Spatially Adaptive Projective Integration Schemes For Stiff Hyperbolic Balance Laws With Spectral Gaps

Koellermeier, Julian; Samaey, Giovanni

The SMAI Journal of computational mathematics, Tome 8 (2022), pp. 295-325.

Stiff hyperbolic balance laws exhibit large spectral gaps, especially if the relaxation term significantly varies in space. Using examples from rarefied gases and the general form of the underlying balance law model, we perform a detailed spectral analysis of the semi-discrete model that reveals the spectral gaps. Based on that, we show the inefficiency of standard time integration schemes expressed by a severe restriction of the CFL number. We then develop the first spatially adaptive projective integration schemes to overcome the prohibitive time step constraints of standard time integration schemes. The new schemes use different time integration methods in different parts of the computational domain, determined by the spatially varying value of the relaxation time. We use our analytical results to derive accurate stability bounds for the involved parameters and show that the severe time step constraint can be overcome. The new adaptive schemes show good accuracy in a numerical test case and can obtain a large speedup with respect to standard schemes.

Classification : 76P05, 35Q20, 35B40, 34E13, 35L02
Mots clés : Projective integration, spatial adaptivity, hyperbolic balance law, moment equations

[1] Anderson, J. D. Fundamentals of Aerodynamics, McGraw-Hill series in aeronautical and aerospace engineering, McGraw-Hill, 1991

[2] Au, J. D.; Torrilhon, M.; Weiss, W. The shock tube experiment in extended thermodynamics, Phys. Fluids, Volume 13 (2001) no. 8, pp. 2423-2432 | Zbl

[3] Bhatnagar, P. L.; Gross, E. P.; Krook, M. A Model for Collision Processes in Gases. 1. Small Amplitude Processes in Charged and Neutral One-Component Systems, Phys. Rev., Volume 94 (1954), pp. 511-525 | DOI | Zbl

[4] Boscarino, S.; Pareschi, L.; Russo, G. A Unified IMEX Runge–Kutta Approach for Hyperbolic Systems with Multiscale Relaxation, SIAM J. Numer. Anal., Volume 55 (2017) no. 4, pp. 2085-2109 | DOI | MR | Zbl

[5] Cai, Z.; Fan, Y.; Li, R. Globally hyperbolic regularization of grad’s moment system in one dimensional space, Commun. Math. Sci., Volume 11 (2013) no. 2, pp. 547-571 | DOI | MR | Zbl

[6] Castro, M. J.; LeFloch, P. G.; Munoz-Ruiz, M. L.; Pares, C. Why many theories of shock waves are necessary: Convergence error in formally path-consistent schemes, J. Comput. Phys., Volume 227 (2008) no. 17, pp. 8107-8129 | DOI | MR | Zbl

[7] Degond, P.; Jin, S.; Mieussens, L. A smooth transition model between kinetic and hydrodynamic equations, J. Comput. Phys., Volume 209 (2005) no. 2, pp. 665-694 | DOI | MR | Zbl

[8] Dumbser, M.; Käser, M.; Toro, E. F. An arbitrary high-order Discontinuous Galerkin method for elastic waves on unstructured meshes - V. Local time stepping and p-adaptivity, Geophys. J. Int., Volume 171 (2007) no. 2, pp. 695-717

[9] Fan, Y.; Koellermeier, J. Accelerating the Convergence of the Moment Method for the Boltzmann Equation Using Filters, J. Sci. Comput., Volume 84 (2020) no. 1, pp. 1-28 | DOI | MR | Zbl

[10] Fan, Y.; Koellermeier, J.; Li, J.; Li, R.; Torrilhon, M. Model Reduction of Kinetic Equations by Operator Projection, J. Stat. Phys., Volume 162 (2016) no. 2, pp. 457-486 | DOI | MR | Zbl

[11] Garcia, A. L.; Bell, J. B.; Crutchfield, W. Y.; Alder, B. J. Adaptive Mesh and Algorithm Refinement Using Direct Simulation Monte Carlo, J. Comput. Phys., Volume 154 (1999) no. 1, pp. 134-155 | DOI | Zbl

[12] Gassner, G.; Dumbser, M.; Hindenlang, F.; Munz, C.-D. Explicit one-step time discretizations for discontinuous Galerkin and finite volume schemes based on local predictors, J. Comput. Phys., Volume 230 (2011) no. 11, pp. 4232-4247 | DOI | MR | Zbl

[13] Gear, C. W.; Kevrekidis, I. G. Projective Methods for Stiff Differential Equations: Problems with Gaps in Their Eigenvalue Spectrum, SIAM J. Sci. Comput., Volume 24 (2003) no. 4, pp. 1091-1106 | DOI | MR | Zbl

[14] Günther, M.; Sandu, A. Multirate generalized additive Runge Kutta methods, Numer. Math., Volume 133 (2016) no. 3, pp. 497-524 | DOI | MR | Zbl

[15] Jin, S. Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations: a review, Riv. Mat. Univ. Parma (N.S.), Volume 3 (2012) no. 2 | MR | Zbl

[16] Klein, R. Scale-Dependent Models for Atmospheric Flows, Annu. Rev. Fluid Mech., Volume 42 (2010) no. 1, pp. 249-274 | DOI | MR | Zbl

[17] Koellermeier, J. Derivation and numerical solution of hyperbolic moment equations for rarefied gas flows, Ph. D. Thesis, RWTH Aachen University (2017) | DOI

[18] Koellermeier, J.; Castro, M. J. High-order Non-Conservative Simulation of Hyperbolic Moment Models, East Asian J. Appl. Math., Volume 11 (2021) no. 3, pp. 435-467 | Zbl

