Communications on Applied Mathematics and Computation >

2025 , Vol. 7 >Issue 6: 2289 - 2338

DOI: https://doi.org/10.1007/s42967-024-00374-1

ORIGINAL PAPERS

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

  • Dinshaw S. Balsara ,
  • Deepak Bhoriya ,
  • Chi-Wang Shu ,
  • Harish Kumar
Expand
  • 1. Physics Department, University of Notre Dame, Notre Dame, USA;
    2. ACMS Department, University of Notre Dame, Notre Dame, USA;
    3. Division of Applied Mathematics, Brown University, Providence, USA;
    4. Department of Mathematics, Indian Institute of Technology, Delhi, India

Received date: 2023-07-17

  Revised date: 2023-12-06

  Online published: 2025-12-24

Supported by

The funding has been acknowledged. DSB acknowledges support via the NSF Grants NSF-19-04774, NSF-AST-2009776, NASA-2020-1241, and NASA-80NSSC22K0628. DSB and HK acknowledge support from a Vajra award, VJR/2018/00129. CWS acknowledges support via AFOSR Grant FA9550-20-1-0055 and NSF Grant DMS-2010107.

Fold

Abstract

Higher order finite difference Weighted Essentially Non-oscillatory (WENO) schemes for conservation laws represent a technology that has been reasonably consolidated. They are extremely popular because, when applied to multidimensional problems, they offer high order accuracy at a fraction of the cost of finite volume WENO or DG schemes. They come in two flavors. There is the classical finite difference WENO (FD-WENO) method (Shu and Osher in J. Comput. Phys. 83: 32–78, 1989). However, in recent years there is also an alternative finite difference WENO (AFD-WENO) method which has recently been formalized into a very useful general-purpose algorithm for conservation laws (Balsara et al. in Efficient alternative finite difference WENO schemes for hyperbolic conservation laws, submitted to CAMC, 2023). However, the FD-WENO algorithm has only very recently been formulated for hyperbolic systems with non-conservative products (Balsara et al. in Efficient finite difference WENO scheme for hyperbolic systems with non-conservative products, to appear CAMC, 2023). In this paper, we show that there are substantial advantages in obtaining an AFD-WENO algorithm for hyperbolic systems with non-conservative products. Such an algorithm is documented in this paper. We present an AFD-WENO formulation in a fluctuation form that is carefully engineered to retrieve the flux form when that is warranted and nevertheless extends to non-conservative products. The method is flexible because it allows any Riemann solver to be used. The formulation we arrive at is such that when non-conservative products are absent it reverts exactly to the formulation in the second citation above which is in the exact flux conservation form. The ability to transition to a precise conservation form when non-conservative products are absent ensures, via the Lax-Wendroff theorem, that shock locations will be exactly captured by the method. We present two formulations of AFD-WENO that can be used with hyperbolic systems with non-conservative products and stiff source terms with slightly differing computational complexities. The speeds of our new AFD-WENO schemes are compared to the speed of the classical FD-WENO algorithm from the first of the above-cited papers. At all orders, AFD-WENO outperforms FD-WENO. We also show a very desirable result that higher order variants of AFD-WENO schemes do not cost that much more than their lower order variants. This is because the larger number of floating point operations associated with larger stencils is almost very efficiently amortized by the CPU when the AFD-WENO code is designed to be cache friendly. This should have great, and very beneficial, implications for the role of our AFD-WENO schemes in the Peta- and Exascale computing. We apply the method to several stringent test problems drawn from the Baer-Nunziato system, two-layer shallow water equations, and the multicomponent debris flow. The method meets its design accuracy for the smooth flow and can handle stringent problems in one and multiple dimensions. Because of the pointwise nature of its update, AFD-WENO for hyperbolic systems with non-conservative products is also shown to be a very efficient performer on problems with stiff source terms.

Key words： Hyperbolic PDEs; Numerical schemes; Conservation laws; Stiff source terms; Finite difference WENO

Cite this article

Dinshaw S. Balsara , Deepak Bhoriya , Chi-Wang Shu , Harish Kumar . Efficient Alternative Finite Difference WENO Schemes for Hyperbolic Systems with Non-conservative Products[J]. Communications on Applied Mathematics and Computation, 2025 , 7(6) : 2289 -2338 . DOI: 10.1007/s42967-024-00374-1

