretization is therefore based on the TR-BDF2 scheme 17,18,19,20 , which is a second order two-stage method. A brief review of the TR-BDF2 method will be given in Section 3, whereas we refer to 19,20 for a detailed analysis of the scheme. ...
... The time discretization is therefore based on the TR-BDF2 scheme 17,18,19,20 , which is a second order two-stage method. A brief review of the TR-BDF2 method will be given in Section 3, whereas we refer to 19,20 for a detailed analysis of the scheme. The solver is implemented in the framework of the open source numerical library deal.II 21 , which supports natively non-conforming ℎ−adaptation. ...
... Notice that, in order to guarantee L-stability, one has to choose = 2 − √ 2 20 . We refer to 19,20 for a more exhaustive discussion on the TR-BDF2 method. We start by considering the equation in system (30) associated to the level set. ...
We propose an implicit discontinuous Galerkin (DG) discretization for incompressible two-phase flows using an artificial compressibility formulation. The conservative level set (CLS) method is employed in combination with a reinitialization procedure to capture the moving interface. A projection method based on the L-stable TR-BDF2 method is adopted for the time discretization of the Navier-Stokes equations and of the level set method. Adaptive mesh refinement (AMR) is employed to enhance the resolution in correspondence of the interface between the two fluids. The effectiveness of the proposed approach is shown in a number of classical benchmarks. A specific analysis on the influence of different choices of the mixture viscosity is also carried out.
... Two new variants of diagonally implicit Runge-Kutta (DIRK) methods are utilised (Butcher & Chen, 2000;Shampine & Reichelt, 1997), which have been shown to be promising for solving stiff constitutive equations. The TR-BDF2 (backward differentiation formulae of order two, with the use of the trapezoidal rule sub-step) has been proposed as it increases the order of the method while still ensuring a stronger form of stability (L-stability) (Hosea & Shampine, 1996). Another promising semi-implicit method is the Rosenbrock method (Shampine & Reichelt, 1997) which provides an efficient method to solve nonlinear differential equations. ...
... The TR-BDF2, first introduced in Hosea & Shampine (1996), is defined as a one-step method that is formed by combining TR in the first sub-step with the use of BDF2 in the second sub-step given below: ...
... where c 2 0, 1 ð Þ is a parameter that determines the stability and characteristics of the method. The value of c ¼ 2 − ffiffi ffi 2 p as derived in Hosea & Shampine (1996), ensures c 2 ¼ 1−c 2−c : When imposing this condition, both stages of the TR-BDF2 method have equal Jacobian matrix. For this particular value, the TR-BDF2 method is considered a DIRK (Kennedy & Carpenter, 2019), along with the Butcher tableau presented in Table 4. ...
Unified constitutive equations have been developed to model the behaviour of metallic materials under various processing conditions. These constitutive equations usually take the form of a set of ordinary differential equations (ODEs), which must be solved thousands of times in a finite element (FE) process simulation. Thus, an efficient and reliable numerical integration method for large systems is crucial for solving this problem. However, in many constitutive equations, numerical stiffness is often present. This means that the stability requirements, rather than the accuracy, constrain the step size. Therefore, certain numerical methods become unsuitable when the required step size becomes unacceptably small. In this study, a series of mathematical analyses was performed to investigate the difficulties in the numerical integration of three sets of unified viscoplastic/creep constitutive equations. Based on an analysis of the current stiffness assessment methods, a novel index was introduced, that enables an accurate assessment of the stiffness of the ODE-type unified constitutive equations. A computational study was also conducted to benchmark several promising implicit numerical integration methods for viscoplastic/creep constitutive equations. This study can assist researchers in metal forming and other fields in choosing appropriate numerical methods when dealing with stiff ODEs.
... with θ denoting the Heaviside function. The second relation in (12) is based on the so-called Bonnet's formula [11] for the curvature and H ϕ denotes the Hessian matrix of ϕ. Finally, we propose here another strategy to evaluate f σ . ...
... , N . Notice that, in order to guarantee L-stability, one has to choose γ = 2 − √ 2 [12]. ...
... Notice that we add a small number η = 10 −10 to the denominator |∇ϕ| so as to avoid division by zero. The same workaround is adopted to evaluate the unit normal vector in (12). The approximation is very robust and no oscillations for the pressure jump arise during the time evolution (see Figure 3). ...
We perform a quantitative assessment of different strategies to compute the contribution due to surface tension in incompressible two-phase flows using a conservative level set (CLS) method. More specifically, we compare classical approaches, such as the direct computation of the curvature from the level set or the Laplace-Beltrami operator, with an evolution equation for the mean curvature recently proposed in literature. We consider the test case of a static bubble, for which an exact solution for the pressure jump across the interface is available, and the test case of an oscillating bubble, showing pros and cons of the different approaches.
... Finally, Equation (1) needs to be solved for the accelerations̈, which then are passed to a numerical time integration scheme for ODEs, e.g. [3], in order to yield velocityȧ nd displacements of the systems for the next time step. ...
... Based on the previous considerations, a recycling method can now be selected to solve Equation (3). Amongst others, the GMRES derivative GCRO-DR and the recycling versions of BiCG or BiCGStab were successfully deployed [6]. ...
The paper deals with the implementation of a Krylov subspace recycling method into the solution of a nonlinear system of equations that arises during the time integration of the equations of motion of a hydrodynamically supported rotor system. The recycling BiCGStab algorithm is used in order to speed up the calculation. Different methods for generating the necessary subspace are pointed out. Additionally, a new method for the approximate determination of the subspace is presented and discussed in the context of the numerical effort and usability w.r.t. the established methods.
... The widely-known high-odd-power force term is used for modeling impact. The solver employed here is a one-step method based on the trapezoidal rule and a backward differentiation formula of order 2 with a free interpolant [44]. ...
... The initial condition is of a slightly perturbed compacton. A stable integration algorithm with a bounded energy-error [44] is employed. ...
Local configurational symmetry in lattice structures may give rise to stationary, compact solutions, even in the absence of disorder and nonlinearity. These compact solutions are related to the existence of flat dispersion curves (bands). Nonlinearity can destabilize such compactons. One common flat-band-generating system is the 1D cross-stitch model, in which compactons were shown to exist for the photonic lattice with Kerr nonlinearity. The compactons exist there already in the linear regime and are not generally destructed by that nonlinearity. Smooth nonlinearity of this kind does not allow doing a complete stability analysis for this chain. We consider a discrete mechanical system with flat dispersion bands, in which the nonlinearity exists due to impact constraints. There, one can use the concept of the saltation matrix for the analytic construction of the monodromy matrix. Besides, we consider a smooth nonlinear lattice with linearly connected massless boxes, each containing two symmetric anharmonic oscillators. In this model, the flat bands and discrete compactons also readily emerge. This system also permits performing comprehensive stability analysis, at least in the anti-continuum limit, due to the reduced number of degrees-of-freedom. In both systems, there exist two types of localization. The first one is the complete localization, and the second one is the more common exponential localization. The latter type is associated with discrete breathers (DBs). Two principal mechanisms for the loss of stability are revealed. The first one is the possible internal instability of the symmetric and/or antisymmetric solution in the individual unit cell of the chain. One can interpret this instability pattern as internal resonance between the compacton and the DB. The other mechanism is global instability related to resonance of the stationary solution with the propagation frequencies.
... which, thanks to Assumption (3.2), is clearly a consistent discretization of (35). Nevertheless, we want to show that the last relation yields a consistent discretization for (33), so as to prove that (60) is a consistent discretization of (36). ...
