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# Solving ordinary differential equations. I: Nonstiff problems. 2nd revised ed., 3rd corrected printing

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... Backward differentiation formulae (BDF) were first introduced by Curtiss and Hirschfelder for the numerical resolution of stiff ordinary differential equations [11]. They have been extensively studied (see [15,16,27] and references therein). In recent years, BDF methods have been proved to be very powerful for the time discretization of semilinear parabolic problems in various situations. ...
... Numerical simulations suggested that the BDF6 method is not quadratically stable. For k ≥ 7, k-step BDF methods are not zero stable [15] and so they cannot be gradient stable [7,Remark 2.8]. ...
... The polynomial δ k is analogous to the "defining polynomial" used for the analysis of multistep schemes (see, e.g., [15,27]), but we use here the variables ∂y n+1 , . . . , ∂y n+k instead of y n , . . . ...
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For a backward differentiation formula (BDF) applied to the gradient flow of a semiconvex function, quadratic stability implies gradient stability. Namely, it is possible to build a Lyapunov functional for the discrete-in-time dynamical system, with a restriction on the time step. The maximum time step which can be derived from quadratic stability has previously been obtained for the BDF1, BDF2 and BDF3 schemes. Here, we compute it for the BDF4 and BDF5 methods. We also prove that the BDF6 scheme is not quadratically stable. Our results are based on the tools developed by Dahlquist and other authors to show the equivalence of A-stability and G-stability.
... To solve the IVP (2) numerically, we employ a RKM, which is a common one-step method to approximate ordinary and differential-algebraic equations [15,16]. More precisely, given a step size h > 0, the solution of the IVP (2) is approximated via the sequence x i ≈ x(t 0 + ih) given by ...
... Having computed the matrix A h , the question that remains to be answered is the quality of the approximation x(ih; x 0 ) − x i , which yields the following well-known definition (cf. [15]). (6). ...
... see, e.g., ( [15], Thm. II.3.6). ...
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Dynamic mode decomposition (DMD) is a popular data-driven framework to extract linear dynamics from complex high-dimensional systems. In this work, we study the system identification properties of DMD. We first show that DMD is invariant under linear transformations in the image of the data matrix. If, in addition, the data are constructed from a linear time-invariant system, then we prove that DMD can recover the original dynamics under mild conditions. If the linear dynamics are discretized with the Runge–Kutta method, then we further classify the error of the DMD approximation and detail that for one-stage Runge–Kutta methods; even the continuous dynamics can be recovered with DMD. A numerical example illustrates the theoretical findings.
... The selection of a new rank r k at each time step allows us to obtain convergence results for step-truncation schemes without assuming (15). We will refer to the schemes (16) and (17) as rank-adaptive B-ST and rank-adaptive SVD-ST, respectively. ...
... In this section, we prove a number of consistency results for step-truncation methods. In particular, we show that the fixed-rank step-truncation method (13) is consistent with the B-TSP method (12), and the rank-adaptive step-truncation methods (16)- (17) are consistent with the fully discrete system (10) (provided the truncation ranks are chosen to satisfy a suitable criterion). Our analysis begins with stating a few known results for the truncation operator T best r . ...
... Consider the following rank-adaptive step-truncation method based on the explicit midpoint rule (see [17,II.1]) ...
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We develop new adaptive algorithms for temporal integration of nonlinear evolution equations on tensor manifolds. These algorithms, which we call step-truncation methods, are based on performing one time step with a conventional time-stepping scheme, followed by a truncation operation onto a tensor manifold. By selecting the rank of the tensor manifold adaptively to satisfy stability and accuracy requirements, we prove convergence of a wide range of step-truncation methods, including explicit one-step and multi-step methods. These methods are very easy to implement as they rely only on arithmetic operations between tensors, which can be performed by efficient and scalable parallel algorithms. Adaptive step-truncation methods can be used to compute numerical solutions of high-dimensional PDEs, which, have become central to many new areas of application such optimal mass transport, random dynamical systems, and mean field optimal control. Numerical applications are presented and discussed for a Fokker-Planck equation with spatially dependent drift on a flat torus of dimension two and four.
... 1. Explicit N RK -stage Runge-Kutta methods ( cf. [14] and references therein). ...
... The approximate number of particles N s of a given plasma species s is a linear function of the vector y whose components are specified in (26). Indeed, the Hermite-DG expansion (14) ...
... x (x) = 10 −3 cos(x), by exciting the Hermite mode C e 1,0,0 in (14). Before proceeding to the numerical study of the conservation properties, we assess the capability of all considered methods to reproduce the correct physical results, in particular, the exponential growth of the electromagnetic whistler wave with the theoretically predicted growth rate from an initial small perturbation. ...
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We study the conservation properties of the Hermite-discontinuous Galerkin (Hermite-DG) approximation of the Vlasov-Maxwell equations. In this semi-discrete formulation, the total mass is preserved independently for every plasma species. Further, an energy invariant exists if central numerical fluxes are used in the DG approximation of Maxwell's equations, while a dissipative term is present when upwind fluxes are employed. In general, traditional temporal integrators might fail to preserve invariants associated with conservation laws (at the continuous or semi-discrete level) during the time evolution. Hence, we analyze the capability of explicit and implicit Runge-Kutta (RK) temporal integrators to preserve such invariants. Since explicit RK methods can only ensure preservation of linear invariants but do not provide any control on the system energy, we develop a novel class of nonlinear explicit RK schemes. The proposed methods can be tuned to preserve the energy invariant at the continuous or semi-discrete level, a distinction that is important when upwind fluxes are used in the discretization of Maxwell's equations since upwind provides a numerical source of energy dissipation that is not present when central fluxes are used. We prove that the proposed methods are able to preserve the energy invariant and to maintain the semi-discrete energy dissipation (if present) according to the discretization of Maxwell's equations. An extensive set of numerical experiments corroborates the theoretical findings. It also suggests that maintaining the semi-discrete energy dissipation when upwind fluxes are used leads to an overall better accuracy of the method relative to using upwind fluxes while forcing exact energy conservation.
... In this section an introduction to Runge-Kutta (RK) methods is given. We will restrict the presentation to the concepts needed to understand the rest of the discussion, for an exhaustive presentation of the subject we refer to [17,18]. ...
... The number of order conditions grows rapidly, for p = 6 there is 37 order conditions and for p = 10 they are 1205. For more details on how to find these conditions we refer to [17,II.2]. ...
... We used a 128 × 128 grid, Re = 100, the starting time is t = 0 and the end time is t = 1, the starting time step for ROCK2 and PIROCK is ∆t = 10 −3 , for RKC it is automatically chosen. The reference solution has been computed with the fourth order RK4 method (see [17,II.1]) using ∆t = 10 −7 and compensated summation. In the figures the symbols represent the different tolerances. ...