References

[1] Andrianov, N., Warnecke, G.: The Riemann problem for the Baer-Nunziato two-phase flow model. J. Comput. Phys. 212, 434–464 (2004)
[2] Baer, M.R., Nunziato, J.W.: A two-phase mixture theory for the deflagration-to-detonation transition (DDT) in reactive granular materials. Int. J. Multiph. Flow 12, 861–889 (1986)
[3] Balsara, D.S.: Total variation diminishing algorithm for adiabatic and isothermal magnetohydrodynamics. Ap. J. Supp. 116, 133–153 (1998)
[4] Balsara, D.S.: Self-adjusting, positivity preserving high order schemes for hydrodynamics and magnetohydrodynamics. J. Comput. Phys. 231, 7504–7517 (2012)
[5] Balsara, D.S., Bhoriya, D., Shu, C.-W., Kumar, H.: Efficient finite difference WENO scheme for hyperbolic systems with non-conservative products. Commun. Appl. Math. Comput. (2023). https://doi.org/10.1007/s42967-023-00275-9
[6] Balsara, D.S., Bhoriya, D., Shu, C.-W., Kumar, H.: Efficient alternative finite difference WENO schemes for hyperbolic conservation laws. (2024). arxiv:2403.01264v1
[7] Balsara, D.S., Garain, S., Florinski, V., Boscheri, W.: An efficient class of WENO schemes with adaptive order for unstructured meshes. J. Comput. Phys. 404, 109062 (2020)
[8] Balsara, D.S., Garain, S., Shu, C.-W.: An efficient class of WENO schemes with adaptive order. J. Comput. Phys. 326, 780–804 (2016)
[9] Balsara, D.S., Rumpf, T., Dumbser, M., Munz, C.-D.: Efficient, high-accuracy ADER-WENO schemes for hydrodynamics and divergence-free magnetohydrodynamics. J. Comput. Phys. 228, 2480 (2009)
[10] Balsara, D.S., Samantaray, S., Subramanian, S.: Efficient WENO-based prolongation strategies for divergence-preserving vector fields. Commun. Appl. Math. Comput. 5(1), 428–484 (2023)
[11] Balsara, D.S., Shu, C.-W.: Monotonicity preserving weighted non-oscillatory schemes with increasingly high order of accuracy. J. Comput. Phys. 160, 405–452 (2000)
[12] Borges, R., Carmona, M., Costa, B., Don, W.S.: An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws. J. Comput. Phys. 227(6), 3101–3211 (2008)
[13] Castro, M., Costa, B., Don, W.S.: High order weighted essentially non-oscillatory WENO-Z schemes for hyperbolic conservation laws. J. Comput. Phys. 230, 1766–1792 (2011)
[14] Castro, M., Gallardo, J.M., Parés, 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. 75(255), 1103–1134 (2006)
[15] Castro, M.J., Pardo, A., Parés, C., Toro, E.F.: On some fast well-balanced first order solvers for nonconservative systems. Math. Comput. 79, 1427–1472 (2010)
[16] Chiochetti, S., Müller, C.: A solver for stiff finite-rate relaxation in Baer-Nunziato two-phase flow models. (2020). arxiv: 2002.09381
[17] Colella, P., Woodward, P.R.: The piecewise parabolic method (PPM) for gas-dynamical simulations. J. Comput. Phys. 54(1), 174–201 (1984)
[18] 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. 431, 110135 (2021)
[19] Dumbser, M., Balsara, D.S.: A new efficient formulation of the HLLEM Riemann solver for general conservative and non-conservative hyperbolic systems. J. Comput. Phys. 304, 275–319 (2016)
[20] 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 86, 405–432 (2013)
[21] Dumbser, M., Castro, M., Parés, C., Toro, E.F.: ADER schemes on unstructured meshes for non-conservative hyperbolic systems: applications to geophysical flows. Comput. Fluids 38, 1731–1748 (2009)
[22] Dumbser, M., Hidalgo, A., Castro, M., Parés, C., Toro, E.F.: FORCE schemes on unstructured meshes II: non-conservative hyperbolic systems. Comput. Methods Appl. Mech. Engrg. 199, 625–647 (2010)
[23] 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. Engrg. 268, 359–387 (2014)
[24] Dumbser, M., Iben, U., Ioriatti, M.: An efficient semi-implicit finite volume method for axially symmetric compressible flows in compliant tubes. Appl. Numer. Math. 89, 24–44 (2015)
[25] Gao, Z., Fang, L.-L., Wang, B.-S., Wang, Y., Don, W.S.: Seventh and ninth orders characteristic-wise alternative WENO finite difference schemes for hyperbolic conservation laws. Comput. Fluids 202, 104519 (2020)