... The employed time discretization method is based on the second order IMEX scheme proposed in [24], for which the coefficients are reported in the Butcher tableaux Table 1 and Table 2 for the explicit and implicit method, respectively. We consider χ = 2− √ 2, so that the implicit part of the IMEX scheme coincides with the TR-BDF2 method [5,35]. Notice also that, as discussed in [47], we take a 32 = 0.5, rather than the value originally chosen in [24], so as to improve the monotonicity of the method. ...
We analyze schemes based on a general Implicit-Explicit (IMEX) time discretization for the compressible Euler equations of gas dynamics, showing that they are asymptotic-preserving(AP) in the low Mach number limit. The analysis is carried out for a general equation of state(EOS). We consider both a single asymptotic length scale and two length scales. We then show that, when coupling these time discretizations with a Discontinuous Galerkin (DG) space discretization with appropriate fluxes, a numerical method effective for a wide range of Mach numbers is obtained. A number of benchmarks for ideal gases and their non-trivial extension to non-ideal EOS validate the performed analysis.
... Trapezoid-Backward Difference (TRBDF2) is a single-step method that uses a Trapezoid step followed by second-order backward difference formula, providing an L-stable method (Hosea & Shampine, 1996). This can be written for index-1 DAEs as: ...
... The main advantage is the higher-order embedded error estimate and lower error constant compared to TRBDF2 for non-stiff and mildly-stiff systems that do not require L-stability. (Hosea & Shampine, 1996) The method can be written for index-1 DAEs as ...
This paper implements efficient numerical methods in Maple to solve index-1 nonlinear Differential Algebraic Equations (DAEs) and stiff Ordinary Differential Equations (ODEs) systems. Single-step methods (like Trapezoid (TR), Implicit-mid point (IMP), Euler-backward (EB), Radau IIA (Rad) methods, TRBDF2, TRX2) and backward-difference formula of order 2 are implemented with adaptive time-stepping methods in Maple to solve index-1 nonlinear DAEs. Maple’s robust and efficient ability to search within a list/set is exploited to identify the sparsity pattern and automatically calculate the analytic Jacobian. The algorithm and implementation are robust and efficient for index-1 DAE problems and scale well for finite difference/finite element discretization of two-dimensional models with system size up to 10,000 nonlinear DAEs and solve the same in a few seconds. The computational efficiency of the proposed algorithm (provided as an open-access code) compares favorably with the commercial solver gPROMs, one of the most commonly used sparse DAE solvers in the industry.
... A variable-step was used in these calculations. The ode23tb algorithm is an implementation of the TR-BDF2 method, which is a combination of trapezoidal and second-order backward differentiation [36]. The purpose of the simulations was to determine the displacement of the tip of the cantilever beam caused by an applied control voltage, of which the values were assumed in advance, to one (unimorph) or two MFC patches (bimorph). ...
... A variablestep was used in these calculations. The ode23tb algorithm is an implementation of the TR-BDF2 method, which is a combination of trapezoidal and second-order backward differentiation [36]. The purpose of the simulations was to determine the displacement of the tip of the cantilever beam caused by an applied control voltage, of which the values were assumed in advance, to one (unimorph) or two MFC patches (bimorph). ...
The subject of this article is an experimental analysis of the control system of a composite-based piezoelectric actuator and an aluminum-based piezoelectric actuator. Analysis was performed for both the unimorph and bimorph structures. To carry out laboratory research, two piezoelectric actuators with a cantilever sandwich beam structure were manufactured. In the first beam, the carrier layer was made of glass-reinforced epoxy composite (FR4), and in the second beam, it was made of 1050 aluminum. A linear mathematical model of both actuators was also developed. A modification of the method of selecting weights in the LQR control algorithm for a cantilever-type piezoelectric actuator was proposed. The weights in the R matrix for the actuator containing a carrier layer made of stiffer material should be smaller than those for the actuator containing a carrier layer made of less stiff material. Additionally, regardless of the carrier layer material, in the case of a bimorph, the weight in the R matrix that corresponds to the control voltage of the compressing MFC patch should be smaller than the weight corresponding to the control voltage of the stretching MFC patch.
... This results in a semidiscrete state with 140 variables. Additionally, integration tolerances are set to ε rel = 10 −3 and ε abs = 10 −7 , adjusted for problem scaling, and handled by a three-stage secondorder ESDIRK method TR-BDF2 (Hosea and Shampine, 1996). We use staggered-direct forward sensitivity analysis to generate sensitivities (Caracotsios and Stewart, 1985). ...
This paper presents a hybrid optimization methodology for parameter estimation of reactive transport systems. Using reduced-order advection-diffusion-reaction (ADR) models, the computational requirements of global optimization with dynamic PDE constraints are addressed by combining metaheuristics with gradient-based optimizers. A case study in preparative liquid chromatography shows that the method achieves superior computational efficiency compared to traditional multi-start methods, demonstrating the potential of hybrid strategies to advance parameter estimation in large-scale, dynamic chemical engineering applications.
... Computational efficiency evaluation Furthermore, numerous equations from the optimal design problem of RO system (as is introduced in the section of "Problem description") are solved using the FFHE and commonly-used computational solvers in MATLAB 27,28,[40][41][42] with respect to 100, 1000, 10000 and 100000 design parameters (or geometrical parameters of feed spacer) respectively (see Table 1). These solvers encompass implementations of classical numerical calculation methods such as the trapezoidal rule, Runge-Kutta method, BDF method, NDF method and others. ...
Large-scale optimal design problems involving nonlinear differential equations are widely applied in modeling such as craft manufacturing, chemical engineering and energy engineering. Herein we propose a fast and flexible holomorphic embedding-based method to solve nonlinear differential equations quickly, and further apply it to handle the industrial application of reverse osmosis desalination. Without solving nonlinear differential equations at each discrete point by a traditional small-step iteration approach, the proposed method determines the solution through an approximation function or approximant within segmented computational domain independently. The results of solving more than 11 million of nonlinear differential equations with various design parameters for the reverse osmosis desalination process indicate that the fast and flexible holomorphic embedding-based method is six-fold faster than several typical solvers in computational efficiency with the same level of accuracy. The proposed computational method in this work has great application potential in engineering design.
... The results are shown in Figure 5. 5. Work-precision diagram for solving discretized two-dimensional Brusselator timedependent PDE. Implicit Solvers: Trapezoid [48], TRBDF2 [23], FBDF [41], QNDF [42], QBDF (alias of QNDF with κ = 0), KenCarp4 [26], Kvaerno5 [30]. For each solver, the Newton's method is drawn in circle and solid line, and the Halley's method is drawn in diamond and dashed line. ...
This paper demonstrates new methods and implementations of nonlinear solvers with higher-order of convergence, which is achieved by efficiently computing higher-order derivatives. Instead of computing full derivatives, which could be expensive, we compute directional derivatives with Taylor-mode automatic differentiation. We first implement Householder's method with arbitrary order for one variable, and investigate the trade-off between computational cost and convergence order. We find that the second-order variant, i.e., Halley's method, to be the most valuable, and further generalize Halley's method to systems of nonlinear equations and demonstrate that it can scale efficiently to large-scale problems. We further apply Halley's method on solving large-scale ill-conditioned nonlinear problems, as well as solving nonlinear equations inside stiff ODE solvers, and demonstrate that it could outperform Newton's method.
... The simulation was carried out in parallel on 16 cores on a Windows machine with an Intel Xeon E5-2660v4 @ 2.0 GHz and 64GB RAM. Each pNMS was solved using Matlab ode23tb, a variable step size solver for stiff differential equations using the trapezoidal rule and backward differentiation 31,32 . We applied the automatic solver settings of Matlab: a maximum solver step size of 0.1 s, a minimum step size of 1 × 10 −11 s, a relative tolerance of 1 × 10 −3 , and disabled zero crossing detection for the ground contact model. ...
