An efficient, second order accurate, universal generalized Riemann problem solver based on the HLLI Riemann solver

Abstract The Riemann problem, and the associated generalized Riemann problem, are increasingly seen as the important building blocks for modern higher order Godunov-type schemes. In the past, building a generalized Riemann problem solver was seen as an intricately mathematical task because the associated Riemann problem is different for each hyperbolic system of interest. This paper changes that situation. The HLLI Riemann solver is a recently-proposed Riemann solver that is universal in that it is applicable to any hyperbolic system, whether in conservation form or with non-conservative products. The HLLI Riemann solver is also complete in the sense that if it is given a complete set of eigenvectors, it represents all waves with minimal dissipation. It is, therefore, very attractive to build a generalized Riemann problem solver version of the HLLI Riemann solver. This is the task that is accomplished in the present paper. We show that at second order, the generalized Riemann problem version of the HLLI Riemann solver is easy to design. Our GRP solver is also complete and universal because it inherits those good properties from original HLLI Riemann solver. We also show how our GRP solver can be adapted to the solution of hyperbolic systems with stiff source terms. Our generalized HLLI Riemann solver is easy to implement and performs robustly and well over a range of test problems. All implementation-related details are presented. Results from several stringent test problems are shown. These test problems are drawn from many different hyperbolic systems, and include hyperbolic systems in conservation form; with non-conservative products; and with stiff source terms. The present generalized Riemann problem solver performs well on all of them.

[1]  Michael Dumbser,et al.  A Simple Extension of the Osher Riemann Solver to Non-conservative Hyperbolic Systems , 2011, J. Sci. Comput..

[2]  Eleuterio F. Toro,et al.  Solvers for the high-order Riemann problem for hyperbolic balance laws , 2008, J. Comput. Phys..

[3]  V. P. Kolgan,et al.  Application of the principle of minimizing the derivative to the construction of finite-difference schemes for computing discontinuous solutions of gas dynamics , 2011, J. Comput. Phys..

[4]  Jiequan Li,et al.  The generalized Riemann problems for compressible fluid flows: Towards high order , 2014, J. Comput. Phys..

[5]  M. J. Castro,et al.  FORCE schemes on unstructured meshes II: Non-conservative hyperbolic systems , 2010 .

[6]  B. V. Leer,et al.  Towards the ultimate conservative difference scheme V. A second-order sequel to Godunov's method , 1979 .

[7]  Michael Dumbser,et al.  High‐order ADER‐WENO ALE schemes on unstructured triangular meshes—application of several node solvers to hydrodynamics and magnetohydrodynamics , 2013, 1310.7256.

[8]  C. Parés.  Numerical methods for nonconservative hyperbolic systems: a theoretical framework. , 2006 .

[9]  李幼升,et al.  Ph , 1989 .

[10]  Michael Dumbser,et al.  Efficient implementation of ADER schemes for Euler and magnetohydrodynamical flows on structured meshes - Speed comparisons with Runge-Kutta methods , 2013, J. Comput. Phys..

[11]  Michael Dumbser,et al.  On Universal Osher-Type Schemes for General Nonlinear Hyperbolic Conservation Laws , 2011 .

[12]  M. Dumbser,et al.  Arbitrary high order discontinuous Galerkin schemes , 2005 .

[13]  Michael Dumbser,et al.  A Novel Solver for the Generalized Riemann Problem Based on a Simplified LeFloch–Raviart Expansion and a Local Space–Time Discontinuous Galerkin Formulation , 2016, J. Sci. Comput..

[14]  P. Raviart,et al.  An asymptotic expansion for the solution of the generalized Riemann problem. Part 2 : application to the equations of gas dynamics , 1989 .

[15]  Kailiang Wu,et al.  A third-order accurate direct Eulerian GRP scheme for the Euler equations in gas dynamics , 2014, J. Comput. Phys..

[16]  Paul R. Woodward,et al.  Extension of the Piecewise Parabolic Method to Multidimensional Ideal Magnetohydrodynamics , 1994 .

[17]  S. Osher,et al.  Efficient implementation of essentially non-oscillatory shock-capturing schemes,II , 1989 .

[18]  Jiequan Li,et al.  An adaptive GRP scheme for compressible fluid flows , 2010, J. Comput. Phys..

[19]  Alfio Borzì,et al.  Multigrid Optimization Schemes for Solving Bose-Einstein Condensate Control Problems , 2007, SIAM J. Sci. Comput..

[20]  P. Woodward,et al.  The Piecewise Parabolic Method (PPM) for Gas Dynamical Simulations , 1984 .

[21]  Tsuyoshi Murata,et al.  {m , 1934, ACML.

[22]  Jiequan Li,et al.  The generalized Riemann problem method for the shallow water equations with bottom topography , 2006 .

[23]  Matania Ben-Artzi,et al.  The generalized Riemann problem for reactive flows , 1989 .

[24]  Gerald Warnecke,et al.  A direct Eulerian GRP scheme for compressible fluid flows , 2006, J. Comput. Phys..

[25]  Sigal Gottlieb David C. Seal Zachary Grant.  A Strong Stability Preserving Analysis for Explicit Multistage Two-Derivative Time-Stepping Schemes Based on Taylor Series Conditions , 2019 .

[26]  Alexander Kurganov,et al.  Well-Balanced Positivity Preserving Central-Upwind Scheme for the Shallow Water System with Friction Terms , 2013 .

[27]  G. D. Maso,et al.  Definition and weak stability of nonconservative products , 1995 .

[28]  Dinshaw S. Balsara,et al.  Self-adjusting, positivity preserving high order schemes for hydrodynamics and magnetohydrodynamics , 2012, J. Comput. Phys..

[29]  P. Raviart,et al.  An asymptotic expansion for the solution of the generalized Riemann problem Part I: General theory , 1988 .