[19] Koellermeier, J.; Rominger, M. Analysis and Numerical Simulation of Hyperbolic Shallow Water Moment Equations, Commun. Comput. Phys., Volume 28 (2020) no. 3, pp. 1038-1084 | DOI | MR | Zbl

[20] Koellermeier, J.; Samaey, G. Projective Integration for Moment Models of the BGK Equation, Computational Science – ICCS 2020 (Lecture Notes in Computer Science), Volume 12142, Springer, 2020, pp. 321-333 | DOI | MR

[21] Koellermeier, J.; Samaey, G. Projective integration schemes for hyperbolic moment equations, Kinet. Relat. Models, Volume 14 (2021) no. 2, p. 353 | DOI | MR | Zbl

[22] Koellermeier, J.; Samaey, G. Software for: Spatially Adaptive Projective Integration Schemes For Stiff Hyperbolic Balance Laws With Spectral Gaps, Zenodo, 2022 (http://doi.org/10.5281/zenodo.6630525)

[23] Koellermeier, J.; Torrilhon, M. Numerical Study of Partially Conservative Moment Equations in Kinetic Theory, Commun. Comput. Phys., Volume 21 (2017) no. 4, pp. 981-1011 | DOI | MR | Zbl

[24] Kolobov, V. I.; Arslanbekov, R. R.; Aristov, V. V.; Frolova, A. A.; Zabelok, S. A. Unified solver for rarefied and continuum flows with adaptive mesh and algorithm refinement, J. Comput. Phys., Volume 223 (2007) no. 2, pp. 589-608 | DOI | Zbl

[25] Lafitte, P.; Lejon, A.; Samaey, G. A High-Order Asymptotic-Preserving Scheme for Kinetic Equations Using Projective Integration, SIAM J. Numer. Anal., Volume 54 (2016) no. 1, pp. 1-33 | DOI | MR | Zbl

[26] Lafitte, P.; Melis, W.; Samaey, G. A high-order relaxation method with projective integration for solving nonlinear systems of hyperbolic conservation laws, J. Comput. Phys., Volume 340 (2017), pp. 1-25 | DOI | MR | Zbl

[27] Lafitte, P.; Samaey, G. Asymptotic-preserving Projective Integration Schemes for Kinetic Equations in the Diffusion Limit, SIAM J. Sci. Comput., Volume 34 (2012) no. 2, p. A579-A602 | DOI | MR | Zbl

[28] Melis, W.; Rey, T.; Samaey, G. Projective Integration for Nonlinear BGK Kinetic Equations, Finite Volumes for Complex Applications VIII - Hyperbolic, Elliptic and Parabolic Problems (Cancès, C; Omnes, P, eds.) (2017), pp. 145-153 | DOI | Zbl

[29] Melis, W.; Rey, T.; Samaey, G. Projective and telescopic projective integration for the nonlinear BGK and Boltzmann equations, SMAI J. Comput. Math., Volume 5 (2019), pp. 53-88 | DOI | Numdam | MR | Zbl

[30] Melis, W.; Samaey, G. Telescopic projective integration for kinetic equations with multiple relaxation times, J. Sci. Comput., Volume 76 (2018), pp. 697-726 | DOI | MR | Zbl

[31] Parés, C. Numerical Methods for Nonconservative Hyperbolic Systems: A Theoretical Framework, SIAM J. Numer. Anal., Volume 44 (2006) no. 1, pp. 300-321 | DOI | MR | Zbl

[32] Pareschi, L.; Russo, G. Implicit–Explicit Runge–Kutta Schemes and Applications to Hyperbolic Systems with Relaxation, J. Sci. Comput., Volume 25 (2005), pp. 129-155 | DOI | MR | Zbl

[33] Schaerer, R. P.; Torrilhon, M. On Singular Closures for the 5-Moment System in Kinetic Gas Theory, Commun. Comput. Phys., Volume 17 (2015) no. 2, pp. 371-400 | DOI | MR | Zbl

[34] Strang, G. On the Construction and Comparison of Difference Schemes, SIAM J. Numer. Anal., Volume 5 (1968) no. 3, pp. 506-517 | DOI | MR | Zbl

[35] Struchtrup, H. Macroscopic Transport Equations for Rarefied Gas Flows: Approximation Methods in Kinetic Theory, Interaction of Mechanics and Mathematics, Springer, 2006

[36] Tcheremissine, F. G. Solution of the Boltzmann Equation in Stiff Regime, Hyperbolic Problems: Theory, Numerics, Applications (Freistühler, H; Warnecke, G, eds.), Birkhäuser (2001), pp. 883-890 | MR

[37] Tiwari, S.; Klar, A. An adaptive domain decomposition procedure for Boltzmann and Euler equations, J. Comput. Appl. Math., Volume 90 (1998) no. 2, pp. 223-237 | DOI | MR | Zbl

[38] Torrilhon, M. Modeling Nonequilibrium Gas Flow Based on Moment Equations, Annu. Rev. Fluid Mech., Volume 48 (2016) no. 1, pp. 429-458 | DOI | MR | Zbl

[39] Tretter, C. Spectral Theory of Block Operator Matrices and Applications, Imperial College Press, 2008, pp. 1-264 | DOI

[40] Van Leemput, P.; Vanroose, W.; Roose, D. Numerical and Analytical Spatial Coupling of a Lattice Boltzmann Model and a Partial Differential Equation, Model Reduction and Coarse-Graining Approaches for Multiscale Phenomena, Springer, 2006, pp. 423-441 | DOI | Zbl

[41] Xiao, T.; Liu, C.; Xu, K.; Cai, Q. A velocity-space adaptive unified gas kinetic scheme for continuum and rarefied flows, J. Comput. Phys., Volume 415 (2020), 109535 | DOI | MR | Zbl