Ankle push-off is important for efficient, human-like walking, and many prosthetic devices mimic push-off using motors or elastic elements. The knee is extended throughout the stance phase and begins to buckle just before push-off, with timing being crucial. However, the exact mechanisms behind this buckling are still unclear. We use a predictive neuromuscular simulation to investigate whether active muscles are required for knee buckling and to what extent ground reaction forces (GRFs) drive it. In a systematic parameter search, we tested how long the knee muscles vastus (VAS), gastrocnemius (GAS), and hamstrings could be deactivated while maintaining a stable gait with impulsive push-off. VAS deactivation up to 35% of the gait cycle resulted in a dynamic gait with increased ankle peak power. GAS deactivation up to 20% of the gait cycle was detrimental to gait efficiency and showed reduced ankle peak power. At the start of knee buckling, the GRF vector is positioned near the knee joint’s neutral axis, assisting in knee flexion. However, this mechanism is likely not enough to drive knee flexion independently. Our findings contribute to the biomechanical understanding of ankle push-off, with applications in prosthetic and bipedal robotic design, and fundamental research on human gait mechanics.
... Recall that TR-BDF2 is A-stable, L-stable but neither algebraically stable nor B-stable [29]. It is second order accurate and belongs to the category of diagonally implicit Runge-Kutta (DIRK) methods. ...
Implicit schemes are popular methods for the integration of time dependent PDEs such as hyperbolic and parabolic PDEs. However the necessity to solve corresponding linear systems at each time step constitutes a complexity bottleneck in their application to PDEs with rough coefficients. We present a generalization of gamblets introduced in \cite{OwhadiMultigrid:2015} enabling the resolution of these implicit systems in near-linear complexity and provide rigorous a-priori error bounds on the resulting numerical approximations of hyperbolic and parabolic PDEs. These generalized gamblets induce a multiresolution decomposition of the solution space that is adapted to both the underlying (hyperbolic and parabolic) PDE (and the system of ODEs resulting from space discretization) and to the time-steps of the numerical scheme.
... We use numerical simulations to illustrate the results of Theorems 6 and 8 and discuss the biological meaning of the results. The numerical simulations are done using the ode23tb solver of Matlab [30] which solves system of stiff ODEs using a trapezoidal rule and second order backward differentiation scheme (TR-BDF2) [6,24]. The values of the parameters used for the numerical simulations are those of Table 1. ...
Controlling pest insects is a challenge of main importance to preserve crop production. In the context of Integrated Pest Management (IPM) programs, we develop a generic model to study the impact of mating disruption control using an artificial female pheromone to confuse males and adversely affect their mating opportunities. Consequently the reproduction rate is diminished leading to a decline in the population size. For more efficient control, trapping is used to capture the males attracted to the artificial pheromone. The model, derived from biological and ecological assumptions, is governed by a system of ODEs. A theoretical analysis of the model without control is first carried out to establish the properties of the endemic equilibrium. Then, control is added and the theoretical analysis of the model enables to identify threshold values of pheromone which are practically interesting for field applications. In particular, we show that there is a threshold above which the global asymptotic stability of the trivial equilibrium is ensured, i.e. the population goes to extinction. Finally we illustrate the theoretical results via numerical experiments.
... See tables 1 and 2 for the explicit and implicit method, respectively. The coefficients of the explicit method were proposed in [25], while the implicit method, also employed in the same paper, coincides indeed with the TR-BDF2 method proposed in [7], [29] and applied to the shallow water and Euler equations in [34]. (20) is certainly possible, we will outline here a more efficient way to implement this method to the discretization of equations (7), that mimics what done above for the simpler θ−method. ...
We propose an extension of the discretization approaches for multilayer shallow water models, aimed at making them more flexible and efficient for realistic applications to coastal flows. A novel discretization approach is proposed, in which the number of vertical layers and their distribution are allowed to change in different regions of the computational domain. Furthermore, semi-implicit schemes are employed for the time discretization, leading to a significant efficiency improvement for subcritical regimes. We show that, in the typical regimes in which the application of multilayer shallow water models is justified, the resulting discretization does not introduce any major spurious feature and allows again to reduce substantially the computational cost in areas with complex bathymetry. As an example of the potential of the proposed technique, an application to a sediment transport problem is presented, showing a remarkable improvement with respect to standard discretization approaches.
... The evolution of the four temperatures in these simulations is plotted in Fig. 3. For the reference solution we take the accurate results produced by the implicit Runge-Kutta solver (Shampine & Hosea 1996), applied directly to the system of equations 61-64. Fig. 3 indicates that the KORAL code, which solves the corresponding equations in their original form during the semi-implicit operator stage, is able to reproduce the Runge-Kutta solutions with very small error. ...
We present a numerical method which evolves a two-temperature, magnetized, radiative, accretion flow around a black hole, within the framework of general relativistic radiation magnetohydrodynamics. As implemented in the code KORAL, the gas consists of two sub-components -- ions and electrons -- which share the same dynamics but experience independent, relativistically consistent, thermodynamical evolution. The electrons and ions are heated independently according to a standard prescription from the literature for magnetohydrodynamical turbulent dissipation. Energy exchange between the particle species via Coulomb collisions is included. In addition, electrons gain and lose energy and momentum by absorbing and emitting synchrotron and bremsstrahlung radiation, and through Compton scattering. All evolution equations are handled within a fully covariant framework in the relativistic fixed-metric spacetime of the black hole. Numerical results are presented for five models of low luminosity black hole accretion. In the case of a model with a mass accretion rate , we find that radiation has a negligible effect on either the dynamics or the thermodynamics of the accreting gas. In contrast, a model with a larger behaves very differently. The accreting gas is much cooler and the flow is geometrically less thick, though it is not quite a thin accretion disk.
... The proposed model's system of DAEs was simulated in MATLAB SIMULINK, utilizing the trapezoidal rule with the second-order backward difference formula (TR-BDF2) (Hosea and Shampine 1996) as the solver. Under the specified process conditions, steady-state values for the flow and composition of each stream in the process were determined, as detailed in Table 1. ...
This study explores controlling a first-and second-generation alcoholic fermentation process with cell recycling, modeled through algebraic differential equations (DAE) with constraints. It evaluates seven controller types: PI, PID, Model Predictive Control (MPC), Neural Network Model Predictive Control (NNMPC), Mixed Model Predictive Control (MMPC), Internal Model Control (IMC), and Linear Quadratic Regulator (LQR). Results show that, while PI and PID controllers can track setpoints, they exhibit slow responses and oscillations. In contrast, MPC controllers respond faster, with both MPC and MMPC demonstrating more robust dynamics, achieving a significant reduction in Integral of the Absolute Error (IAE) across various disturbances , notably an 87% reduction in regulatory scenarios. NNMPC outperforms PI and PID but exhibits overshoot and oscillations, and lacks robustness for servo-type problems. However, MMPC showcases comparable or superior performance to MPC, surpassing other controllers in robustness, especially the IMC and LQR in the regulatory problems. NNMPC maintains a simulation time of under 4 seconds, whereas MPC incurs a computational cost 1,000 times higher. Integrating NNMPC's optimal increment as an initial estimate in MPC reduces computational time by up to 79.7%. These findings highlight the ANN's effectiveness in addressing complex control challenges, especially when integrated with MPC. MMPC offers a superior balance between accuracy, robustness, and computational efficiency, serving as a promising solution for reducing computational costs in MPC-type controllers.
... for time step δt, time t n = nδt for n ∈ N * , known state y n at time t n , unknown state y n+1 to be solved for, and embedded solutionŷ n+1 computed for the time step update to be described further below. We specifically utilize the second order L-stable implicit Runge-Kutta (RK) method TR-BDF2 [6,28] due to its stiff accuracy, L-stability, high (2nd) stage order, and first-order embedded method. The TR-BDF method with parameter γ = 2 − √ 2 has s = 3 stages and is given by Butcher tableaux ...
