ADER finite volume schemes for nonlinear reaction--diffusion equations

Authors:
Eleuterio F. Toro;Arturo Hidalgo
Affiliations:
Laboratory of Applied Mathematics, Department of Civil and Environmental Engineering. University of Trento, Trento, Italy;Departamento de Matemática Aplicada y Métodos Informáticos, ETS Ingenieros de Minas, Universidad Politécnica de Madrid, Madrid, Spain
Venue:
Applied Numerical Mathematics
Year:
2009

Citing 12
Cited 2

Uniformly high order accurate essentially non-oscillatory schemes, 111

Journal of Computational Physics
Adaptive finite element methods for parabolic problems IV: nonlinear problems

SIAM Journal on Numerical Analysis
A Nonlinear Mixed Finite Eelement Method for a Degenerate Parabolic Equation Arising in Flow in Porous Media

SIAM Journal on Numerical Analysis
On WAF-type schemes for multidimensional hyperbolic conservation laws

Journal of Computational Physics
Computational Differential Equations

Computational Differential Equations
ADER: Arbitrary High Order Godunov Approach

Journal of Scientific Computing
Axisymmetric vortex method for low-mach number, diffusion-controlled combustion

Journal of Computational Physics
A Finite Volume Scheme for Two-Phase Immiscible Flow in Porous Media

SIAM Journal on Numerical Analysis
Local discontinuous Galerkin methods for nonlinear dispersive equations

Journal of Computational Physics
Derivative Riemann solvers for systems of conservation laws and ADER methods

Journal of Computational Physics
A contribution to the construction of diffusion fluxes for finite volume and discontinuous Galerkin schemes

Journal of Computational Physics
Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws

Journal of Computational Physics

ADER Schemes for Nonlinear Systems of Stiff Advection---Diffusion---Reaction Equations

Journal of Scientific Computing
Adaptive ADER Methods Using Kernel-Based Polyharmonic Spline WENO Reconstruction

SIAM Journal on Scientific Computing

Quantified Score

Hi-index	0.00

Visualization

Abstract

We construct finite volume schemes of arbitrary order of accuracy in space and time for solving nonlinear reaction-diffusion partial differential equations. The numerical schemes, written in conservative form, result from extending the Godunov and the ADER frameworks, both originally developed for approximating solutions to hyperbolic equations. The task is to define numerical fluxes and numerical sources. In the ADER approach, numerical fluxes are computed from solutions to the Derivative Riemann Problem (DRP) (or generalized Riemann problem, or high-order Riemann problem), the Cauchy problem in which the initial conditions either side of the interface are smooth functions, polynomials of arbitrary degree, for example. We propose, and systematically asses, a general DRP solver for nonlinear reaction-diffusion equations and construct corresponding finite volume schemes of arbitrary order of accuracy. Schemes of 1st to 10-th order of accuracy in space and time are implemented and systematically assessed, with particular attention paid to their convergence rates. Numerical examples are also given.