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Total Variation Diminishing Runge-Kutta Schemes.
Abstract
In this paper we further explore a class of high order Total Variation Diminishing (TVD) Runge-Kutta time discretization initialized in Shu & Osher (1988), suitable for solving hyperbolic conservation laws with stable spatial discretizations. We illustrate with numerical examples that non-TVD but linearly stable Runge-Kutta time discretization can generate oscillations even for TVD spatial discretization, verifying the claim that TVD Runge-Kutta methods are important for such applications. We then explore the issue of optimal TVD Runge-Kutta methods for second, third and fourth order, and for low storage Runge-Kutta methods.
... The conservation laws for mass, momentum, and energy along with the evolution of deviatoric stresses were spatially discretized using a fifth-order weighted essentially non-oscillatory (WENO) shock-capturing scheme [74][75][76] and time integration was performed using a third-order total-variationdiminishing Runge-Kutta scheme. 77 The interfacial conditions between HMX and the pore surface were modeled by applying the free-surface conditions at the interface using a modified ghost-fluid method. [59][60][61]78 Detailed descriptions of the numerical algorithms, levelset implementation, and interface treatment are available in previous publications, 60,79 where the methods were validated for a variety of multi-material problems including pore collapse. ...
Microstructures of energetic materials (EMs) exhibit defects including pores, cracks, inclusions, and delaminated interfaces, all of which act as sites for energy localization under shock loading. Reactions are triggered at these sites and can couple with shocks, leading to detonation. Convoluted and elongated pores or cracks in energetic crystals can significantly enhance or mitigate EM sensitivity and must be factored into micro-structure aware reactive burn models. Here, we advance the state of modeling and physical understanding of the response of elongated pores in cyclotetramethylene-tetranitramine (HMX) to shock loading by employing: (1) updated atomistics-consistent models to show that continuum calculations with such models produce pore collapse and hotspots that closely reproduce molecular dynamics (MD) results; (2) high-order numerical methods to accurately capture shock and interfacial dynamics; and (3) grid resolution that resolves all relevant scales in the physics of elasto-viscoplastic deformation of the material under high strain-rate loading, down to a lower limit set by molecular/statistical-mechanical considerations. These high physical and numerical fidelity calculations demonstrate that continuum predictions are in agreement with atomistic calculations for various orientations of an elongated pore (penny-shape crack). Furthermore, such continuum simulations, particularly for micrometer-scale pores and cracks, can be performed at much smaller computational cost than MD calculations. This paper examines the emergence of shear bands and their impact on pore collapse and hotspot intensity for various orientations of a nm-scale pore. Then, the collapse of a micron-sized pore (inaccessible to MD) is studied to obtain insights into how the shear band and pore-collapse dynamics changes (or not) as the size of the pore increases by several orders of magnitude. The work provides confidence in the recently advanced atomistics-consistent model set for HMX and also provides new physical details of elongated pore-shock interaction that will be of interest to the energetic materials community.
... The hydrodynamic steps (77) and (79) are solved with a third-order strong stability preserving Runge-Kutta method (SSPRK3) [49] with a flow time step ∆t f . For the strong-form chemistry sub-step (78), a second order Runge-Kutta (RK2) method is used with chemistry sub-steps of size ∆t c < ∆t f . ...
High-order methods have recently been shown to be an effective tool for high-fidelity flow computations like direct numerical simulations and large eddy simulations because of their
strong balance between accuracy and computational cost. In this work, a high-order discontinuous Galerkin spectral element method (DGSEM) is developed to solve the chemically reacting Navier-Stokes equations. To handle the disparate length and time scales associated with these equations, we develop a novel method which combines the spectral accuracy of the SEM with the flexibility of the DG approach. The framework, implemented in the spectral element code Nek5000, is well suited to capture turbulence in smooth regions of the flow, while maintaining numerical stability in the presence of shocks. An entropy-residual based artificial viscosity is added to smooth shocked regions of flow, and a positivity-preserving limiter is implemented to suppress non-physical oscillations. These enhancements support
the numerical stability of the hydrodynamic sub-step, which is decoupled from the chemistry integration through a second-order operator splitting method. A series of smooth and discontinuous validation cases are presented in increasing physical and computational complexity for both inviscid and viscous flows. In particular, simulations of canonical one-dimensional and two-dimensional detonations are performed, and the high-order numerical results are validated against available literature data. Additional validation studies are carried out for classical three-dimensional numerical simulations of incompressible and compressible turbulent flows.