We present a novel spatial discretization for the Cahn-Hilliard equation including transport. The method is given by a mixed discretization for the two elliptic operators, with the phase field and chemical potential discretized in discontinuous Galerkin spaces, and two auxiliary flux variables discretized in a divergence-conforming space. This allows for the use of an upwind-stabilized discretization for the transport term, while still ensuring a consistent treatment of structural properties including mass conservation and energy dissipation. Further, we couple the novel spatial discretization to an adaptive time stepping method in view of the Cahn-Hilliard equation's distinct slow and fast time scale dynamics. The resulting implicit stages are solved with a robust preconditioning strategy, which is derived for our novel spatial discretization based on an existing one for continuous Galerkin based discretizations. Our overall scheme's accuracy, robustness, efficient time adaptivity as well as structure preservation and stability with respect to advection dominated scenarios are demonstrated in a series of numerical tests.
... It is worth mentioning that for this value of θ the stability function (4.2) is identical to that of the TR-BDF2 method, a familiar combination of the trapezoidal rule and the second-order backward differentiation formula introduced by Bank et al. [2] and subsequently studied by e.g. Hosea & Shampine [11]. The TR-BDF2 method has been advocated for the numerical approximation of one-asset American option values and their Greeks Delta and Gamma by Le Floc'h [23]. ...
In this paper we consider the numerical solution of the two-dimensional time-dependent partial integro-differential complementarity problem (PIDCP) that holds for the value of American-style options under the two-asset Kou jump-diffusion model. Following the method of lines (MOL), we derive an efficient numerical method for the pertinent PIDCP. Here, for the discretization of the nonlocal double integral term, an extension is employed of the fast algorithm by Toivanen (2008) in the case of the one-asset Kou jump-diffusion model. For the temporal discretization, we study a useful family of second-order diagonally implicit Runge-Kutta (DIRK) methods. Their adaptation to the semidiscrete two-dimensional Kou PIDCP is obtained by means of an effective iteration introduced by d'Halluin, Forsyth & Labahn (2004) and d'Halluin, Forsyth & Vetzal (2005). Ample numerical experiments are presented showing that the proposed numerical method achieves a favourable, second-order convergence behaviour to the American two-asset option value as well as to its Greeks Delta and Gamma.
... Similarly, the Butcher tableau for the TR-BDF2 method approach is shown in (4) and represents a similar meaning of (3) and θ is a constant [15]. After the computation of each step, the step size h k+1 for the next step can be obtained by calculating the prediction error ∆y. ...
... As commonly done in numerical models for atmospheric physics, we resort to an operator splitting approach. The diffusion model is treated with the implicit part of the IMEX method, which corresponds to the TR-BDF2 scheme (Hosea and Shampine 1996;Orlando et al. 2022). The nonlinear diffusivity has the form ...
We systematically validate the static local mesh refinement capabilities of a recently proposed implicit–explicit discontinuous Galerkin scheme implemented in the framework of the deal.II library. Non‐conforming meshes are employed in atmospheric flow simulations to increase the resolution around complex orography. The proposed approach is fully mass and energy conservative and allows local mesh refinement in the vertical and horizontal direction without relaxation at the internal coarse/fine mesh boundaries. A number of numerical experiments based on classical benchmarks with idealized as well as more realistic orography profiles demonstrate that simulations with the locally refined mesh are stable for long lead times and that no spurious effects arise at the interfaces of mesh regions with different resolutions. Moreover, correct values of the momentum flux are retrieved and the correct large‐scale orographic response is reproduced. Hence, large‐scale orography‐driven flow features can be simulated without loss of accuracy using a much lower total amount of degrees of freedom.
... Модель РП відповідає Т-подібній схемі на рис. 1, а і наступним рівнянням напруг: u rLC1 =u gu m ; u rLC2 =u m -u b ; u g =u rLC1 +u rLC2 +u b , де u rLC1 , u rLC2 -напруги на послідовних резонансних ланцюгах, u gвихідна напруга інвертору, u m -напруга на еквівалентному RL-ланцюзі намагнічування, u b -напруга на вході випрямляча. Під час моделювання обрано вирішувач Ode23tb [30]. Блоки моделювання послідовного резонансного ланцюга Series rLC-branch1 і Series rLC-branch2 на рис. ...
The work presents the calculations of the control characteristics of the full-bridge resonant converter with a series resonant LLC circuit and frequency control by two methods - the first harmonic method and the superposition method. The theoretical results were verified by the analytical-structural modeling method. The power circuit of the resonant converter for the analysis of electromagnetic processes is replaced by a linear T-shaped circuit with two series resonant RLC-circuits and equivalent generators of rectangular voltages, which simulate a transistor inverter and a diode rectifier in the quasi-continuous current mode. Analytical-structural modeling method consists in partly analytical and partly structural ways of building a numerical model of the resonant converter in the form of the simulation model in the MATLAB-Simulink environment. Linear structural links of the model are created on the basis of integral equations of circles. Non-linear links are created based on the non-linear functions and causal relationships. The structural model based on these links takes into account the nonlinearity of the elements of the power circuit of the resonant converter and is based on simpler mathematical expressions compared to the equivalent mathematical model of the resonant converter. The structural model corresponds to the idea of the resonant converter in the form of the resonant circuit with independent equivalent voltage generators and allows to adjust the magnetic coupling coefficient between the transformer windings and simulate processes with arbitrary control functions of equivalent generators. The peculiarity of the use of the superposition method for calculating the static characteristics of the resonant converter is the need to match the voltage phases of the equivalent generators of the equivalent circuit during the changes of the operating frequency or relative load voltage. The dependence of the input voltage of the rectifier, which is simulated by the second equivalent generator, on the processes of the power circuit of the real resonant converter, determines the conditions for matching (adjusting) the phases of the equivalent generators. References 30, figures 5.
... 54 However, these approaches do not always guarantee the correct identification of the 55 mechanistic part, and the outcomes depend on the specific regularization term used [28]. 56 To the best of our knowledge, the identifiability analysis of the mechanistic parameters 57 in a HNODE model has not been investigated in the literature so far. ...
Parameter estimation is one of the central problems in computational modeling of biological systems. Typically, scientists must fully specify the mathematical structure of the model, often expressed as a system of ordinary differential equations, to estimate the parameters. This process poses significant challenges due to the necessity for a detailed understanding of the underlying biological mechanisms. In this paper, we present an approach for estimating model parameters and assessing their identifiability in situations where only partial knowledge of the system structure is available. The partially known model is extended into a system of Hybrid Neural Ordinary Differential Equations, which captures the unknown portions of the system using neural networks.
Integrating neural networks into the model structure introduces two primary challenges for parameter estimation: the need to globally explore the search space while employing gradient-based optimization, and the assessment of parameter identifiability, which may be hindered by the expressive nature of neural networks. To overcome the first issue, we treat biological parameters as hyperparameters in the extended model, exploring the parameter search space during hyperparameter tuning. The second issue is then addressed by an a posteriori analysis of parameter identifiability, computed by introducing a variant of a well-established approach for mechanistic models. These two components are integrated into an end-to-end pipeline that is thoroughly described in the paper. We assess the effectiveness of the proposed workflow on test cases derived from three different benchmark models. These test cases have been designed to mimic real-world conditions, including the presence of noise in the training data and various levels of data availability for the system variables.