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In this master thesis we have compared different second order stabilized explicit Runge-Kutta methods when applied to the incompressible Navier-Stokes equations by means of a projection method and a differential algebraic approach. We explored the stability and accuracy properties of the RKC, ROCK2 and PIROCK schemes when coupled with the projection and the differential algebraic approach. PIROCK has shown unexpected instabilities, ROCK2 resulted to be the most efficient and versatile Runge-Kutta method taken into account. The differential algebraic approach sounds computationally costly but it exhibits better accuracy and a larger stability region. These properties make it more efficient than the projection method. The theory presented in the first chapters is supported by numerical experiments.
... F b is the body force and F c and T c are the contact force and torque summed over all the contacts of the particle. Provided the forces and torques are given, this is a system of ordinary differential equations that can be solved by various numerical schemes, such as the monograph of Hairer and Wanner [22]. The Hertz-Mindlin model [23,24], a combination of a non-linear spring model and a dashpot model, is adopted for the contact forces and consequently, the torques. ...
... where v ref is the reference velocity at the center of the blade as defined in Eq. (22) and l s the distance between the blade center and the free surface. The dimensionless shear rate _ γ e in Eq. (26) becomes dependent on the rotation speed ω, ...
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Mixing granular materials plays versatile roles in many engineering fields. In this study, we focus on an underwater mixing process as a novel application for offshore mining. We apply the discrete element method (DEM), which was augmented by a lubrication model, to evaluate the mechanical responses of the mixing process under water. Further, a non-dimensional parameter is introduced to evaluate the mixing resistance. The DEM simulation is first calibrated by a mixing experiment involving colored sand immersed in water. A parametric study is then conducted to evaluate the geometric and operational factors involved in the mixing process, including the blade angle, penetration depth, filling depth, container-blade aspect ratio, model scale, and rotation speed. Their influences on the mechanical responses of the mixing process are evaluated, i.e., the resultant torques on the mixer head, the resultant forces on the ground and side wall of the container, the effective particle masses mobilized by mixing, and the proposed mixing resistance parameter. It was found that the influences range from insignificant to highly nonlinear, and vary considerably among the evaluated factors and steady-state responses. Especially, an optimal blade angle range is observed in terms of minimal mixing resistance. These findings can contribute to the optimization of the mixer geometry, selection of the rotation speed, estimation of the motor power, and economic design for numerical and laboratory experiments with scaled-down models.
... For this particular example, the Parareal update (12), with PP 1 as shown in (16), is ...
... The time step size of the fine implicit Euler propagator is set to δt = 10 −5 and the coarse solver is chosen to perform one time step per window and thus has time step size ∆T = 1/N . The Parareal algorithm is iterated until the l 2 norm of the difference between the PP 1 components of the solution at the end of interval I n−1 and the initial condition of I n for all T n is below a relative tolerance of 5 · 10 −4 and an absolute tolerance of 10 −10 (see error norm of [16]). ...
Preprint
This article proposes modifications of the Parareal algorithm for its application to higher index differential algebraic equations (DAEs). It is based on the idea of applying the algorithm to only the differential components of the equation and the computation of corresponding consistent initial conditions later on. For differential algebraic equations with a special structure as e.g. given in flux-charge modified nodal analysis, it is shown that the usage of the implicit Euler method as a time integrator suffices for the Parareal algorithm to converge. Both versions of the Parareal method are applied to numerical examples of nonlinear index 2 differential algebraic equations.
... We set [11, p. 11], [20,Theorem 10.5], [29] (clearly, the matrix A + A H is self-adjoint) ...
... , 8, of the function q as the zeroes of the function v from (21); thus, implicitly, λ k , k = 1, . . . , 8, are the poles of the function r t from (20). On the left Fig. 1, these points are marked by the sign ⊕. ...
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Let $A$ be a square complex matrix; $z_1$, ..., $z_{N}\in\mathbb C$ be arbitrary (possibly repetitive) points of interpolation; $f$ be an analytic function defined on a neighborhood of the convex hull of the union of the spectrum $\sigma(A)$ of the matrix $A$ and the points $z_1$, ..., $z_{N}$; and the rational function $r=\frac uv$ (with the degree of the numerator $u$ less than $N$) interpolates $f$ at these points (counted according to their multiplicities). Under these assumptions estimates of the kind $$\bigl\Vert f(A)-r(A)\bigr\Vert\le \max_{t\in[0,1];\mu\in\text{convex hull}\{z_1,z_{2},\dots,z_{N}\}}\biggl\Vert\Omega(A)[v(A)]^{-1} \frac{\bigl(vf\bigr)^{{(N)}} \bigl((1-t)\mu\mathbf1+tA\bigr)}{N!}\biggr\Vert,$$ where $\Omega(z)=\prod_{k=1}^N(z-z_k)$, are proposed. As an example illustrating the accuracy of such estimates, an approximation of the impulse response of a dynamic system obtained using the reduced-order Arnoldi method is considered, the actual accuracy of the approximation is compared with the estimate based on this paper.
... In this thesis, the symplectic Euler, as well as several explicit Runge-Kutta (RK) schemes have been utilized for numerical integration: RK32, RK38, RK4, Fehlberg45, Merson45, Dormand-Prince5, and Dormand-Prince853 (Dormand and Prince, 1980; Hairer et al., 1993). With the exception of the latter, these methods were implemented as numerical solvers employing butcher tables for configuration. ...
... Internally it uses order 5 and 3 for error measure and step size optimization. It exhibits a good trade off between accuracy and run time performance, when there is a need for high accuracy (Hairer et al., 1993) 4 . The work shows in the artificial test cases of the circle and rectangle, that the high order integration improves the circle, but cannot handle abrupt changes in curvature, i.e. the sharp corners of the rectangle. ...
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Point cloud data structures are widely used due to their application in data capturing, e.g. by depth cameras and laser scanners, and in physically based particle simulation. Typically, point cloud algorithms operate heavily in local neighborhoods and employ accelerating spatial data structures, such as grids or trees. In this thesis a second order tensor view on point cloud neighborhoods, similar to a covariance, is introduced and analyzed. Based thereon, computational performance was optimized, continuous geometries, such as lines, were reconstructed, and point cloud visualization was enhanced. Strategies to optimize the tensor computation were explored by utilizing different spatial data structures on CPU and/or GPU: kd-tree, octree, uniform grid cell hashing and sorting, curvilinear grid filling, and a screen space based technique. These are tailored to different use cases, i.e. full numerical computation, visualization of fine grain point clouds, and visual optimization by vertex clustering. The grid hashing yielded a speed up of about 500 compared to a naïve parallel tree based implementation, also taking advantage of heterogeneous hardware systems. An algorithm to reconstruct lines in noisy point clouds was developed and analyzed in depth. An eigenvector streamline integration of a vector field – derived from mixing eigenvectors at optimal neighborhood radii and angular weighted directions – reconstructs lines by merging multiple line lets. Therefore, a multi scale tensor was introduced to identify optimal radii, to detect start points for the integration, and to compute a novel noise rate. A comparison to a recent reconstruction method showed that it can compete in terms of quality as well as performance and, moreover, supports 3D line reconstructions. A new visualization method for the multi scale tensor via color mapped images is utilized to reveal and optimize different choices within the tensor computation, i.e. weighting functions and points of reference. Further, point cloud data were explored by a single scale tensor visualization approach stemming from diffusion tensor magnetic resonance imaging. Examples are demonstrated on data of airborne light detection and ranging laser scans and particle simulations of an evolving cosmos including star birth modeling in gaseous regions of galaxy clusters.