[26] Gerolymos, G.A., Sénéchal, D., Vallet, I.: Very high order WENO schemes. J. Comput. Phys. 228, 8481–8524 (2009)
[27] Harten, A., Engquist, B., Osher, S., Chakravarthy, S.: Uniformly high order essentially non-oscillatory schemes III. J. Comput. Phys. 71, 231–303 (1987)
[28] Henrick, A.K., Aslam, T.D., Powers, J.M.: Mapped weighted essentially non-oscillatoriy schemes: achieving optimal order near critical points. J. Comput. Phys. 207, 542–567 (2006)
[29] Jiang, G.-S., Shu, C.-W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126, 202–228 (1996)
[30] Jiang, Y., Shu, C.-W., Zhang, M.: An alternative formulation of finite difference ENO schemes with Lax-Wendroff time discretization for conservation laws. SIAM J. Sci. Comput. 35(2), A1137–A1160 (2013)
[31] Jiang, Y., Shu, C.-W., Zhang, M.: Free-stream preserving finite-difference schemes on curvilinear meshes. Methods Appl. Anal. 21(1), 001–030 (2014)
[32] 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. 231, 3561–3586 (2012)
[33] Liu, X.-D., Osher, S., Chan, T.: Weighted essentially non-oscillatory schemes. J. Comput. Phys. 115, 200–212 (1994)
[34] Merriman, B.: Understanding the Shu-Osher conservative finite difference form. J. Sci. Comput. 19(1/2/3), 309 (2003)
[35] Pareschi, L., Russo, G.: Implicit-explicit Runge-Kutta schemes and applications to hyperbolic systems with relaxation. J. Sci. Comput. 25, 129 (2005)
[36] Pelanti, M., Bouchut, F., Mangeney, A.: A Roe-type scheme for two-phase shallow granular flows over variable topography. Math. Model. Numer. Anal. 42, 851–885 (2008)
[37] Pitman, E.B., Le, L.: A two-fluid model for avalanche and debris flows. Philos. Trans. R. Soc. A 363, 1573–1601 (2005)
[38] Rhebergen, S., Bokhove, O., van der Vegt, J.J.W.: Discontinuous Galerkin finite element methods for hyperbolic nonconservative partial differential equations. J. Comput. Phys. 227, 1887–1922 (2008)
[39] Saurel, R., Abgrall, R.: A multiphase Godunov method for compressible multifluid and multiphase flows. J. Comput. Phys. 150, 425–467 (1999)
[40] Schwendeman, D.W., Wahle, C.W., Kapila, A.K.: The Riemann problem and a high-resolution Godunov method for a model of compressible two-phase flow. J. Comput. Phys. 212, 490–526 (2006)
[41] Shu, C.-W.: High order weighted essentially non-oscillatory schemes for convection dominated problems. SIAM Rev. 51, 82–126 (2009)
[42] Shu, C.-W.: Essentially non-oscillatory and weighted essentially non-oscillatory schemes. Acta Numer. 29, 701–762 (2020)
[43] Shu, C.-W., Osher, S.J.: Efficient implementation of essentially non-oscillatory shock capturing schemes. J. Comput. Phys. 77, 439–471 (1988)
[44] Shu, C.-W., Osher, S.J.: Efficient implementation of essentially non-oscillatory shock capturing schemes II. J. Comput. Phys. 83, 32–78 (1989)
[45] Spiteri, R.J., Ruuth, S.J.: A new class of optimal high-order strong-stability-preserving time-stepping schemes. SIAM J. Numer. Anal. 40, 469–491 (2002)
[46] Spiteri, R.J., Ruuth, S.J.: Non-linear evolution using optimal fourth-order strong-stability-preserving Runge-Kutta methods. Math. Comput. Simul 62, 125–135 (2003)
[47] Tokareva, S.A., Toro, E.F.: HLLC-type Riemann solver for the Baer-Nunziato equations of compressible two-phase flow. J. Comput. Phys. 229, 3573–3604 (2010)
[48] Woodward, P., Colella, P.: The numerical simulation of two-dimensional fluid flow with strong shocks. J. Comput. Phys. 54, 115–173 (1984)
[49] Zheng, F., Shu, C.-W., Qiu, J.: A high order conservative finite difference scheme for compressible two-medium flows. J. Comput. Phys. 445, 110597 (2021)
[50] Zhu, J., Qiu, J.: A new fifth order finite difference WENO scheme for solving hyperbolic conservation laws. J. Comput. Phys. 318, 110–121 (2016)
[51] Zhu, J., Shu, C.-W.: A new type of multi-resolution WENO schemes with increasingly higher order of accuracy. J. Comput. Phys. 375, 659–683 (2018)
Options
Outlines

/

〈 〉

Copyright © Editorial By Communications on Applied Mathematics and Computation
沪ICP备09014157