Author summary
Parameter estimation is a central challenge in modeling biological systems. Typically, scientists calibrate the parameters by aligning model predictions with measured data once the model structure is defined. Our paper introduces a workflow that leverages the integration between mechanistic modeling and machine learning to estimate model parameters when the model structure is not fully known. We focus mainly on analyzing the identifiability of the model parameters, which measures how confident we can be in the parameter estimates given the available experimental data and partial mechanistic understanding of the system. We assessed the effectiveness of our approach in various in silico scenarios. Our workflow represents a first step to adapting traditional methods used in fully mechanistic models to the scenario of hybrid modeling.
... For the integration of f (u(t)), the TR-BDF2 scheme [28] is used. The implementation of this scheme can be found in [29,30]. The sparsity of the ODEs is exploited to reduce the computation time. ...
The paper presents the problem of coupling the gas flow dynamics in pipelines with the thermodynamics of hydrogen solubility in steel for the estimation of the fracture toughness. In particular, the influence of hydrogen blended natural gas transmission on hydrogen solubility and, consequently, on fracture toughness is investigated with a focus on the L485ME low-alloy steel grade. Hydraulic simulations are conducted to obtain the pressure and temperature conditions in the pipeline. The hydrogen content is calculated from Sievert's law and, as a consequence, the fracture toughness of the base metal and heat-affected zone is estimated. Experimental data is used to define hydrogen-assisted crack size propagation in steel as well as to a plane strain fracture toughness. The simulations are conducted for a real natural gas transmission system and compared against the threshold stress intensity factor. The results showed that the computed fracture toughness for the heat-affected zone significantly decreases for all natural gas and hydrogen blends. The applied methodology allows for identification of the hydrogen-induced embrittlement susceptibility of pipelines constructed from thermomechanically rolled tubes worldwide most commonly used for gas transmission networks in the last few decades.
... A review of these methods is presented in [5], where a variant of this more general, self-adjusting multi-rate approach was introduced. The proposal in [5] was tailored on the specific implicit TR-BDF2 method [4], [12], along with a general stability analysis for one-step multi-rate methods. General multi-rate Runge-Kutta methods were introduced in [8]. ...
We present an approach for the efficient implementation of self- adjusting multi-rate Runge-Kutta methods and we extend the previously available stability analyses of these methods to the case of an arbitrary number of sub-steps for the active components. We propose a physically motivated model problem that can be used to assess the stability of different multi-rate versions of standard Runge-Kutta methods and the impact of different interpolation methods for the latent variables. Finally, we present the results of several numerical experiments, performed with implementations of the proposed methods, in the framework of the OpenModelica open-source modelling and simulation software, which demonstrate the efficiency gains deriving from the use of the proposed multi-rate approach for physical modelling problems with multiple time scales.
... We based our implementation of an implicit solver on the implicit adaptive scheme TR-BDF2 (trapezoidal rule / backward differentiation formula) [48], [49] provided by the ode23tb function from Matlab. This function is capable of solving stiff systems of differential equations that can be written in the form ...
In recent years, the use of HVDC cables has grown exponentially. One of the main challenges that remains concerns the space charge accumulation inside the insulating materials. A better understanding of the mechanisms governing this phenomenon is essential to improve the performance of HVDC systems. Numerical simulations are often employed to achieve this goal. For this reason, it is important to perform them in an efficient way. In this work, we test several numerical techniques, aiming to assess which one is the best to use for fast and reliable simulations. We consider a well-known bipolar dynamic model from the literature for our simulations. The model considers a single level of deep traps and is implemented in a one-dimensional Cartesian coordinate system, considering a thin specimen of polymeric material. We compare three different time discretization methods: a fully explicit, a semi-implicit, and a fully implicit approaches. For the advective flux discretization, we compare the first-order upwind scheme (FOU) with a second-order upwind scheme coupled with the Koren flux limiter (SOU/KL). Regarding the computation of the polarization current, we introduce a simple approach using Sato’s equation and compare it with the well-established approach based on the total current density.
... The SLH code is based on a finite-volume discretisation and offers both explicit and implicit time steppers via the method of lines. In this work, we use the ESDIRK23 implicit stepper (Hosea & Shampine 1996). The bulk Mach numbers reported in Sect. ...
There is strong observational evidence that the convective cores of intermediate-mass and massive main sequence stars are substantially larger than those predicted by standard stellar-evolution models. However, it is unclear what physical processes cause this phenomenon or how to predict the extent and stratification of stellar convective boundary layers. Convective penetration is a thermal-timescale process that is likely to be particularly relevant during the slow evolution on the main sequence. We use our low-Mach-number Seven-League Hydro code to study this process in 2.5D and 3D geometries. Starting with a chemically homogeneous model of a 15\,M10^3 to 10^60.09 to 0.44$ pressure scale heights, which is broadly compatible with observations.
... Some examples of methods that have been developed to improve the stability region of the BDF schemes are the MEBDF [2,7,8] or the TIAS [3,9] schemes, that are in fact multi-stage methods A-stable up to order 4 and 6, respectively, or the second derivative method proposed by Enright [1,10] that has obtained a single-stage method A-stable up to order 4 by using a higher derivative of the solution. Another way to improve the stability properties of the BDF schemes was found with the Composite-Backward Differentiation Formulae (C-BDF) [4,11,12], which are schemes that inherit the L-stability property of the second-order accurate BDF scheme and that, to the knowledge of the author, have been mainly used until now for electromagnetic transient simulations [13] and for thermal radiative diffusion problems [14], and only recently has their potential in solving structural mechanics [15] and fluid dynamics [16] problems been investigated. Furthermore, note that in the above cited references, the C-BDF schemes have been used without the Richardson Extrapolation (RE) technique. ...
In this work we investigate the effectiveness of the Backward Euler-Backward Differentiation Formula (BE-BDF2) in solving unsteady compressible inviscid and viscous flows. Furthermore, to improve its accuracy and its order of convergence, we have equipped this time integration method with the Richardson Extrapolation (RE) technique. The BE-BDF2 scheme is a second-order accurate, A-stable, L-stable and self-starting scheme. It has two stages: the first one is the simple Backward Euler (BE) and the second one is a second-order Backward Differentiation Formula (BDF2) that uses an intermediate and a past solution. The RE is a very simple and powerful technique that can be used to increase the order of accuracy of any approximation process by eliminating the lowest order error term(s) from its asymptotic error expansion. The spatial approximation of the governing Navier–Stokes equations is performed with a high-order accurate discontinuous Galerkin (dG) method. The presented numerical results for canonical test cases, i.e., the isentropic convecting vortex and the unsteady vortex shedding behind a circular cylinder, aim to assess the performance of the BE-BDF2 scheme, in its standard version and equipped with RE, by comparing it with the ones obtained by using more classical methods, like the BDF2, the second-order accurate Crank–Nicolson (CN2) and the explicit third-order accurate Strong Stability Preserving Runge–Kutta scheme (SSP-RK3).
... Photons are treated phenomenologically by the addition of a photon term included in the rate equations (see the Appendix for further details). Using rates reported in Ref. 24, the equations of motion were solved numerically using an implicit Runge-Kutta method with a backward differentiation method 30,31 to simulate the time evolution of a diamond maser. ...
Masers—the microwave analog of lasers—are coherent microwave sources that can act as oscillators or quantum-limited amplifiers. Masers have historically required high vacuum and cryogenic temperatures to operate, but recently, masers based on diamond have been demonstrated to operate at room temperature and pressure, opening a route to new applications as ultra-low noise microwave amplifiers. For these new applications to become feasible at a mass scale, it is important to optimize diamond masers by minimizing their size and maximizing the power of signals that can be amplified. Here, we develop and numerically solve an extended rate equation model to present a detailed phenomenology of masing dynamics and determine the optimal properties required for the copper cavity, dielectric resonator, and gain medium in order to develop portable maser devices. We conclude by suggesting how the material parameters of the diamond gain media and dielectric resonators used in diamond masers can be optimized, and how rate equation models could be further developed to incorporate the effects of temperature and nitrogen concentration on spin lifetimes.