... The convergence result is based on studying stability and consistency, using the procedure called Lady Windermere's fan from [21,Section II.3], however, these two issues cannot be separated as in most convergence proofs, since this would lead to sub-optimal error estimates. Instead, the error is rewritten using recursion formula which, using the parabolic smoothing property (see, e.g., [14,Theorem 4.6 (c)]), leads to an induction process to ensure that the numerical solution stays within a strip around the exact solution. ...
... The proof of our main result is based on a recursive expression for the global error, which involves the local error and some nonlinear error terms. The recursive formula is obtained using a procedure which is sometimes called Lady Windermere's fan [21,Section II.3]; our approach is inspired by [37], [45,Chapter 3]. The local errors are weighted by T (τ ) j , therefore a careful accumulation estimate-heavily relying on the parabolic smoothing property-is required. ...
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We derive a numerical method, based on operator splitting, to abstract parabolic semilinear boundary coupled systems. The method decouples the linear components which describe the coupling and the dynamics in the bulk and on the surface, and treats the nonlinear terms by approximating the integral in the variation of constants formula. The convergence proof is based on estimates for a recursive formulation of the error, using the parabolic smoothing property of analytic semigroups and a careful comparison of the exact and approximate flows. Numerical experiments, including problems with dynamic boundary conditions, reporting on convergence rates are presented.
... These well-known stiff solvers have been tested to be practically valuable for differentialalgebraic problems [5,9,10] and for the hyperbolic systems with multiscale relaxation [1]. Also, the variable-step versions are computationally efficient in capturing the multi-scale time behaviors [2,4,[10][11][12][13]15] via adaptive time-stepping strategies. ...
... On the other hand, the stability and convergence analysis of variable-step BDF methods are difficult and remain incomplete, cf. [2,4,[6][7][8][9]15], because they involve multiple degrees of freedom (independent time-step sizes). ...
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We prove that the two-step backward differentiation formula (BDF2) method is stable on arbitrary time grids; while the variable-step BDF3 scheme is stable if almost all adjacent step ratios are less than 2.553. These results relax the severe mesh restrictions in the literature and provide a new understanding of variable-step BDF methods. Our main tools include the discrete orthogonal convolution kernels and an elliptic-type matrix norm.
... The numerical integration of eq.(1) can be performed using the Runge Kutta (RK) methods. The explicit RK schemes possess the advantage of ease of implementation, while achieving up to sixth-order accuracy [1]. The traditional explicit RK methods do not preserve numerical energy [2] and result in the violation of energy conser-vation at each time step of integration. ...
... The time discretization was performed using the fourth and sixth order RRK methods. The fourth order scheme implemented was the classical RK method, while the sixth order scheme was the Verner's method [1]. The numerical study was performed on a one dimensional grid ∈ [−5, 5] and = 1, with a Gaussian initial condition ...
Preprint
Linear hyperbolic partial differential equations (PDEs) are known to conserve energy in the absence of a source term. For example, the solution of the advection equation at time is the time-shifted function of the initial condition. The numerical solutions of hyperbolic PDEs obtained using the traditional Runge Kutta temporal schemes do not conserve the numerical energy at each time step of integration. The recently-introduced Relaxation Runge-Kutta schemes utilize the relaxation parameter , which ensure energy conservation at each time step. Mimetic methods satisfy a discrete form the extended Gauss's divergence theorem and therefore satisfy a global conservation law. In this paper, we investigate the application of the high order Mimetic spatial discretization methods in combination with the Relaxation Runge-Kutta schemes for hyperbolic PDEs. Numerical examples are shown to illustrate the energy preservation of these schemes.
... Before constructing our LSEs, let us introduce the well-known Adams method in numerical analysis for ODEs (see, e.g., Butcher [2], Hairer et al. [6], Hairer and Wanner [7] and Iserles [8]), which is the combinations of two methods as preditor-corrector pair, says, the Adams-Bashforth and the Adams-Moulton formulae. For instance, to compute an approximate valuex t k of the solution of (1.2) at t = t k , we firstly prepare a predictor x * t k given by Adams-Bashforth method with ℓ = 1, 2, . . . ...
... , t k−ℓ ) is the Lagrange interpolating polynomial through the points (s, g(s)), s = t k , . . . , t k−ℓ (see, e.g., Section III.1 in Hairer et al. [6]). In particular, substituting g ≡ 1, we have ...
Preprint
We consider stochastic differential equations (SDEs) driven by small L\'evy noise with some unknown parameters, and propose a new type of least squares estimators based on discrete samples from the SDEs. To approximate the increments of a process from the SDEs, we shall use not the usual Euler method, but the Adams method, that is, a well-known numerical approximation of the solution to the ordinary differential equation appearing in the limit of the SDE. We show the consistency of the proposed estimators as well as the asymptotic distribution in a suitable observation scheme. We also show that our estimators can be better than the usual LSE based on the Euler method in the finite sample performance.
... Applying a Runge-Kutta method to the above equation, one gets an induction of the form y n+1 = R(p, q)y n , with p = hλ and q = hµ. We define the stability domain of a Runge-Kutta method applied to (13) ...
... Equation (13) can be seen as the test equation for linear SDEs with the iµ replacing the noise. Hence, inspired by SK-ROCK [4], we consider the following stability polynomial ...
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We introduce a novel second order family of explicit stabilized Runge-Kutta-Chebyshev methods for advection-diffusion-reaction equations which outperforms existing schemes for relatively high Peclet number due to its favorable stability properties and explicitly available coefficients. The construction of the new schemes is based on stabilization using second kind Chebyshev polynomials first used in the construction of the stochastic integrator SK-ROCK. We propose an adaptive algorithm to implement the new scheme that is able to automatically select the suitable step size, number of stages, and damping parameter at each integration step. Numerical experiments that illustrate the efficiency of the new methods are presented.
... In the course of the paper, we will see that the conservation of mechanical stability properties by numerical schemes (not to be confused with numerical stability properties) is closely related to their ability to exactly keep energy and momentum as invariants of motion. The development of energy-momentum conserving schemes is a field of active research [9,10]. We may distinguish between projective methods and schemes which directly preserve the energy and momentum. ...
... follows from the closed form solution of the matrix differential equation (9). Without loss of generality, we set A I K (0) = I . ...
