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

Abstract

In this article we extend the high order ADER finite volume schemes introduced for stiff hyperbolic balance laws by Dumbser, Enaux and Toro (J. Comput. Phys. 227:3971–4001, 2008) to nonlinear systems of advection–diffusion–reaction equations with stiff algebraic source terms. We derive a new efficient formulation of the local space-time discontinuous Galerkin predictor using a nodal approach whose interpolation points are tensor-products of Gauss–Legendre quadrature points. Furthermore, we propose a new simple and efficient strategy to compute the initial guess of the locally implicit space-time DG scheme: the Gauss–Legendre points are initialized sequentially in time by a second order accurate MUSCL-type approach for the flux term combined with a Crank–Nicholson method for the stiff source terms. We provide numerical evidence that when starting with this initial guess, the final iterative scheme for the solution of the nonlinear algebraic equations of the local space-time DG predictor method becomes more efficient. We apply our new numerical method to some systems of advection–diffusion–reaction equations with particular emphasis on the asymptotic preserving property for linear model systems and the compressible Navier–Stokes equations with chemical reactions.