Efficient Finite Difference WENO Scheme for Hyperbolic Systems with Non-conservative Products

Abstract

Higher order finite difference weighted essentially non-oscillatory (WENO) schemes have been constructed for conservation laws. For multidimensional problems, they offer a high order accuracy at a fraction of the cost of a finite volume WENO or DG scheme of the comparable accuracy. This makes them quite attractive for several science and engineering applications. But, to the best of our knowledge, such schemes have not been extended to non-linear hyperbolic systems with non-conservative products. In this paper, we perform such an extension which improves the domain of the applicability of such schemes. The extension is carried out by writing the scheme in fluctuation form. We use the HLLI Riemann solver of Dumbser and Balsara (J. Comput. Phys. 304: 275–319, 2016) as a building block for carrying out this extension. Because of the use of an HLL building block, the resulting scheme has a proper supersonic limit. The use of anti-diffusive fluxes ensures that stationary discontinuities can be preserved by the scheme, thus expanding its domain of the applicability. Our new finite difference WENO formulation uses the same WENO reconstruction that was used in classical versions, making it very easy for users to transition over to the present formulation. For conservation laws, the new finite difference WENO is shown to perform as well as the classical version of finite difference WENO, with two major advantages: (i) It can capture jumps in stationary linearly degenerate wave families exactly. (ii) It only requires the reconstruction to be applied once. Several examples from hyperbolic PDE systems with non-conservative products are shown which indicate that the scheme works and achieves its design order of the accuracy for smooth multidimensional flows. Stringent Riemann problems and several novel multidimensional problems that are drawn from compressible Baer-Nunziato multiphase flow, multiphase debris flow and two-layer shallow water equations are also shown to document the robustness of the method. For some test problems that require well-balancing we have even been able to apply the scheme without any modification and obtain good results. Many useful PDEs may have stiff relaxation source terms for which the finite difference formulation of WENO is shown to provide some genuine advantages.

Keywords

Hyperbolic PDEs / Numerical schemes / Non-conservative products / Stiff source terms / Finite difference WENO

References

[1.] Andrianov N, Warnecke G. The Riemann problem for the Baer-Nunziato two-phase flow model. J. Comput. Phys., 2004, 212: 434-464

[2.] Baer MR, Nunziato JW. A two-phase mixture theory for the deflagration-to-detonation transition (DDT) in reactive granular materials. Int. J. Multiphase Flow, 1986, 12: 861-889

[3.] Balsara DS. Self-adjusting, positivity preserving high order schemes for hydrodynamics and magnetohydrodynamics. J. Comput. Phys., 2012, 231: 7504-7517

[4.] Balsara DS, Garain S, Florinski V, Boscheri W. An efficient class of WENO schemes with adaptive order for unstructured meshes. J. Comput. Phys., 2020, 404

[5.] Balsara DS, Garain S, Shu C-W. An efficient class of WENO schemes with adaptive order. J. Comput. Phys., 2016, 326: 780-804

[6.] Balsara DS, Rumpf T, Dumbser M, Munz C-D. Efficient, high-accuracy ADER-WENO schemes for hydrodynamics and divergence-free magnetohydrodynamics. J. Comput. Phys., 2009, 228: 2480

[7.] Balsara DS, Samantaray S, Subramanian S. Efficient WENO-based prolongation strategies for divergence-preserving vector fields. Commun. Appl. Math. Comput., 2023, 5: 428-484

[8.] Balsara DS, Shu C-W. Monotonicity preserving weighted non-oscillatory schemes with increasingly high order of accuracy. J. Comput. Phys., 2000, 160: 405-452

[9.] Bdzil JB, Kapila A, Hennessey MP. Shock structure for the seven-equation, two-phase continuum-mixture model. Combust. Theory Model., 2021, 25(6): 970-1001

[10.] Berberich JP, Käppeli R, Chandrashekar P, Klingenberg C. High order discretely well-balanced methods for arbitrary hydrostatic atmospheres. Commun. Comput. Phys., 2021, 30(3): 666-708

[11.] Borges R, Carmona M, Costa B, Don WS. An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws. J. Comput. Phys., 2008, 227(6): 3101-3211

[12.] Boscheri W, Balsara DS. High order direct Arbitrary-Lagrangian-Eulerian (ALE) PNPM schemes on unstructured meshes. J. Comput. Phys., 2019, 398: 108899

[13.] Castro M, Costa B, Don WS. High order weighted essentially non-oscillatory WENO-Z schemes for hyperbolic conservation laws. J. Comput. Phys., 2011, 230: 1766-1792

[14.] Castro M, Gallardo JE, Pares C. High order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products, applications to shallow-water systems. Math. Comput., 2006, 75(255): 1103-1134