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The stability properties of a freely rotating rigid body are governed by the intermediate axis theorem, i.e., rotation around the major and minor principal axes is stable whereas rotation around the intermediate axis is unstable. The stability of the principal axes is of importance for the prediction of rockfall. Current numerical schemes for 3D rockfall simulation, however, are not able to correctly represent these stability properties. In this paper an extended intermediate axis theorem is presented, which not only involves the angular momentum equations but also the orientation of the body, and we prove the theorem using Lyapunov’s direct method. Based on the stability proof, we present a novel scheme which respects the stability properties of a freely rotating body and which can be incorporated in numerical schemes for the simulation of rigid bodies with frictional unilateral constraints. In particular, we show how this scheme is incorporated in an existing 3D rockfall simulation code. Simulations results reveal that the stability properties of rotating rocks play an essential role in the run-out length and lateral spreading of rocks.
... The matrices B 1 , B 2 are Toeplitz because of the use of a constant step size. Indeed, this property is maintained when replacing implicit Euler by implicit multistep methods [26] such as BDF (backward differentiation formula). When using Runge-Kutta integrators [26], we still get banded Toeplitz matrices B 1 , B 2 but -as we will see in Section 4 -we will have to embed A, M into larger matrices in order to arrive at a matrix equation of the form (1). ...
... Indeed, this property is maintained when replacing implicit Euler by implicit multistep methods [26] such as BDF (backward differentiation formula). When using Runge-Kutta integrators [26], we still get banded Toeplitz matrices B 1 , B 2 but -as we will see in Section 4 -we will have to embed A, M into larger matrices in order to arrive at a matrix equation of the form (1). ...
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This work is concerned with linear matrix equations that arise from the space-time discretization of time-dependent linear partial differential equations (PDEs). Such matrix equations have been considered, for example, in the context of parallel-in-time integration leading to a class of algorithms called ParaDiag. We develop and analyze two novel approaches for the numerical solution of such equations. Our first approach is based on the observation that the modification of these equations performed by ParaDiag in order to solve them in parallel has low rank. Building upon previous work on low-rank updates of matrix equations, this allows us to make use of tensorized Krylov subspace methods to account for the modification. Our second approach is based on interpolating the solution of the matrix equation from the solutions of several modifications. Both approaches avoid the use of iterative refinement needed by ParaDiag and related space-time approaches in order to attain good accuracy. In turn, our new approaches have the potential to outperform, sometimes significantly, existing approaches. This potential is demonstrated for several different types of PDEs.
... which depend on the quotient χ of the generating polynomials of the multi-step method [32]. Popular examples are the backward differentiation formulas (BDF) of order p ≤ 2 with ...
... In this work a fourth degree Runge-Kutta integration ⃗ ( + ∆ ) = rk4(⃗ , ⃗ ̇, , ∆ ) [19] is used to integrate Equation (32) and simulate the system. ...
Conference Paper
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Within the wide field of self-assembly, the self-reconfiguring chain has unique potential for reliable and repeatable assembly of 3-dimensional structures as demonstrated by protein biosynthesis. This potential could be translated to self-reconfiguring robots by utilizing magnetic forces between the chain components as a driving force for the folding process. Due to the constraints introduced by the joints between the chain components, simulation of the dynamics of longer chains is computationally intensive and challenging. This paper presents a novel analytical approach to formulate the Newton-Euler dynamics of a self-reconfiguring chain in a single vectorised differential equation. The vector-ised differential equation allows for a convenient implementation of a parallel processing architecture using Single Instruction Multiple Data (SMID) or Graph-ical Processing Unit (GPU) computation and as a result, can improve simulation time of rigid body chains. Properties of existing interpretations of the Newton-Euler and Euler-Lagrange algorithms are discussed in their efficiency to compute the dynamics of rigid body chains. Finally, GPU and SMID supported simulation, based on the vectorised Newton-Euler equations described in this paper, are compared , showing a significant improvement in computation time using GPU architecture for long chains with a certain chain geometry.
... A sophisticated adaption of this operator splitting was derived and applied for the Bloch-McConnell equations. Third, exact error metrics based on the new exact solver are exploited in extensive numerical experiments, analyzing and comparing the broad set of typical solution methods including matrix exponentials [44,45], asymmetric operator splitting, the Implicit Euler scheme [46,47], the Crank-Nicolson method [48] and a Runge-Kutta method [49]. To accommodate reproducible research, parts of the code used to generate the results for this paper are available at https://github.com/GrafChristina/BlochSim. ...
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Purpose To introduce new solution methods for the Bloch and Bloch-McConnell equations and compare them quantitatively to different known approaches. Theory and Methods A new exact solution per time step is derived by means of eigenvalues and generalized eigenvectors. Fast numerical solution methods based on asymmetric and symmetric operator splitting, which are already known for the Bloch equations, are extended to the Bloch-McConnell equations. Those methods are compared to other numerical methods including spin domain, one-step and multi-step methods, and matrix exponential. Error metrics are introduced based on the exact solution method, which allows to assess the accuracy of each solution method quantitatively for arbitrary example data. Results Accuracy and performance properties for nine different solution methods are analyzed and compared in extensive numerical experiments including various examples for non-selective and slice-selective MR imaging applications. The accuracy of the methods heavily varies, in particular for short relaxation times and long pulse durations. Conclusion In absence of relaxation effects, the numerical results confirm the rotation matrices approach as accurate and computationally efficient Bloch solution method. [Otherwise, as well as for the Bloch-McConnell equations, symmetric operator splitting methods are recommended due to their excellent numerical accuracy paired with efficient run time.]
... Such methods rely on the error estimation built into the integrator to adapt the time step and cross the surface without needing to compute its precise location, as long as there is a test available to decide which side of the surface the integrator is on. There may be more rejected steps near S and a negative impact on accuracy and efficiency as can be seen in [10] in Figure 6.3 and related discussion. ...
Preprint
We introduce conservative integrators for long term integration of piecewise smooth systems with transversal dynamics and piecewise smooth conserved quantities. In essence, for a piecewise dynamical system with piecewise defined conserved quantities such that its trajectories cross transversally to its interface, we combine Mannshardt's transition scheme and the Discrete Multiplier Method to obtain conservative integrators capable of preserving conserved quantities up to machine precision and accuracy order. We prove that the order of accuracy of the integrators is preserved after crossing the discontinuity in the case of codimension one number of conserved quantities. Numerical examples illustrate the preservation of accuracy order.
... The ODE system (15) can now be integrated by any classical numerical integrator ( [HNW93]) without having to solve the Kepler's equation (2) at each integration step. In the following sections, we will use a Taylor's integrator ( [JZ05a]) that we briefly recall for self-consistency in Appendix B. We have used Taylor's method because it can produce solutions with high accuracy (say 10´3 0 ), since it can easily increase the orders of the Taylor's expansions in order to provide good enough trajectory values. ...