... In recent years, more and more research has been achieved on schemes with multiple sub-steps. Among them, we have several popular ones such as the TR-BDF2 scheme [3,4,47,61] (also known as the Bathe scheme in some contexts). This scheme is a predictor-corrector scheme that uses the implicit trapezoidal rule (Crank-Nicolson) for the first substep by takingγ t as the step and the second-order backward differentiation formula for the second one, using the data at t n and calculated previously for the first sub-step. ...
This work focuses on the numerical performance of HHT-α and TR-BDF2 schemes for dynamic frictionless unilateral contact problems between an elastic body and a rigid obstacle. Nitsche’s method, the penalty method, and the augmented Lagrangian method are considered to handle unilateral contact conditions. Analysis of the convergence of an opposed value of the parameter α~ for the HHT-α method is achieved. The mass redistribution method has also been tested and compared with the standard mass matrix. Numerical results for 1D and 3D benchmarks show the functionality of the combinations of schemes and methods used.
... (more specifically OrdinaryDiffEq.jl). Some of these (such as QNDF and TRBDF2) are competitive with lsoda and CVODE, hence these additional solvers were also benchmarked [72,73]. All benchmarks were carried out on the MIT supercloud HPC [74]. ...
We introduce Catalyst.jl, a flexible and feature-filled Julia library for modeling and high-performance simulation of chemical reaction networks (CRNs). Catalyst supports simulating stochastic chemical kinetics (jump process), chemical Langevin equation (stochastic differential equation), and reaction rate equation (ordinary differential equation) representations for CRNs. Through comprehensive benchmarks, we demonstrate that Catalyst simulation runtimes are often one to two orders of magnitude faster than other popular tools. More broadly, Catalyst acts as both a domain-specific language and an intermediate representation for symbolically encoding CRN models as Julia-native objects. This enables a pipeline of symbolically specifying, analyzing, and modifying CRNs; converting Catalyst models to symbolic representations of concrete mathematical models; and generating compiled code for numerical solvers. Leveraging ModelingToolkit.jl and Symbolics.jl, Catalyst models can be analyzed, simplified, and compiled into optimized representations for use in numerical solvers. Finally, we demonstrate Catalyst’s broad extensibility and composability by highlighting how it can compose with a variety of Julia libraries, and how existing open-source biological modeling projects have extended its intermediate representation.
The no-insulation high-temperature superconducting (NI-HTS) coil technology is a promising field of application of HTS tapes, which has gained popularity in recent years. Compared to conventional insulated coils, NI-HTS coils have a better ability to cope with quenches, given the possibility for current and heat to redistribute towards adjacent turns in presence of a hot-spot. In recent years, the authors developed a nonlinear circuit model to compute current distribution and AC losses in NI-HTS coils (named CALYPSO). This model describes the currents flowing from turn to turn due to the NI configuration, as well as the magnetization currents arising in each tape. However, applying this model to coils composed of a large number of turns results in a high computational burden. This work presents an in-depth discussion of the reasons for the long computation time and the solutions and code improvements implemented to tackle this issue. Additionally, a comparison between the losses predicted by the code and those measured on straight REBCO tapes is presented. The model is then applied to investigate the electrodynamics of a NI pancake coil including both magnetization currents and radial currents. The impact of surface contact resistivity between turns on the delay between the magnetic field along the coil axis and the transport current is analyzed, showing the details of the current distribution between turns and inside individual tapes.
This paper proposes a new type of high power rapid disconnected permanent magnet eddy current friction torque limiter (RDPMEFTL) in response to the existing situation of the friction torque limiter’s short life and poor reliability for overload protection devices for coal mine machinery. The RDPMEFTL’s basic principle is to combine the permanent magnetic eddy current transmission mechanism with friction clutch, and to use the permanent magnetic eddy current structure to control the friction torque limiter’s opening and closing in order to achieve the transmission system’s overload protection. Firstly, the basic structure and working principle of the permanent magnet friction torque limiter are described, and then a two-dimensional analytical model of the slotted permanent magnet drive structure is given based on the sub-domain method, and the distributions of its air-gap magnetic field and conductor currents are investigated, so then we obtain the characteristics of the changes of its electromagnetic axial force and torque with the slip and air-gap; Secondly, the electromagnetic-mechanical coupling model of the whole drive system was established by combining the Karnopp friction model with the electromagnetic model, and the dynamics and electromagnetic response of the starting, overloading and stopping processes were analyzed. Compared with the dry friction torque limiter, the RDPMEFTL will effectively improve the reliability and service life of the drive system, reduce the corresponding downtime maintenance work, and increase the productivity. Finally, tests were conducted with a small prototype and the test results were compared with the results of the analytical solution and 3D finite element analysis, and they were basically consistent.
Parameter estimation is one of the central challenges in computational biology. In this paper, we present an approach to estimate model parameters and assess their identifiability in cases where only partial knowledge of the system structure is available. The partially known model is embedded into a system of hybrid neural ordinary differential equations, with neural networks capturing unknown system components. Integrating neural networks into the model presents two main challenges: global exploration of the mechanistic parameter space during optimization and potential loss of parameter identifiability due to the neural network flexibility. To tackle these challenges, we treat biological parameters as hyperparameters, allowing for global search during hyperparameter tuning. We then conduct a posteriori identifiability analysis, extending a well-established method for mechanistic models. The pipeline performance is evaluated on three test cases designed to replicate real-world conditions, including noisy data and limited system observability.
As renewable sources and power electronic devices become more prevalent in the power system, the characteristics of power system are now shaped by power electronics, which presents new challenges for the acceleration algorithm used in electromagnetic transient (EMT) simulation. In this context, this paper refines five key technical points of interpolation algorithms and proposes an improved interpolation algorithm. In particular, the Padé approximation is adopted and combined with numerical integration methods to establish a discrete model for accuracy and stability improvement. The switching theorem is used for re-initialization, which further reduces the switching error and reflects more realistic switching transients. Moreover, the algorithm's stability is theoretically proved through Lyapunov stability analysis. With the improved interpolation algorithm, switching transients can be accurately simulated while maintaining high simulation efficiency and stability. The performance of the proposed method is validated by both simulation and experimental tests.
X-pinch is one of the efficient sources of radiation due to its capability to generate high-energy-density plasmas. While existing 3D magneto-hydrodynamics (MHD) codes could be utilized to model such plasma, conducting full 3D numerical simulations is often expensive and time-consuming. In this study, we present a numerical framework capable of flexible adoption of models, based on the concept of the fractional step method where we separated the solution methods for different parts of the system. Numerical schemes were chosen primarily for their generality and flexibility, independent of specific characteristics of the target system, such as strict hyperbolicity. Specifically, we adopt the explicit relaxation scheme of the advection part and the implicit TR-BDF2 method of the source part. The developed framework is applied up to the 2D (r, z) cylindrical coordinate system and validated against representative benchmark problems for the MHD system, including (1) the Brio and Wu shock tube in both ideal and resistive settings, (2) the Orszag–Tang vortex, and (3) the magnetized Noh problem. Finally, we present a preliminary simulation of X-pinch.
Rare-earth (Re)Ba2Cu3O7 x (ReBCO) no-insulation (NI) coil is widely concerned due to its excellent electro magnetic and thermal properties. However, the presence of the turn-to-turn shunts in NI coils leads to that complexity of numerical simulation is increased. In this paper, a modified J model is proposed and the corresponding explicit–implicit hybrid algorithm is designed to calculate NI coils. The numerical results are in good agreement with the experimental data and the circuit model. The homogenization model is also proposed to simulate the large-scale NI coils in the background magnets. The modified J model has good accuracy and fast calculation speed, which can also be used to solve electromagnetic fields of insulation coils efficiently.
