A universal centred high-order method based on implicit Taylor series expansion with fast second order evolution of spatial derivatives

In this paper, a centred universal high-order finite volume method for solving hyperbolic balance laws is presented. The scheme belongs to the family of ADER methods where the Generalized Riemann Problems (GRP) is a building block. The solution to these problems is carried through an implicit Taylor series expansion, which allows the scheme to works very well for stiff source terms. A von Neumann stability analysis is carried out to investigate the range of CFL values for which stability and accuracy are balanced. The scheme implements a centred, low dissipation approach for dealing with the advective part of the system which profits from small CFL values. Numerical tests demonstrate that the present scheme can solve, efficiently, hyperbolic balance laws in both conservative and non-conservative form as well. An empirical convergence rate assessment shows that the expected theoretical orders of accuracy are achieved up to the fifth order.

[1]  Dinshaw S. Balsara,et al.  An efficient, second order accurate, universal generalized Riemann problem solver based on the HLLI Riemann solver , 2018, J. Comput. Phys..

[2]  Z. Xin,et al.  The relaxation schemes for systems of conservation laws in arbitrary space dimensions , 1995 .

[3]  On a General Definition of the Godunov Method for Nonconservative Hyperbolic Systems. Application to Linear Balance Laws , 2006 .

[4]  E. Toro,et al.  Solution of the generalized Riemann problem for advection–reaction equations , 2002, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[5]  Eleuterio F. Toro,et al.  ADER: Arbitrary-Order Non-Oscillatory Advection Schemes , 2001 .

[6]  Eleuterio F. Toro,et al.  Derivative Riemann solvers for systems of conservation laws and ADER methods , 2006, J. Comput. Phys..

[7]  Eleuterio F. Toro,et al.  Low-dissipation centred schemes for hyperbolic equations in conservative and non-conservative form , 2020, J. Comput. Phys..

[8]  F. Bouchut.  Construction of BGK Models with a Family of Kinetic Entropies for a Given System of Conservation Laws , 1999 .

[9]  M. Dumbser,et al.  An arbitrary high-order Discontinuous Galerkin method for elastic waves on unstructured meshes — III. Viscoelastic attenuation , 2007 .

[10]  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..

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

[12]  S. Kawashima.  Asymptotic stability of Maxwellians of the discrete Boltzmann equation , 1987 .

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

[14]  Lucas O. Müller,et al.  Hyperbolic reformulation of a 1D viscoelastic blood flow model and ADER finite volume schemes , 2014, J. Comput. Phys..

[15]  Michael Dumbser,et al.  Well-Balanced High-Order Centred Schemes for Non-Conservative Hyperbolic Systems. Applications to Shallow Water Equations with Fixed and Mobile Bed , 2009 .

[16]  Michael Dumbser,et al.  Comparison of solvers for the generalized Riemann problem for hyperbolic systems with source terms , 2012, J. Comput. Phys..

[17]  Dinshaw S. Balsara,et al.  A simplified Cauchy-Kowalewskaya procedure for the implicit solution of generalized Riemann problems of hyperbolic balance laws , 2019, Computers & Fluids.

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

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

[20]  E. Toro,et al.  The Derivative Riemann Problem for the Baer–Nunziato Equations , 2008 .

[21]  Andreas Öchsner,et al.  Maxima—A Computer Algebra System , 2019 .

[22]  Carlos Parés Madroñal,et al.  Numerical methods for nonconservative hyperbolic systems: a theoretical framework , 2006, SIAM J. Numer. Anal..

[23]  Michael Dumbser,et al.  Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws , 2008, J. Comput. Phys..

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

[25]  E. F. Toro,et al.  The Riemann Problem: Solvers and Numerical Fluxes , 2016 .

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

[27]  S. Osher,et al.  Uniformly high order accuracy essentially non-oscillatory schemes III , 1987 .

[28]  Tai-Ping Liu.  Nonlinear resonance for quasilinear hyperbolic equation , 1987 .

[29]  Michael Dumbser,et al.  ADER-WENO finite volume schemes with space-time adaptive mesh refinement , 2012, J. Comput. Phys..

[30]  S. Kružkov.  FIRST ORDER QUASILINEAR EQUATIONS IN SEVERAL INDEPENDENT VARIABLES , 1970 .

[31]  A. Bressan.  Hyperbolic Systems of Conservation Laws , 1999 .

[32]  Eleuterio F. Toro,et al.  Implicit, semi-analytical solution of the generalized Riemann problem for stiff hyperbolic balance laws , 2015, J. Comput. Phys..

[33]  E. Toro.  Riemann Solvers and Numerical Methods for Fluid Dynamics , 1997 .

[34]  Randall J. LeVeque,et al.  A study of numerical methods for hyperbolic conservation laws with stiff source terms , 1990 .

[35]  Tai-Ping Liu.  Admissible solutions of hyperbolic conservation laws , 1981 .