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We consider the dissipative spin-orbit problem in Celestial Mechanics, which describes the rotational motion of a triaxial satellite moving on a Keplerian orbit subject to tidal forcing and "drift". Our goal is to construct quasi-periodic solutions with fixed frequency, satisfying appropriate conditions. With the goal of applying rigorous KAM theory, we compute such quasi-periodic solution with very high precision. To this end, we have developed a very efficient algorithm. The first step is to compute very accurately the return map to a surface of section (using a high order Taylor's method with extended precision). Then, we find an invariant curve for the return map using recent algorithms that take advantage of the geometric features of the problem. This method is based on a rapidly convergent Newton's method which is guaranteed to converge if the initial error is small enough. So, it is very suitable for a continuation algorithm. The resulting algorithm is quite efficient. We only need to deal with a one dimensional function. If this function is discretized in $N$ points, the algorithm requires $O(N \log N)$ operations and $O(N)$ storage. The most costly step (the numerical integration of the equation along a turn) is trivial to parallelize. The main goal of the paper is to present the algorithms, implementation details and several sample results of runs. We also present both a rigorous and a numerical comparison of the results of averaged and not averaged models.
... and define the map G e -Ψ e˝Pe˝Ψ´1 e which can be computed accurately by numerical integrators such as [HNW93,JZ05]. ...
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We provide evidence of the existence of KAM quasi-periodic attractors for a dissipative model in Celestial Mechanics. We compute the attractors extremely close to the breakdown threshold. We consider the spin-orbit problem describing the motion of a triaxial satellite around a central planet under the simplifying assumption that the center of mass of the satellite moves on a Keplerian orbit, the spin-axis is perpendicular to the orbit plane and coincides with the shortest physical axis. We also assume that the satellite is non-rigid; as a consequence, the problem is affected by a dissipative tidal torque that can be modeled as a time-dependent friction, which depends linearly upon the velocity. Our goal is to fix a frequency and compute the embedding of a smooth attractor with this frequency. This task requires to adjust a drift parameter. The goal of this paper is to provide numerical calculations of the condition numbers and verify that, when they are applied to the numerical solutions, they will lead to the existence of the torus for values of the parameters extremely close to the parameters of breakdown. Computing reliably close to the breakdown allows to discover several interesting phenomena, which we will report in [CCGdlL20a]. The numerical calculations of the condition numbers presented here are not completely rigorous, since we do not use interval arithmetic to estimate the round off error and we do not estimate rigorously the truncation error, but we implement the usual standards in numerical analysis (using extended precision, checking that the results are not affected by the level of precision, truncation, etc.). Hence, we do not claim a computer-assisted proof, but the verification is more convincing that standard numerics. We hope that our work could stimulate a computer-assisted proof.
... Since the forms (40) are all polynomial, there is no other approximation, and thus any appreciable error in the moment equations (48) is the result of the moment closure. We solve the system (48) numerically using an implicit BDF solver [7,11,36]. As described analytically, numerical solution using the zero closure quickly leads to negative, non-physical moments. ...
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Stochastic chemical kinetics at the single-cell level give rise to heterogeneous populations of cells even when all individuals are genetically identical. This heterogeneity can lead to nonuniform behaviour within populations, including different growth characteristics, cell-fate dynamics, and response to stimuli. Ultimately, these diverse behaviours lead to intricate population dynamics that are inherently multiscale: the population composition evolves based on population-level processes that interact with stochastically distributed single-cell states. Therefore, descriptions that account for this heterogeneity are essential to accurately model and control chemical processes. However, for real-world systems such models are computationally expensive to simulate, which can make optimisation problems, such as optimal control or parameter inference, prohibitively challenging. Here, we consider a class of multiscale population models that incorporate population-level mechanisms while remaining faithful to the underlying stochasticity at the single-cell level and the interplay between these two scales. To address the complexity, we study an order-reduction approximations based on the distribution moments. Since previous moment-closure work has focused on the single-cell kinetics, extending these techniques to populations models prompts us to revisit old observations as well as tackle new challenges. In this extended multiscale context, we encounter the previously established observation that the simplest closure techniques can lead to non-physical system trajectories. Despite their poor performance in some systems, we provide an example where these simple closures outperform more sophisticated closure methods in accurately, efficiently, and robustly solving the problem of optimal control of bioproduction in a microbial consortium model.
... We note that estimates in L ∞ (0, t F ; H) ∩ L p (0, t F ; V), for some p > 1, are standard in the PDE and numerical-PDE literature [40,44,59,60,27,24,39,51,50,3]. On the other hand, to our knowledge, the numerical ODE literature [31,55,10,32] has not focused on a priori bounds in Bochner-type norms, space-time compactness, or convergence without regularity assumptions for problems of growing dimensionality (i.e. discretization of evolutionary PDEs). ...
Preprint
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We study diagonally implicit Runge-Kutta (DIRK) schemes when applied to abstract evolution problems that fit into the Gelfand-triple framework. We introduce novel stability notions that are well-suited to this setting and provide simple, necessary and sufficient, conditions to verify that a DIRK scheme is stable in our sense and in Bochner-type norms. We use several popular DIRK schemes in order to illustrate cases that satisfy the required structural stability properties and cases that do not. In addition, under some mild structural conditions on the problem we can guarantee compactness of families of discrete solutions with respect to time discretization.
... The system of equations (6) has been numerically integrated with an explicit Runge-Kutta-Dormand-Prince method [42,43], with the error tolerance set at 10 −8 . The three invariants of the inviscid and unforced system are conserved within the error tolerance. ...
Article
The chaotic dynamics of a low-order Galerkin truncation of the two-dimensional magnetohydrodynamic system, which reproduces the dynamics of fluctuations described by nearly incompressible magnetohydrodynamic in the plane perpendicular to a background magnetic field, is investigated by increasing the external forcing terms. Although this is the case closest to two-dimensional hydrodynamics, which shares some aspects with the classical Feigenbaum scenario of transition to chaos, the presence of magnetic fluctuations yields a very complex interesting route to chaos, characterized by the splitting into multiharmonic structures of the field amplitudes, and a mixing of phase-locking and free phase precession acting intermittently. When the background magnetic field lies in the plane, the system supports the presence of Alfvén waves thus lowering the nonlinear interactions. Interestingly enough, the dynamics critically depends on the angle between the direction of the magnetic field and the reference system of the wave vectors. Above a certain critical angle, independently from the external forcing, a breakdown of the phase locking appears, accompanied with a suppression of the chaotic dynamics, replaced by a simple periodic motion.
... These mathematical models are usually very complex and the numerical analysis is essential in order to obtain quantitative and qualitative information of the solution. A well-known procedure to carry out the numerical approximations is the so-called method of lines which is based on separating the spatial approximation, carried out by finite elements or other classical methods, from the time integration which is made with schemes for ordinary differential schemes [1,2]. ...
Article
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We avoid as as much as possible the order reduction of Rosenbrock methods when they are applied to nonlinear partial differential equations by means of a similar technique to the one used previously by us for the linear case. For this we use a suitable choice of boundary values for the internal stages. The main difference from the linear case comes from the difficulty to calculate those boundary values exactly in terms of data. In any case, the implementation is cheap and simple since, at each stage, just some additional terms concerning those boundary values and not the whole grid must be added to what would be the standard method of lines.