... 52-54 A 3rd-order, total-variation-diminishing Runge-Kutta scheme is used for time-integration. 55 To define and advect material interfaces, a narrow-band level set-based tracking method was utilized. The use of a level set function allows for the capability to handle large deformations of interfaces, which are inherent to shock-induced pore collapse events. ...
Previous works [Herrin et al., J. Appl. Phys. 136(13), 135901 (2024), Nguyen et al., J. Appl. Phys. 136(11), 114902 (2024)] obtained atomistics-consistent material models for two common energetic crystals, HMX (1,3,5,7-Tetranitro-1,3,5,7-tetrazocane) and RDX (1,3,5-Trinitro-1,3,5-triazinane) such that pore collapse calculations adhered closely to molecular dynamics (MD) results on key features of energy localization, particularly the appearance of shear bands, shapes of the collapsing pores, and the transition from viscoplastic to hydrodynamic collapse. However, only one pore size (of 50 nm diameter) was studied and some important aspects such as temperature distributions in the hotspot were found to be inconsistent with the atomistic models. One potential issue was noted but not resolved adequately in those works, namely, the grid resolution that should be employed in the meso-scale calculations for various pore sizes and shock strengths. Conventional computational mechanics guidelines for selecting meshes as fine as possible, balancing computational effort, accuracy, and grid independence, were shown not to produce physically consistent features associated with shear localization. Here, we examine the physics of pore collapse, shear band evolution and structure, and hotspot formation for both HMX and RDX; we then evaluate under what conditions atomistics-consistent models yield “physically correct” (considering MD as “ground truth”) hotspots for a range of pore diameters, from nm to micrometers, and for a wide range of shock strengths. The study provides insights into the effects of pore size and shock strength on pore collapse and hotspots, identifying aspects such as size-independent behaviors, and proportion of energy contained in shear as opposed to jet impact-heated regions of the hotspot.
... In this framework, the spatial discretization of the momentum equations is realized by the thirdorder accurate FVMS3 scheme [43] which uses a merged compact stencil to get rid of the difficulty in selecting admissible cells in the reconstruction process. For pressure-velocity coupling, the fractional step approach [18,3] is adopted, while time advancement is handled using a third-order Total Variation Diminishing (TVD) Runge-Kutta scheme [8]. The resulting numerical scheme significantly improves the numerical accuracy and dissipation and effectively reduces the dependence of numerical solutions on the grid quality, which makes it well-suited for high-Reynolds-number flows in complex geometries. ...
This study employs high-fidelity numerical simulations to investigate the influence of appendages on the turbulent flow dynamics and far-field acoustic radiation of the SUBOFF submarine model at a Reynolds number of Re = 1.2 × 10^7. Utilizing a third-order numerical scheme combined with wall-modeled large eddy simulation (WMLES) and the Ffowcs Williams-Hawkings (FW-H) acoustic analogy, the hydrodynamic and acoustic behaviors of an appended SUBOFF configuration are compared to those of a bare hull. A computational grid of 103 million cells resolves the intricate flow interactions, while 648 hydrophones positioned 500 diameters from the model capture far-field acoustic signatures. Key results reveal that appendages significantly amplify hydrodynamic and acoustic disturbances. Flow separations and vortex shedding at appendage junctions elevate pressure-induced drag contributions, contrasting the viscous-dominated drag of the bare hull. The sail-hull interaction intensifies local surface pressure fluctuations, increasing power spectral density (PSD) amplitudes by up to an order of magnitude. In the far field, the appended SUBOFF generates sound pressure levels approximately 20 dB higher than the bare hull, with distinct dipole directivity patterns and peak noise levels (85.10 dB) observed on the central plane. Appendages also disrupt wake symmetry, introducing complex vortical structures such as horseshoe and necklace vortices. These findings demonstrate the critical influence of appendages on hydrodynamic and acoustic behavior, filling a gap in turbulence noise research for complex underwater geometries and providing a vital foundation for the noise reduction optimization of advanced underwater vehicles.
