Reformulations for general advection-diffusion-reaction equations and locally implicit ADER schemes

Following Cattaneo's original idea, in this article we first present two relaxation formulations for time-dependent, non-linear systems of advection-diffusion-reaction equations. Such formulations yield time-dependent non-linear hyperbolic balance laws with stiff source terms. Then we present a locally implicit version of the ADER method to solve these stiff systems to high accuracy. The new ingredient of the numerical methodology is a locally implicit solution of the generalised Riemann problem. We illustrate the formulations and the resulting numerical approach by solving the compressible Navier-Stokes equations.

[1]  D. Longsine,et al.  Simultaneous rayleigh-quotient minimization methods for Ax=λBx , 1980 .

[2]  Martin Käser,et al.  Adaptive Methods for the Numerical Simulation of Transport Processes , 2003 .

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

[4]  E. Lemmon,et al.  Viscosity and Thermal Conductivity Equations for Nitrogen, Oxygen, Argon, and Air , 2004 .

[5]  R. Nickalls.  A new approach to solving the cubic: Cardan’s solution revealed , 1993, The Mathematical Gazette.

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

[7]  Gabriel Nagy,et al.  The behavior of hyperbolic heat equations' solutions near their parabolic limits , 1994 .

[8]  T. O. Lewis,et al.  Quadratic Matrix Equations , 1974 .

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

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

[11]  Chun-Hua Guo,et al.  On a quadratic matrix equation associated with an M‐matrix , 2003 .

[12]  S. Osher,et al.  Uniformly high order accurate essentially non-oscillatory schemes, 111 , 1987 .

[13]  Jimmie D. Lawson,et al.  A Birkhoff contraction formula with applications to Riccati Equations , 2007, 2007 46th IEEE Conference on Decision and Control.

[14]  Fermín Navarrina,et al.  A finite element formulation for a convection-diffusion equation based on Cattaneo's law , 2007 .

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

[16]  Michael Dumbser,et al.  Arbitrary high order PNPM schemes on unstructured meshes for the compressible Navier–Stokes equations , 2010 .

[17]  Hiroaki Nishikawa,et al.  A first-order system approach for diffusion equation. I: Second-order residual-distribution schemes , 2007, J. Comput. Phys..

[18]  Eleuterio F. Toro,et al.  ARBITRARILY ACCURATE NON-OSCILLATORY SCHEMES FOR A NONLINEAR CONSERVATION LAW , 2002 .

[19]  Eleuterio F. Toro,et al.  ADER finite volume schemes for nonlinear reaction--diffusion equations , 2009 .

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

[21]  Fermín Navarrina,et al.  Un nuevo enfoque para el tratamiento de los términos difusivos de la ecuación de convección-difusión en el método de Garlekin discontinuo , 2007 .

[22]  Shi Jin.  A Convex Entropy for a Hyperbolic System with Relaxation , 1996 .

[23]  Hiroaki Nishikawa.  New-Generation Hyperbolic Navier-Stokes Schemes: O(1=h) Speed-Up and Accurate Viscous/Heat Fluxes , 2011 .

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

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

[26]  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.

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

[28]  E. Toro,et al.  Numerical and analytical study of an atherosclerosis inflammatory disease model , 2014, Journal of mathematical biology.

[29]  Michael Dumbser,et al.  High-Order Unstructured One-Step PNPM Schemes for the Viscous and Resistive MHD Equations , 2009 .

[30]  Hiroaki Nishikawa,et al.  A first-order system approach for diffusion equation. II: Unification of advection and diffusion , 2010, J. Comput. Phys..

[31]  Armin Iske,et al.  ADER schemes on adaptive triangular meshes for scalar conservation laws , 2005 .

[32]  James R. Scott.  Solving ODE Initial Value Problems With Implicit Taylor Series Methods , 2000 .

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

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

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

[36]  Michael Dumbser,et al.  Building Blocks for Arbitrary High Order Discontinuous Galerkin Schemes , 2006, J. Sci. Comput..

[37]  Tai-Ping Liu.  Hyperbolic conservation laws with relaxation , 1987 .

[38]  Eleuterio F. Toro,et al.  Solver for the Generalized Riemann Problem for Balance Laws with Stiff Source Terms: The Scalar Case , 2012 .

[39]  E. Toro,et al.  An arbitrary high-order Discontinuous Galerkin method for elastic waves on unstructured meshes - V. Local time stepping and p-adaptivity , 2007 .

[40]  Michael Dumbser,et al.  Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems , 2007, J. Comput. Phys..

[41]  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..

[42]  Claus-Dieter Munz,et al.  ADER: A High-Order Approach for Linear Hyperbolic Systems in 2D , 2002, J. Sci. Comput..

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

[44]  Lucas O. Müller,et al.  Well‐balanced high‐order solver for blood flow in networks of vessels with variable properties , 2013, International journal for numerical methods in biomedical engineering.

[45]  Alan J. Laub,et al.  On the Iterative Solution of a Class of Nonsymmetric Algebraic Riccati Equations , 2000, SIAM J. Matrix Anal. Appl..

[46]  Michael Dumbser,et al.  ADER Schemes for Nonlinear Systems of Stiff Advection–Diffusion–Reaction Equations , 2011, J. Sci. Comput..

[47]  Eleuterio F. Toro,et al.  ADER schemes for scalar non-linear hyperbolic conservation laws with source terms in three-space dimensions , 2005 .

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

[49]  P. Roe.  CHARACTERISTIC-BASED SCHEMES FOR THE EULER EQUATIONS , 1986 .

[50]  Michael Dumbser,et al.  ADER discontinuous Galerkin schemes for aeroacoustics , 2005 .