... which in general represents a first order inhomogeneous linear differential equation with nonconstant coefficient matrix A(t) and forcing term B(t). Such systems are well-understood from a theoretical point of view [3,20] and there exist a variety of numerical methods for solving them [15,16,14,22]. Although efficient numerical methods are readily available for the n x -dimensional system in x(t) (1), the potentially high-dimensional nature of the associated sensitivity problem (5) renders such methods inefficient in that setting. ...
Preprint
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In this paper we develop a new method for numerically approximating sensitivities in parameter-dependent ordinary differential equations (ODEs). Our approach, intended for situations where the standard forward and adjoint sensitivity analysis become too computationally costly for practical purposes, is based on the Peano-Baker series from control theory. We give a representation, using this series, for the sensitivity matrix $\boldsymbol{S}$ of an ODE system and use the representation to construct a numerical method for approximating $\boldsymbol{S}$. We prove that, under standard regularity assumptions, the error of our method scales as $O(\Delta t ^2 _{max})$, where $\Delta t _{max}$ is the largest time step used when numerically solving the ODE. We illustrate the performance of the method in several numerical experiments, taken from both the systems biology setting and more classical dynamical systems. The experiments show the sought-after improvement in running time of our method compared to the forward sensitivity approach. For example, in experiments involving a random linear system, the forward approach requires roughly $\sqrt{n}$ longer computational time, where $n$ is the dimension of the parameter space, than our proposed method.
... We remark that, in the last decades the ODE is profoundly and successfully used in divers scientific branches. Thus, there are many books and software available for numerically solving such equations [6,12]. ...
Article
A model for the amount of pollution in lakes connected with some rivers is introduced. In this model, it is supposed the density of pollution in a lake has memory. The model leads to a system of fractional differential equations. This system is transformed into a system of Volterra integral equations with memory kernels. The existence and regularity of the solutions are investigated. A high-order numerical method is introduced and analyzed and compared with an explicit method based on the regularity of the solution. Validation examples are supported, and some models are simulated and discussed.
... We remark that, in the last decades the ODE is profoundly and successfully used in divers scientific branches. Thus, there are many books and software available for numerically solving such equations [6,12]. ...
Article
A model for the amount of pollution in lakes connected with some rivers is introduced. In this model, it is supposed the density of pollution in a lake has memory. The model leads to a system of fractional differential equations. This system is transformed into a system of Volterra integral equations with memory kernels. The existence and regularity of the solutions are investigated. A high-order numerical method is introduced and analyzed and compared with an explicit method based on the regularity of the solution. Validation examples are supported, and some models are simulated and discussed.
... Collocation methods and Radau IIA. Collocation methods [27,II.7] are time integrators based on polynomial quadrature that can also be expressed as fully-implicit Runge-Kutta methods. Well-known examples include the A-stable Gauss methods [8,Sec. ...
Preprint
In this paper we generalize the polynomial time integration framework to additively partitioned initial value problems. The framework we present is general and enables the construction of many new families of additive integrators with arbitrary order-of-accuracy and varying degree of implicitness. In this first work, we focus on a new class of implicit-explicit polynomial block methods that are based on fully-implicit Runge-Kutta methods with Radau nodes. We show that the new integrators have improved stability compared to existing IMEX Runge-Kutta methods, while also being more computationally efficient due to recent developments in preconditioning techniques for solving the associated systems of nonlinear equations. For PDEs on periodic domains where the implicit component is trivial to invert, we will show how parallelization of the right-hand-side evaluations can be exploited to obtain significant speedup compared to existing serial IMEX Runge-Kutta methods. For parallel (in space) finite-element discretizations, the new methods obtain accuracy several orders of magnitude lower than existing IMEX Runge-Kutta methods, and/or obtain a given accuracy as much as 16 times faster in terms of computational runtime.
... The BDF schemes are widely used for stiff or differential-algebraic problems [7,8]. Recently, they were also applied for simulating hyperbolic systems with multiscale relaxation [1] and stiff kinetic equations [5]. ...
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This is one of our series works on discrete energy analysis of the variable-step BDF schemes. In this part, we present stability and convergence analysis of the third-order BDF (BDF3) schemes with variable steps for linear diffusion equations, see e.g. [SIAM J. Numer. Anal., 58:2294-2314] and [Math. Comp., 90: 1207-1226] for our previous works on the BDF2 scheme. To this aim, we first build up a discrete gradient structure of the variable-step BDF3 formula under the condition that the adjacent step ratios are less than 1.4877, by which we can establish a discrete energy dissipation law. Mesh-robust stability and convergence analysis in the $L^2$ norm are then obtained. Here the mesh robustness means that the solution errors are well controlled by the maximum time-step size but independent of the adjacent time-step ratios. We also present numerical tests to support our theoretical results.
... In this work, we consider the following generalization of the classical SIR model (adapted from the model from Book by Hairer et al. [15] p. 295; [19]), which we will call generalized SIR model with constant time delays: ...
Article
The COVID-19 disease has forced countries to make a considerable collaborative effort between scientists and governments to provide indicators to suitable follow-up the pandemic’s consequences. Mathematical modeling plays a crucial role in quantifying indicators describing diverse aspects of the pandemic. Consequently, this work aims to develop a clear, efficient, and reproducible methodology for parameter optimization, whose implementation is illustrated using data from three representative regions from Chile and a suitable generalized SIR model together with a fitted positivity rate. Our results reproduce the general trend of the infected’s curve, distinguishing the reported and real cases. Finally, our methodology is robust, and it allows us to forecast a second outbreak of COVID-19 and the infection fatality rate of COVID-19 qualitatively according to the reported dead cases.
... If assumption (4.8) is not satisfied, then the sequence ( u n ) satisfies the linear recurrence relation (4.10). The roots of the characteristic equation are simple roots, namely 1 and k −1 complex numbers with modulus less than 1, because the k-step BDF scheme is strictly zero-stable for k ≤ 6 [65,123]. Thus, ( u n ) converges to a constant value M with an exponential rate. ...
Preprint
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We review space and time discretizations of the Cahn-Hilliard equation which are energy stable. In many cases, we prove that a solution converges to a steady state as time goes to infinity. The proof is based on Lyapunov theory and on a Lojasiewicz type inequality. In a few cases, the convergence result is only partial and this raises some interesting questions. Numerical simulations in two and three space dimensions illustrate the theoretical results. Several perspectives are discussed.
... The time step size of the fine trapezoidal rule propagator is set to δt = 10 −5 and the coarse solver is chosen to perform one time step per window and thus has time step size T = 1/N. The Parareal algorithm is iterated until the l 2 norm of the difference between the PP 1 components of the solution at the end of interval I n−1 and the initial condition of I n for all T n is below a relative tolerance of 5 · 10 −8 and an absolute tolerance of 10 −15 (see error norm of [16]). ...