In recent years the advent of the no-insulation high-temperature superconducting (NI-HTS) coils technology, has increased the need for new simulation tools to accurately predict their response in typical operating conditions. Detailed knowledge of the current distribution in NI-HTS coils is essential to compute the magnetic field profile generated by the coil, and investigate the losses in electrodynamic transient. Of the many circuit models proposed in the literature, few can describe the behavior of both the screening currents in the tape and the radial currents. This work aims to provide all the elements to create a lumped parameter circuit model suitable for an in-depth analysis of the current distribution inside a NI-HTS coil. The lumped parameters circuit model proposed in this work can compute the current distribution in both the radial and azimuthal directions inside the coil as well as the screening currents arising inside the REBCO tapes. To compute the distribution of the screening currents and their impact on AC losses, the tape composing each turn is discretized across its width with several sub-elements, each represented through different circuit components The model was compared with 2D FEM simulation and an analytical formula to asses its ability in evaluating the screening currents in the case of a single tape. Although this work is aimed at the study of layer wound NI-HTS coils, the model described can be applied to the study of tapes, stacks of tapes, or pancake coils just by modifying the topology of the network.
An implicit variable-step BDF2 scheme is established for solving the space fractional Cahn-Hilliard equation derived from a gradient flow in the negative order Sobolev space , . The Fourier pseudo-spectral method is applied for the spatial approximation. The space fractional Cahn-Hilliard model poses significant challenges in theoretical analysis for variable time-stepping algorithms compared to the classical model, primarily due to the introduction of the fractional Laplacian. This issue is settled by developing a general discrete Hölder inequality involving the discretization of the fractional Laplacian. Subsequently, the unique solvability and the modified energy dissipation law are theoretically guaranteed. We further rigorously provided the convergence of the fully discrete scheme by utilizing the newly proved discrete Young-type convolution inequality to deal with the nonlinear term. Numerical examples with various interface widths and mobility are conducted to show the accuracy and the energy decay for different orders of the fractional Laplacian. In particular, we demonstrate that the adaptive time-stepping strategy, compared with the uniform time steps, captures the multiple time scale evolutions of the solution in simulations.
In this study, the simulation of a vacuum forming process employing a micromechanical inspired viscoelastic–viscoplastic model is investigated. In the vacuum forming process, a plastic sheet is heated above the glass transition temperature and subsequently forced into a mold by applying a vacuum. The model consists of a generalized Maxwell model combined with an dissipative element in series. Each Maxwell element incorporates a hyperelastic element in series with a viscous element based on a hyperbolical law. While the generalized Maxwell model considers the relaxation due to molecular alignment, the additional viscous element is a modification based on the approach of Bergström and thus considers molecular chain reptation. The model is designed with the aim to converge to the generalized linear Maxwell model in the limit of small deformation. Furthermore, the viscous modeling is temperature activated and follows the Williams–Landel–Ferry approach in the limit of linear viscoelasticity. To simulate rheological standard experiments, a physical‐network‐based implementation into Simscape is presented. To validate the performance of the model in thermoforming, it is implemented into Fortran programming language for finite element simulation with Abaqus/Explicit. It can be shown that the simulation is able to predict the thickness in high correlation with experimental results.
A class of linear implicit methods for numerical solution of stiff ODE's is presented. These require only occasional calculation of the Jacobian matrix while maintaining stability. Especially, an effective second order stable algorithm with automatic stepsize control is designed and tested.
This paper describes a new adaptive method that has been developed to give improved efficiency for solving differential equations where the degree of stiffness varies during the course of the integration or is not known beforehand. The method is a modification of the theta method, in which the new adaptive strategy is to automatically select the value of theta and to switch between functional iteration and Newton iteration for the solution of the nonlinear equations arising at each integration step. The criteria for selecting theta and for switching are established by optimising the permissible step size.The performance of the adaptive methods is demonstrated on a range of test problems including one arising from the method of lines solution of a convection-dominated partial differential equation. In some cases the new approach halves the amount of computational work.
A class of linear implicit methods for numerical solution of stiff ODE's is presented. These require only occasional calculation of the Jacobian matrix while maintaining stability. Especially, an effective second order stable algorithm with automatic stepsize control is designed and tested.
This paper describes mathematical and software developments for a suite of programs for solving ordinary differential equations in MATLAB.
The mathematical problem: discrete variable methods The computational problem: basic methods Convergence and stability: stability for large step sizes Error estimation and control: stiff problems Problems References Appendix Some mathematical tools Index
VODE is a new initial value ODE solver for stiff and nonstiff systems. It uses variable-coefficient Adams-Moulton and Backward Differentiation Formula (BDF) methods in Nordsieck form, as taken from the older solvers EPISODE and EPISODEB, treating the Jacobian as full or banded. Unlike the older codes, VODE has a highly flexible user interface that is nearly identical to that of the ODEPACK solver LSODE.
In the process, several algorithmic improvements have been made in VODE, aside from the new user interface. First, a change in stepsize and/or order that is decided upon at the end of one successful step is not implemented until the start of the next step, so that interpolations performed between steps use the more correct data. Second, a new algorithm for setting the initial stepsize has been included, which iterates briefly to estimate the required second derivative vector. Efficiency is often greatly enhanced by an added algorithm for saving and reusing the Jacobian matrix J, as it occurs in the Newton matrix, under certain conditions. As an option, this Jacobian-saving feature can be suppressed if the required extra storage is prohibitive. Finally, the modified Newton iteration is relaxed by a scalar factor in the stiff case, as a partial correction for the fact that the scalar coefficient in the Newton matrix may be out of date.
The fixed-leading-coefficient form of the BDF methods has been studied independently, and a version of VODE that incorporates it has been developed. This version does show better performance on some problems, but further tuning and testing are needed to make a final evaluation of it.
Like its predecessors, VODE demonstrates that multistep methods with fully variable stepsizes and coefficients can outperform fixed-step-interpolatory methods on problems with widely different active time scales. In one comparison test, on a one-dimensional diurnal kinetics-transport problem with a banded internal Jacobian, the run time for VODE was 36 percent lower than that of LSODE without the J-saving algorithm and 49 percent lower with it. The fixed-leading-coefficient version ran slightly faster, by another 12 percent without J-saving and 5 percent with it. All of the runs achieved about the same accuracy.
In this paper, we present an overview of the physical principles and numerical methods used to solve the coupled system of nonlinear partial differential equations that model the transient behavior of silicon VLSI device structures. We also describe how the same techniques are applicable to circuit simulation. A composite linear multistep formula is introduced as the time-integration scheme. Newton-iterative methods are exploited to solve the nonlinear equations that arise at each time step. We also present a simple data structure for nonsymmetric matrices with symmetric nonzero structures that facilitates iterative or direct methods with substantial efficiency gains over other storage schemes. Several computational examples, including a CMOS latchup problem, are presented and discussed.
This paper describes some problems that are encountered in the implementation of a class of Singly Diagonally Implicit Runge-Kutta (SDIRK) methods. The contribution to the local error from the local truncation error and the residual error from the algebraic systems involved are analysed. A section describes a special interpolation formula. This is used as a prediction stage in the iterative solution of the algebraic equations. A strategy for computing a starting stepsize is presented. The techniques are applied to numerical examples.
A generalization of a composite linear multistep method [2] is developed and applied to the approximate integration of systems of ordinary differential equations. The proposed scheme is second-order accurate and L-stable. An algorithm, based on the integration formula derived in this paper, is applied to approximate the solutions of a number of standard test problems. The numerical results indicate that the method is competitive with other fixed-order methods particularly in terms of computational overhead and could provide the basis for efficient temporal integration in the semidiscretization of time dependent partial differential equations.