... This approach yields a numerical solution that automatically satisfies the S 2 constraint. We adopt the TVDRK methodology [32,33,11,12], which constructs higher-order numerical solutions using convex combinations of elementary forward Euler-type building blocks. This construction gives rise to a class of straightforward, high-order, and efficient explicit numerical schemes for solving ODEs on S 2 . ...
We propose implicit integrators for solving stiff differential equations on unit spheres. Our approach extends the standard backward Euler and Crank-Nicolson methods in Cartesian space by incorporating the geometric constraint inherent to the unit sphere without additional projection steps to enforce the unit length constraint on the solution. We construct these algorithms using the exponential map and spherical linear interpolation (SLERP) formula on the unit sphere. Specifically, we introduce a spherical backward Euler method, a projected backward Euler method, and a second-order symplectic spherical Crank-Nicolson method. While all methods require solving a system of nonlinear equations to advance the solution to the next time step, these nonlinear systems can be efficiently solved using Newton's iterations. We will present several numerical examples to demonstrate the effectiveness and convergence of these numerical schemes. These examples will illustrate the advantages of our proposed methods in accurately capturing the dynamics of stiff systems on unit spheres.
In this paper, we propose a new sixth-order weighted essentially non-oscillatory (WENO) scheme for solving compressible aerodynamics. As with the previous six-order central upwind WENO scheme, this present scheme is a convex combination of four linear reconstructions. The stencil type is the same as WENO-Z6 scheme (Hu FX 2018), although the first three upwind stencils nominally use three cell values, the original secondary reconstruction has been modified to have fourth-order accuracy by adding a third-order correction term involving five cell values. In addition, corresponding smoothness indicators are provided, the new schemes are called WENO-MS-JS6 and WENO-MS-Z6. And by using the new reference smoothness indicator, the WENO-MS-Z6 scheme achieved the optimal convergence accuracy order in the smooth region including the first-order critical point (). Several numerical examples demonstrate that the new schemes have better resolution and the same robustness as the recently developed sixth order WENO schemes.
This study presents a comparative analysis of high-resolution shock-capturing schemes in two-dimensional magnetohydrodynamic (MHD) simulations of magnetic flux emergence in the solar atmosphere. We evaluate four distinct reconstruction techniques based on recent improvements to the weighted essentially nonoscillatory (WENO) and targeted essentially nonoscillatory schemes. While these schemes have proven successful for the Euler equations of gas dynamics, their effectiveness in MHD simulations remains relatively unexplored. Our implementation combines the Harten–Lax–van Leer-discontinuities approximate Riemann solver for accurate flux computations, the generalized Lagrangian multiplier method for divergence control, and a third-order strong stability preserving the Runge–Kutta scheme for time integration. Numerical experiments reveal that these advanced schemes provide significant improvements in both accuracy and robustness in capturing complex MHD phenomena such as magnetosonic waves, MHD shocks, and magnetic buoyancy-driven instabilities. Among the tested methods, IMWENO-P proves to be the most physically consistent, effectively reproducing energy redistribution and compression patterns in line with theoretical predictions. These findings offer valuable insights into the strengths and limitations of each approach for simulating magnetic flux emergence dynamics.
A method of second-order accuracy is described for integrating the equations of ideal compressible flow. The method is based on the integral conservation laws and is dissipative, so that it can be used across shocks. The heart of the method is a one-dimensional Lagrangean scheme that may be regarded as a second-order sequel to Godunov's method. The second-order accuracy is achieved by taking the distributions of the state quantities inside a gas slab to be linear, rather than uniform as in Godunov's method. The Lagrangean results are remapped with least-squares accuracy onto the desired Euler grid in a separate step. Several monotonicity algorithms are applied to ensure positivity, monotonicity and nonlinear stability. Higher dimensions are covered through time splitting. Numerical results for one-dimensional and two-dimensional flows are presented, demonstrating the efficiency of the method. The paper concludes with a summary of the results of the whole series “Towards the Ultimate Conservative Difference Scheme.”
A class of new explicit second order accurate finite difference schemes for the computation of weak solutions of hyperbolic conservation laws is presented. These highly nonlinear schemes are obtained by applying a nonoscillatory first order accurate scheme to an appropriately modified flux function. The so-derived second order accurate schemes achieve high resolution while preserving the robustness of the original nonoscillatory first order accurate scheme. Numerical experiments are presented to demonstrate the performance of these new schemes.