Article
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This article proposes modifications of the Parareal algorithm for its application to higher index differential algebraic equations (DAEs). It is based on the idea of applying the algorithm to only the differential components of the equation and the computation of corresponding consistent initial conditions later on. For differential algebraic equations with a special structure as, e.g. given in flux-charge modified nodal analysis, it is shown that the usage of the implicit Euler method as a time integrator suffices for the Parareal algorithm to converge. Both versions of the Parareal method are applied to numerical examples of nonlinear index 2 differential algebraic equations.
... To prove this stability estimate, we use the following ODE estimate, see also [8,Theorem 10.2]. ...
Preprint
We consider a particle system with uniform coupling between a macroscopic component and individual particles. The constraint for each particle is of full rank, which implies that each movement of the macroscopic component leads to a movement of all particles and vice versa. Skeletal muscle tissues share a similar property which motivates this work. We prove convergence of the mean-field limit, well-posedness and a stability estimate for the mean-field PDE.
... The difference of coefficients between products and reactants species; The underlying ODEs are generated automatically by the simulation library which is an integral part of the PetriNuts framework and is used by Spike, Snoopy and Marcie. A numerical solution of the obtained ODEs can be found by applying different solvers; a classification can be found in [HNW93,HW96]. The basics steps of deterministic simulation are presented by Algorithm 3. ...
Thesis
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Reproducibility of simulation experiments is still a significant challenge and has attracted considerable attention in recent years. One cause of this situation is bad habits of the scientific community. Many results are published without data or source code, and only a textual description of the simulation set-up is provided. Other causes are: no complete simulation set-up, no proper output data analysis and inconsistency of published data, which makes it impossible to compare results. The progress of computational modelling, amount of data and complexity of models requires designing experiments in such a way that ensures reproducibility. A textual description does not provide all the needed details. A computer code is more reliable than a textual description. It is the precise specification that describes a simulation configuration, model, etc. When computer code, data, models and all parameters are provided, the simulation results become reproducible. The main goal of this thesis is to develop a tool that ensures reproducibility and efficient execution of simulation experiments, often involving many individual simulation runs. The tool should support a wide range of application scenarios, where the typical scenario is simulation of biochemical reaction networks, which are represented as (coloured) Petri nets interpreted in the stochastic, continuous or hybrid paradigm. The model to be simulated can be given in various formats, including SBML. The result is a command line tool called Spike, which can be used for various scenarios, including benchmarking, simulation of adaptive models and parameter optimization. It builds on a human-readable configuration script SPC, supporting the efficient specification of multiple model configurations as well as multiple simulator configurations in a single configuration file.
... which depend on the quotient χ of the generating polynomials of the multi-step method [32]. Popular examples are the backward differentiation formulas (BDF) of order p ≤ 2 with ...
Article
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Time-domain Boundary Element Methods (BEM) have been successfully used in acoustics, optics and elastodynamics to solve transient problems numerically. However, the storage requirements are immense, since the fully populated system matrices have to be computed for a large number of time steps or frequencies. In this article, we propose a new approximation scheme for the Convolution Quadrature Method powered BEM, which we apply to scattering problems governed by the wave equation. We use $${\mathscr {H}}^2$$ H 2 -matrix compression in the spatial domain and employ an adaptive cross approximation algorithm in the frequency domain. In this way, the storage and computational costs are reduced significantly, while the accuracy of the method is preserved.
... We now describe the algorithm used, which follows standard procedures in the numerical integration of ordinary differential equations (ODEs). For the implementation of the method, we used a variable formula strategy as described in [14,§ III.5] for the fully nonlinear BDF formulae, but with explicit treatment of the convective term as specified in (20). In particular we notice that the local error estimate of the method of order k is given by ...
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This paper studies fully discrete finite element approximations to the Navier-Stokes equations using inf-sup stable elements and grad-div stabilization. For the time integration two implicit-explicit second order backward differentiation formulae (BDF2) schemes are applied. In both the laplacian is implicit while the nonlinear term is explicit, in the first one, and semi-implicit, in the second one. The grad-div stabilization allow us to prove error bounds in which the constants are independent of inverse powers of the viscosity. Error bounds of order r in space are obtained for the L 2 error of the velocity using piecewise polynomials of degree r to approximate the velocity together with second order bounds in time, both for fixed time step methods and for methods with variable time steps. A CFL-type condition is needed for the method in which the nonlinear term is explicit relating time step and spatial mesh sizes parameters.
... To ensure the stability of the k-step scheme, the norm of the matrix A in equation (39) is not more than 1 (see Chapter III.4, Lemma 4.4 in [32]). is can be satisfied if the eigenvalues eig(A) of the matrix A make |eig(A)| ≤ 1 and the eigenvalues are simple if |eig(A)| � 1. In addition, the eigenvalues of A satisfy the root condition by Definition 1. ...
Article
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For backward stochastic differential equations (BSDEs), we construct variable step size Adams methods by means of Itô–Taylor expansion, and these schemes are nonlinear multistep schemes. It is deduced that the conditions of local truncation errors with respect to Y and Z reach high order. The coefficients in the numerical methods are inferred and bounded under appropriate conditions. A necessary and sufficient condition is given to judge the stability of our numerical schemes. Moreover, the high-order convergence of the schemes is rigorously proved. The numerical illustrations are provided.
... • The BDF2 method is A-stable and L-stable such that it would be more suitable than Crank-Nicolson type schemes for solving the stiff dissipative problems, see e.g. [3,6,15]. • The nonuniform grid and adaptive time-stepping techniques [12,[18][19][20]24] are powerful in capturing the multi-scale behaviors and accelerating the long-time simulations of phase field models including the CH model. • The convergence theory of variable-steps BDF2 scheme remains incomplete for nonlinear parabolic equations. ...
Article
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The two-step backward differential formula (BDF2) with unequal time-steps is applied to construct an energy stable convex-splitting scheme for the Cahn-Hilliard model. We focus on the numerical influences of time-step variations by using the recent theoretical framework with the discrete orthogonal convolution kernels. Some novel discrete convolution embedding inequalities with respect to the orthogonal convolution kernels are developed such that a concise $L^2$ norm error estimate is established at the first time under an updated step-ratio restriction $0 <r_k:=\tau_k/\tau_{k-1}\leq r_{\mathrm{user}}$, where $r_{\mathrm{user}}$ can be chosen by the user such that $r_{\mathrm{user}}<4.864$. The stabilized convex-splitting BDF2 scheme is shown to be mesh-robustly convergent in the sense that the convergence constant (prefactor) in the error estimate is independent of the adjoint time-step ratios. The suggested method is proved to preserve a modified energy dissipation law at the discrete levels if $0<r_k\le r_{\mathrm{user}}$, such that it is mesh-robustly stable in an energy norm. On the basis of ample tests on random time meshes, a useful adaptive time-stepping strategy is applied to efficiently capture the multi-scale behaviors and to accelerate the long-time simulation approaching the steady state.