A numerical method applied to the differential equation y′=λy can lead to a solution of the form , where z and w = λh satisfy an equation P(w, z) = 0. In general P(w,z) is a polynomial in both w and z, of degree M in w and N in z. Existing multistep and Runge-Kuta methods correspond to the cases M = 1 and N = 1 respectively. New methods are fottd by taking MS≧2N≧2. The approach here is first to find a suitable polynomial P(w,z) with the desired stability properties, and then to find a process which leads to this polynomial. Third- and fourth-order A- and L-stable processes are given of the semi-explicit arid linearly implicit types.
The stiffness in some systems of nonlinear differential equations is shown to be characterized by single stiff equations of the form The stability and accuracy of numerical approximations to the solution , obtained using implicit one-step integration methods, are studied. An S-stability property is introduced for this problem, generalizing the concept of A-stability. A set of stiffly accurate one-step methods is identified and the concept of stiff order is defined in the limit . These additional properties are enumerated for several classes of A-stable one-step methods, and are used to predict the behaviour of numerical solutions to stiff nonlinear initial-value problems obtained using such methods. A family of methods based on a compromise between accuracy and stability considerations is recommended for use on practical problems.
Researchers investigating Runge-Kutta methods use truncation error coefficients to derive formulas and assess their quality. In this paper Albrecht's recurrence for generating order conditions is refined to produce truncation error coefficients. The refined recurrence provides a fast and relatively simple way of calculating truncation error coefficients of the Albrecht expansion for general Runge-Kutta methods.
To be A-stable, and possibly useful for stiff systems, a Runge–Kutta formula must be implicit. There is a significant computational advantage in diagonally implicit formulae, whose coefficient matrix is lower triangular with all diagonal elements equal. We derive new, strongly S-stable diagonally implicit Runge–Kutta formulae of order 2 in 2 stages and of order 3 in 3 stages, and show that it is impossible for a strongly S-stable diagonally implicit method to attain order 4 in 4 stages. Merely A-stable diagonally implicit formulae, of order 3 in 2 stages and of order 4 in 3 stages, were previously known; we prove that no 4-stage method of this type has order 5. We describe a computer program for stiff differential equations which uses these methods, and compare them to each other and to the GEAR package.
This chapter focuses on the applications of EPISODE. The EPISODE program is a package of FORTRAN subroutines aimed at the automatic solution of problems of the form, , with minimum effort required in the face of potential difficulties in the problem. The program implements both a generalized Adams method, well suited for nonstiff problems, and a generalized backward differentiation formula, well suited for stiff problems. Both methods are of implicit linear multistep type. The EPISODE package consists of eight FORTRAN subroutines, to be combined with the user's calling program and subroutines DIFFUN and PEDERV. EPISODE has been written in a manner intended to maximize its portability among various computer installations. The language used is almost entirely ANSI Standard FORTRAN. Also, a new approach is taken toward the problem of single versus double precision. To generate the single precision version of the EPISODE package, it is applied to a converter subroutine, supplied with the package and also written in standard FORTRAN, which reads the special flags and makes appropriate changes.
Implicit formulas are quite popular for the solution of ODEs. They seem to be necessary for the solution of stiff problems. Every code based on an implicit formula must deal with certain tasks studied in this paper: (i) a choice of basic variable has to be made. The literature is extremely confusing as to what the possibilities are and the consequences of the choice. These matters are clarified. (ii) A test for convergence of the method for solving the implicit equations must be made. Ways of improving the reliability of this test are studied. (iii) Deciding when to form a new approximate Jacobian and/or iteration matrix is a crucial issue for the efficiency of a code. New insight which suggests rather specific actions will be developed. The paper closes with an interesting numerical example.
The stopping criterion in the iterative solution of the non-linear equations that arise in connection with the use of implicit
methods for the solution of stiff systems of ordinary differential equations is studied. The conclusion is that there must
be a consistent choice between the type of error estimator and the vector used for the test of convergence and stopping criterion.
When solving ordinary differential equations numerically, the local error is estimated at each step. In the classical situation of ‘small’ step sizes, it is clear what is required of the error estimators. Stiff problems are solved with ‘large’ step sizes. The quality of error estimators is studied in this situatuion, and it is shown how to modify unsatisfactory estimators so as to improve them greatly. Several formulas from the literature are treated as examples.
The simulation of a large-scale gas transmission network involves the numerical solution of a large system of initial valued, stiff algebraic/differential equations. Rapid changes in the solution are present due to the disturbances generated by the varying consumer demand and the operation of network controlling devices such as compressors. This paper discusses the design of an efficient variable-step integrator for the solution of the problem. Two sets of strategies are presented for implementing the variable-step integrator; one for the implicit numerical method such as the diagonally implicit Runge–Kutta methods, and the other for the linearly implicit Rosenbrock-type method. The performance of the numerical methods implemented are compared with the British Gas simulation program PAN on a number of large, realistic transmission networks.
Implicit formulas are quite popular for the solution of ODEs. They seem to be necessary for the solution of stiff problems. An acceptance test must be made of the approximate solution of the algebraic equations arising in the evaluation of an implicit formula. A reliable way to make this test is presented. It offers many practical advantages over current tests.
This paper describes a technique for comparing numerical methods that have been designed to solve stiff systems of ordinary differential equations. The basis of a fair comparison is discussed in detail. Measurements of cost and reliability are made over a collection of 25 carefully selected problems. The problems have been designed to show how certain major factors affect the performance of a method.
The technique is applied to five methods, of which three turn out to be quite good, including one based on backward differentiation formulas, another on second derivative formulas, and a third on extrapolation. However, each of the three has a weakness of its own, which can be identified with particular problem characteristics.
Carroll, J., A composite integration scheme for the numerical solution of systems of parabolic PDEs in one space dimension, Journal of Computational and Applied Mathematics 46 (1993) 327–343.We apply the numerical method of lines to approximate the solution of a system of parabolic partial differential equations in one space dimension. By approximating the spatial derivatives by discrete values of the solution at a set of mesh points in space, we use these approximations to represent the partial differential equations at each mesh point, giving a semidiscrete system of ordinary differential equations in the time direction. The resulting ODE system is integrated numerically using a second-order L-stable composite integration scheme (the author, this journal, 1989) using variable stepsize integration. An implementation of the composite scheme is applied to approximate the solution of some test problems selected from the literature using a uniform spatial mesh. We present the results of a number of numerical experiments which allow an appraisal of the proposed approach with published works.
Modified Rosenbrock-Wanner methods for systems of stiff ordinary differential equations, Dissertation (1982); also Rept. CS-83-06: A preliminary assessment of the local error estimates of some Rosenbrock-type methods
- H Zedan
H. Zedan, Modified Rosenbrock-Wanner methods for systems of stiff ordinary differential equations, Dissertation (1982); also Rept. CS-83-06: A preliminary assessment of the local error estimates of some Rosenbrock-type methods, School of Mathematics, Computer Science, University of Bristol, Bristol (1983).
Modified Rosenbrock-Wanner methods for systems of stiff ordinary differential equations
- Zedan
- L F Shampine
- M W Reichelt
- Suite
L. F. Shampine and M. W. Reichelt, \The MATLAB ODE Suite", Math.
Dept., Rept. No. 94-6, Southern Methodist University, Dallas, TX (1994).
Dissertation (1982) and Rept. CS-83-06; A preliminary assessment of the local error estimates of some Rosenbrock-type methods
- H Zedan
H. Zedan,\Modi ed Rosenbrock-Wanner methods for systems of sti ordinary
di erential equations", Dissertation (1982) and Rept. CS-83-06; A preliminary
assessment of the local error estimates of some Rosenbrock-type methods,
School of Math., Comput. Sci., Univ. of Bristol, U.K. (1983).