Implicit two-step Runge-Kutta methods are studied. It will be shown that these methods require fewer stages to achieve the same order as one-step Runge-Kutta methods, which means the two-step methods are potentially more efficient than one-step methods. Order conditions are derived and examples of two-step one-stage methods of order 2 and two-step two-stage methods of order 4 are presented. Stability properties of these methods with respect to y′ = ay are studied and A-stable two-step methods of order 2 are characterized. Two-step two-stage methods of order 4 which are A-stable are found by an extensive computer search. Semi-implicit two-stage methods of order 4 were also constructed. This is in contrast to the situation encountered in the Runge-Kutta theory where the unique two-stage method of order 4 is not semi-implicit.
A method of second-order accuracy is described for integrating the equations of ideal compressible flow. The method is based on the integral conservation laws and is dissipative, so that it can be used across shocks. The heart of the method is a one-dimensional Lagrangean scheme that may be regarded as a second-order sequel to Godunov's method. The second-order accuracy is achieved by taking the distributions of the state quantities inside a gas slab to be linear, rather than uniform as in Godunov's method. The Lagrangean results are remapped with least-squares accuracy onto the desired Euler grid in a separate step. Several monotonicity algorithms are applied to ensure positivity, monotonicity and nonlinear stability. Higher dimensions are covered through time splitting. Numerical results for one-dimensional and two-dimensional flows are presented, demonstrating the efficiency of the method. The paper concludes with a summary of the results of the whole series “Towards the Ultimate Conservative Difference Scheme.”
In the computation of conservation laws u, + f(u)s = 0, TVD (total-variation diminishing) schemes have been very successful. But there is a severe disadvantage of all TVD schemes: They must degenerate locally to first-order accuracy at nonsonic critical points. In this paper we describe a procedure to obtain TVB (total-variation-bounded) schemes which are of uniformly high-order accuracy in space including at critical points. Together with a TVD high-order time discretization (discussed in a separate paper), we may have globally high-order in space and time TVB schemes. Numerical examples are provided to illustrate these schemes.
This is the second paper in a series in which we construct and analyze a class of TVB (total variation bounded) discontinuous Galerkin finite element methods for solving conservation laws ut+ ∑i=1d(fi(u))xi=0. In this paper we present a general framework of the methods, up to any order of formal accuracy, using scalar one-dimensional initial value and initial-boundary problems as models. In these cases we prove TVBM (total variation bounded in the means), TVB, and convergence of the schemes. Numerical results using these methods are also given. Extensions to systems and/or higher dimensions will appear in future papers.
This is the second paper in a series in which we construct and analyze a class of TVB (total variation bounded) discontinuous Galerkin finite element methods for solving conservation laws . In this paper we present a general framework of the methods, up to any order of formal accuracy, using scalar one-dimensional initial value and initial-boundary problems as models. In these cases we prove TVBM (total variation bounded in the means), TVB, and convergence of the schemes. Numerical results using these methods are also given. Extensions to systems and/or higher dimensions will appear in future papers.
In this paper we study the two-dimensional version of the Runge-Kutta Local Projection Discontinuous Galerkin (RKDG) methods, already defined and analyzed in the one-dimensional case. These schemes are defined on general triangulations. They can easily handle the boundary conditions, verify maximum principles, and are formally uniformly high-ordrr accurate. Preliminary numerical results showing the performance of the schemes on a variety of initial-boundary value problems are shown.
A special algorithm is derived for pseudo-Runge-Kutta methods that is useful in stepsize control. The new method is designed to enable the change of stepsize without new functions evaluations. This formula can also provide dense output.
In the computation of conservation laws , TVD (total-variation-diminishing) schemes have been very successful. But there is a severe disadvantage of all TVD schemes: They must degenerate locally to first-order accuracy at nonsonic critical points. In this paper we describe a procedure to obtain TVB (total-variation-bounded) schemes which are of uniformly high-order accuracy in space including at critical points. Together with a TVD high-order time discretization (discussed in a separate paper), we may have globally high-order in space and time TVD schemes. Numerical examples are provided to illustrate these schemes.