[30]  Manuel Jesús Castro Díaz,et al.  A Class of Computationally Fast First Order Finite Volume Solvers: PVM Methods , 2012, SIAM J. Sci. Comput..

[31]  Eleuterio F. Toro,et al.  ADER schemes for three-dimensional non-linear hyperbolic systems , 2005 .

[32]  Kailiang Wu,et al.  A Direct Eulerian GRP Scheme for Spherically Symmetric General Relativistic Hydrodynamics , 2016, SIAM J. Sci. Comput..

[33]  Joseph Falcovitz,et al.  Generalized Riemann Problems in Computational Fluid Dynamics , 2003 .

[34]  M. J. Castro,et al.  ADER schemes on unstructured meshes for nonconservative hyperbolic systems: Applications to geophysical flows , 2009 .

[35]  M. Brio,et al.  An upwind differencing scheme for the equations of ideal magnetohydrodynamics , 1988 .

[36]  Michael Dumbser,et al.  Multidimensional Riemann problem with self-similar internal structure. Part II - Application to hyperbolic conservation laws on unstructured meshes , 2015, J. Comput. Phys..

[37]  Zhongfeng Sun,et al.  Remark on the generalized Riemann problem method for compressible fluid flows , 2007, J. Comput. Phys..

[38]  Michael Dumbser,et al.  A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes , 2008, J. Comput. Phys..

[39]  Zachary Grant,et al.  Explicit Strong Stability Preserving Multistage Two-Derivative Time-Stepping Schemes , 2015, J. Sci. Comput..

[40]  Dinshaw S. Balsara,et al.  Notes on the Eigensystem of Magnetohydrodynamics , 1996, SIAM J. Appl. Math..

[41]  Michael Dumbser,et al.  Efficient, high accuracy ADER-WENO schemes for hydrodynamics and divergence-free magnetohydrodynamics , 2008, Journal of Computational Physics.

[42]  Michael Dumbser,et al.  A new efficient formulation of the HLLEM Riemann solver for general conservative and non-conservative hyperbolic systems , 2016, J. Comput. Phys..

[43]  Manuel Jesús Castro Díaz,et al.  High order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. Applications to shallow-water systems , 2006, Math. Comput..

[44]  Carlos Parés,et al.  A HLLC scheme for nonconservative hyperbolic problems. Application to turbidity currents with sediment transport , 2013 .

[45]  Eleuterio F. Toro,et al.  ADER: Arbitrary High Order Godunov Approach , 2002, J. Sci. Comput..

[46]  Matania Ben-Artzi,et al.  Computation of reactive duct flows in external fields , 1990 .

[47]  Yue Wang,et al.  Thermodynamical effects and high resolution methods for compressible fluid flows , 2017, J. Comput. Phys..

[48]  Dinshaw S. Balsara,et al.  Multidimensional Riemann problem with self-similar internal structure. Part I - Application to hyperbolic conservation laws on structured meshes , 2014, J. Comput. Phys..

[49]  Dinshaw S. Balsara,et al.  Computational electrodynamics in material media with constraint-preservation, multidimensional Riemann solvers and sub-cell resolution - Part I, second-order FVTD schemes , 2017, J. Comput. Phys..

[50]  Michael Dumbser,et al.  A family of HLL-type solvers for the generalized Riemann problem , 2017, Computers & Fluids.

[51]  Dongsu Ryu,et al.  Numerical magnetohydrodynamics in astrophysics: Algorithm and tests for multidimensional flow , 1995 .

[52]  Rémi Abgrall,et al.  A comment on the computation of non-conservative products , 2010, J. Comput. Phys..

[53]  Claus R. Goetz,et al.  Approximate solutions of generalized Riemann problems for nonlinear systems of hyperbolic conservation laws , 2015, Math. Comput..

[54]  Michael Dumbser,et al.  Lagrangian ADER-WENO finite volume schemes on unstructured triangular meshes based on genuinely multidimensional HLL Riemann solvers , 2013, J. Comput. Phys..

[55]  Eleuterio F. Toro,et al.  Towards Very High Order Godunov Schemes , 2001 .

[56]  J. Falcovitz,et al.  A second-order Godunov-type scheme for compressible fluid dynamics , 1984 .

[57]  Huazhong Tang,et al.  A direct Eulerian GRP scheme for relativistic hydrodynamics: Two-dimensional case , 2012, J. Comput. Phys..

[58]  Manuel Jesús Castro Díaz,et al.  Why many theories of shock waves are necessary: Convergence error in formally path-consistent schemes , 2008, J. Comput. Phys..

[59]  Eleuterio F. Toro,et al.  Reformulations for general advection-diffusion-reaction equations and locally implicit ADER schemes , 2014, J. Comput. Phys..

[60]  I. Bohachevsky,et al.  Finite difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics , 1959 .

[61]  Luz Mu.  GODUNOV METHOD FOR NONCONSERVATIVE HYPERBOLIC SYSTEMS , 2007 .

[62]  Bernd Einfeld.  On Godunov-type methods for gas dynamics , 1988 .

[63]  C. Cattaneo,et al.  Sulla Conduzione Del Calore , 2011 .

[64]  Jiequan Li,et al.  Hyperbolic balance laws: Riemann invariants and the generalized Riemann problem , 2007, Numerische Mathematik.

[65]  J. Falcovitz,et al.  An upwind second-order scheme for compressible duct flows , 1986 .

[66]  P. Roe,et al.  On Godunov-type methods near low densities , 1991 .

[67]  Dinshaw S. Balsara,et al.  Multidimensional Riemann problem with self-similar internal structure - part III - a multidimensional analogue of the HLLI Riemann solver for conservative hyperbolic systems , 2017, J. Comput. Phys..