[15.] Castro MJ, Pardo A, Parés C, Toro EF. On some fast well-balanced first order solvers for nonconservative systems. Math. Comput., 2010, 79: 1427-1472

[16.] Chiochetti, S., Müller, C.: A solver for stiff finite-rate relaxation in Baer-Nunziato two-phase flow models. Fluid Mech. Appl. 121, 31–44 (2020). https://doi.org/10.1007/978-3-030-33338-6_3

[17.] Coquel F, Marmignon C, Rai P, Renac F. An entropy stable high-order discontinuous Galerkin spectral element method for the Baer-Nunziato two-phase flow model. J. Comput. Phys., 2021, 431

[18.] Cravero I, Semplice M. On the accuracy of WENO and CWENO reconstructions of third order on nonuniform meshes. J. Sci. Comput., 2016, 67(3): 1219-1246

[19.] Delchini MO, Ragusa JC, Berry RA. Viscous regularization for the non-equilibrium seven-equation two-phase flow model. J. Sci. Comput., 2016, 69: 764-804

[20.] Dumbser M, Balsara DS. A new, efficient formulation of the HLLEM Riemann solver for general conservative and non-conservative hyperbolic systems. J. Comput. Phys., 2016, 304: 275-319

[21.] Dumbser M, Boscheri W. High-order unstructured Lagrangian one-step WENO finite volume schemes for non-conservative hyperbolic systems: applications to compressible multi-phase flows. Comput. Fluids, 2013, 86: 405-432

[22.] Dumbser M, Castro M, Parés C, Toro EF. ADER schemes on unstructured meshes for non-conservative hyperbolic systems: applications to geophysical flows. Comput. Fluids, 2009, 38: 1731-1748

[23.] Dumbser M, Hidalgo A, Castro M, Parés C, Toro EF. FORCE schemes on unstructured meshes II: non-conservative hyperbolic systems. Comput. Methods Appl. Mech. Eng., 2010, 199: 625-647

[24.] Dumbser M, Hidalgo A, Zanotti O. High-order space-time adaptive ADER-WENO finite volume schemes for non-conservative hyperbolic systems. Comput. Methods Appl. Mech. Eng., 2014, 268: 359-387

[25.] Dumbser M, Iben U, Ioriatti M. An efficient semi-implicit finite volume method for axially symmetric compressible flows in compliant tubes. Appl. Numer. Math., 2015, 89: 24-44

[26.] Dumbser M, Käser M. Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems. J. Comput. Phys., 2007, 221: 693-723

[27.] Friedrichs O. Weighted essentially non-oscillatory schemes for the interpolation of mean values on unstructured grids. J. Comput. Phys., 1998, 144: 194-212

[28.] Gerolymos GA, Sénéchal D, Vallet I. Very high order WENO schemes. J. Comput. Phys., 2009, 228: 8481-8524

[29.] Gonthier KA, Powers JM. A numerical investigation of transient detonation in granulated material. Shock Waves, 1996, 6(4): 183-195

[30.] Gonthier KA, Powers JM. A high resolution numerical method for a two-phase model of deflagration-to-detonation transition. J. Comput. Phys., 2000, 163(2): 376-433

[31.] Grosheintz-Laval L, Käppeli R. Well-balanced finite volume schemes for nearly steady adiabatic flows. J. Comput. Phys., 2020, 423

[32.] Harten A, Engquist B, Osher S, Chakravarthy S. Uniformly high order essentially non-oscillatory schemes III. J. Comput. Phys., 1987, 71: 231-303

[33.] Hennessey MP, Kapila A, Schwendeman DW. An HLLC-type Riemann solver and high-resolution Godunov method for a two-phase model of reactive flow with general equations of state. J. Comput. Phys., 2020, 405

[34.] Henrick AK, Aslam TD, Powers JM. Mapped weighted essentially non-oscillatory schemes: achieving optimal order near critical points. J. Comput. Phys., 2006, 207: 542-567

[35.] Jiang G-S, Shu C-W. Efficient implementation of weighted ENO schemes. J. Comput. Phys., 1996, 126: 202-228

[36.] Kapila A, Menikoff R, Bdzil JB, Son SF, Stewart DS. Two-phase modeling of deflagration-to-detonation transition in granular materials: reduced equations. Phys. Fluids, 2001, 13(10): 3002-3024

[37.] Käppeli R. Well-balanced methods for computational astrophysics. Living Rev. Comput. Astrophys., 2022, 8(2): 1-88

[38.] Käppeli R, Mishra S. Well-balanced schemes for the Euler equations with gravitation. J. Comput. Phys., 2014, 259: 199-219

[39.] Käser M, Iske A. ADER schemes on adaptive triangular meshes for scalar conservation laws. J. Comput. Phys., 2005, 205: 486-508