... Pour cela, elle a été confrontée utilise les équations de Navier-Stokes en instationnaire (URANS). La partie solveur dans [5], une approche quasi-statique est implicite [8]. Cette méthode améliore la robustesse de la convergence par rapport à un solveur statique classique utilisant une méthode de Newton-Raphson. ...
Conference Paper
This is a French conference reserved for members of the ATMA association. The subject of this conference is mechanics through the maritime and aeronautical fields
... Then by taking the H 1 τ -norm for the above two and the induction for the boundedness of U n and U n+ 1 2 (or the Lady Windermere's fan argument [38]), we get e n H 1 τ ≤ Ch 2 for 0 ≤ n ≤ T0 h which implies e n L ∞ τ ≤ Ch 2 . By the fact U (t, t) = u(t), we conclude U n (t n /ε) − u(t n ) X ≤ Ch 2 . ...
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In this paper, we consider a class of highly oscillatory Hamiltonian systems which involve a scaling parameter $\varepsilon\in(0,1]$. The problem arises from many physical models in some limit parameter regime or from some time-compressed perturbation problems. The solution of the model exhibits rapid temporal oscillations with $\mathcal{O}(1)$-amplitude and $\mathcal{O}(1/\varepsilon)$-frequency, which makes classical numerical methods inefficient. We apply the two-scale formulation approach to the problem and propose two new time-symmetric numerical integrators. The methods are proved to have the uniform second order accuracy for all $\varepsilon$ at finite times and some near-conservation laws in long times. Numerical experiments on a H\'{e}non-Heiles model, a nonlinear Schr\"{o}dinger equation and a charged-particle system illustrate the performance of the proposed methods over the existing ones.
... To demonstrate the superiority of the proposed exponential methods over the classical methods, we measure the computing times of different methods. Adams linear multistep methods (see, e.g., [28]), exponential multistep methods, and exponential Rosenbrock multistep methods are compared in terms of computational times. The CPU times in seconds when all of the methods achieve a common error tolerance 10 −8 are recorded in Table 1. ...
Article
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Stiff delay differential equations are frequently utilized in practice, but their numerical simulations are difficult due to the complicated interaction between the stiff and delay terms. At the moment, only a few low-order algorithms offer acceptable convergent and stable features. Exponential integrators are a type of efficient numerical approach for stiff problems that can eliminate the influence of stiffness on the scheme by precisely dealing with the stiff term. This study is concerned with two exponential multistep methods of Adams type for stiff delay differential equations. For semilinear delay differential equations, applying the linear multistep method directly to the integral form of the equation yields the exponential multistep method. It is shown that the proposed k-step method is stiffly convergent of order k. On the other hand, we can follow the strategy of the Rosenbrock method to linearize the equation along the numerical solution in each step. The so-called exponential Rosenbrock multistep method is constructed by applying the exponential multistep method to the transformed form of the semilinear delay differential equation. This method can be easily extended to nonlinear delay differential equations. The main contribution of this study is that we show that the k-step exponential Rosenbrock multistep method is stiffly convergent of order k+1 within the framework of a strongly continuous semigroup on Banach space. As a result, the methods developed in this study may be utilized to solve abstract stiff delay differential equations and can be served as time matching methods for delay partial differential equations. Numerical experiments are presented to demonstrate the theoretical results.
... fast Fourier transform, frequency analysis, integration, and interpolation). Moreover, TRIP natively integrates polynomial dynamical systems through Adams PECE and DOPRI8 methods (Hairer et al. 1993;Prince & Dormand 1981). Finally, TRIP provides a dedicated procedural language that supports loops, conditional statements, and function and structure definition, which permits the symbolic and numerical kernels to be interfaced in autonomous programs. ...
Article
Uncertainty quantification plays an important role in problems that involve inferring a parameter of an initial value problem from observations of the solution. Conrad et al. (Stat Comput 27(4):1065–1082, 2017) proposed randomisation of deterministic time integration methods as a strategy for quantifying uncertainty due to the unknown time discretisation error. We consider this strategy for systems that are described by deterministic, possibly time-dependent operator differential equations defined on a Banach space or a Gelfand triple. Our main results are strong error bounds on the random trajectories measured in Orlicz norms, proven under a weaker assumption on the local truncation error of the underlying deterministic time integration method. Our analysis establishes the theoretical validity of randomised time integration for differential equations in infinite-dimensional settings.
Article
We review space and time discretizations of the Cahn-Hilliard equation which are energy stable. In many cases, we prove that a solution converges to a steady state as time goes to infinity. The proof is based on Lyapunov theory and on a Lojasiewicz type inequality. In a few cases, the convergence result is only partial and this raises some interesting questions. Numerical simulations in two and three space dimensions illustrate the theoretical results. Several perspectives are discussed.
Chapter
Parareal is a widely studied parallel-in-time method that can achieve meaningful speedup on certain problems. However, it is well known that the method typically performs poorly on non-diffusive equations. This paper analyzes linear stability and convergence for IMEX Runge-Kutta Parareal methods on non-diffusive equations. By combining standard linear stability analysis with a simple convergence analysis, we find that certain Parareal configurations can achieve parallel speedup on non-diffusive equations. These stable configurations possess low iteration counts, large block sizes, and a large number of processors. Numerical examples using the nonlinear Schrödinger equation demonstrate the analytical conclusions.
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The one‐leg, two‐step time discretization proposed by Dahlquist, Liniger and Nevanlinna is second order and variable step G‐stable. G‐stability for systems of ordinary differential equations (ODEs) corrresponds to unconditional, long time energy stability when applied to the Navier–Stokes equations (NSEs). In this report, we analyze the method of Dahlquist, Liniger and Nevanlinna as a variable step, time discretization of the Navier–Stokes equations. We prove that the kinetic energy is bounded for variable time‐steps, show that the method is second‐order accurate, characterize its numerical dissipation and prove error estimates. The theoretical results are illustrated by several numerical tests.
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Рассмотрен вопрос о сосуществовании аттракторов нейродинамической модели с запаздыванием, представляющей собой систему двух специальным образом связанных дифференциально-разностных уравнений. Разработан алгоритм оценки показателей Ляпунова для рассматриваемой системы.
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Soft Robotics o�ers an opportunity to fabricate more adapt- able systems to be used outside industrial areas. For practical applica- tions of soft robots, their active parts should achieve high forces and considerable displacements keeping the structures lightweight, modular, and easy to fabricate. Consequently, robust and predictable motion actu- ators with easy manufacture are required. This paper applies the origami concept to address a 3D printed low-weight modular soft pneumatic lin- ear actuator denoted ORI-Structure. The presented structure has a high contraction rate (up to 52.5%) and can lift 1161.64� its weight. Addi- tionally, we present a more complex module with 3 DOF formed with the ORI-Structure. The module can displace linearly with a high-contraction rate (up to 52.5%), and it can rotate in two axes up to 42�. In both cases, the module exhibits high-lifting capabilities (611.53� its weight). In both cases, simulation and experiments are introduced to describe the design parameters and their performances.
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