[40.] Kupka F, Happenhofer N, Higueras I, Koch O. Total-variation-diminishing implicit-explicit Runge-Kutta methods for the simulation of double-diffusive convection in astrophysics. J. Comput. Phys., 2012, 231: 3561-3586

[41.] LeVeque RJ. Balancing source terms and flux gradients in high-resolution Godunov methods: the quasi-steady wave propagation algorithm. J. Comput. Phys., 1998, 146: 346-365

[42.] Levy D, Puppo G, Russo G. Compact central WENO schemes for multidimensional conservation laws. SIAM J. Sci. Comput., 2000, 22: 656-672

[43.] Liu X-D, Osher S, Chan T. Weighted essentially non-oscillatory schemes. J. Comput. Phys., 1994, 115: 200-212

[44.] Pao, S.P., Salas, M.D.: A numerical study of two-dimensional shock-vortex interaction. In: AIAA Paper 81-1205 14th Fluid and Plasma Dynamics Conference. American Institute of Aeronautics and Astronautics, Reston (1981)

[45.] Pareschi L, Russo G. Implicit-explicit Runge-Kutta schemes and applications to hyperbolic systems with relaxation. J. Sci. Comput., 2005, 25: 129

[46.] Pelanti M, Bouchut F, Mangeney A. A Roe-type scheme for two-phase shallow granular flows over variable topography. Math. Model. Numer. Anal., 2008, 42: 851-885

[47.] Peshkov I, Pavelka M, Romenski E, Grmela M. Continuum mechanics and thermodynamics in the Hamilton and the Godunov-type formulations. Continuum Mech. Thermodyn., 2018, 30: 1343-1378

[48.] Peshkov I, Romenski E. A hyperbolic model for viscous Newtonian flows. Continuum Mech. Thermodyn., 2016, 28: 85-104

[49.] Pitman EB, Le L. A two-fluid model for avalanche and debris flows. Philos. Trans. R. Soc. A, 2005, 363: 1573-1601

[50.] Powers JM. Two-phase viscous modeling of compaction of granular materials. Phys. Fluids, 2004, 16(8): 2975-2990

[51.] Rhebergen S, Bokhove O, van der Vegt JJW. Discontinuous Galerkin finite element methods for hyperbolic nonconservative partial differential equations. J. Comput. Phys., 2008, 227: 1887-1922

[52.] Saurel R, Abgrall R. A multiphase Godunov method for compressible multifluid and multiphase flows. J. Comput. Phys., 1999, 150: 425-467

[53.] Schwendeman DW, Wahle CW, Kapila AK. The Riemann problem and a high-resolution Godunov method for a model of compressible two-phase flow. J. Comput. Phys., 2006, 212: 490-526

[54.] Semplice M, Coco A, Russo G. Adaptive mesh refinement for hyperbolic systems based on third-order compact WENO reconstruction. J. Sci. Comput., 2016, 66(2): 692-724

[55.] Shu C-W. High order weighted essentially non-oscillatory schemes for convection dominated problems. SIAM Rev., 2009, 51: 82-126

[56.] Shu C-W. Essentially non-oscillatory and weighted essentially non-oscillatory schemes. Acta Numer., 2020, 29: 701-762

[57.] Shu C-W, Osher SJ. Efficient implementation of essentially non-oscillatory shock capturing schemes. J. Comput. Phys., 1988, 77: 439-471

[58.] Shu C-W, Osher SJ. Efficient implementation of essentially non-oscillatory shock capturing schemes II. J. Comput. Phys., 1989, 83: 32-78

[59.] Spiteri RJ, Ruuth SJ. A new class of optimal high-order strong-stability-preserving time-stepping schemes. SIAM J. Anal., 2002, 40: 469-491

[60.] Spiteri RJ, Ruuth SJ. Non-linear evolution using optimal fourth-order strong-stability-preserving Runge-Kutta methods. Math. Comput. Simul., 2003, 62: 125-135

[61.] Tokareva SA, Toro EF. HLLC-type Riemann solver for the Baer-Nunziato equations of compressible two-phase flow. J. Comput. Phys., 2010, 229: 3573-3604

[62.] Woodward P, Colella P. The numerical simulation of two-dimensional fluid flow with strong shocks. J. Comput. Phys., 1984, 54: 115-173

[63.] Zhu J, Qiu J. A new fifth order finite difference WENO scheme for solving hyperbolic conservation laws. J. Comput. Phys., 2016, 318: 110-121

[64.] Zhu J, Shu C-W. A new type of multi-resolution WENO schemes with increasingly higher order of accuracy. J. Comput. Phys., 2018, 375: 659-683

Funding

National Science Foundation(DMS-2010107); National Aeronautics and Space Administration(80NSSC22K0628); Air Force Office of Scientific Research(FA9550-20-1-0055); ASCLS Education and Research Fund(Varja)