Explicit Finite Volume Schemes of Arbitrary High Order of Accuracy for Hyperbolic Systems with Stiff Source Terms

In this article we propose a new class of finite volume schemes of arbitrary accuracy in space and time for systems of hyperbolic balance laws with stiff source terms. The new class of schemes is based on a three stage procedure. First, in order to achieve high order accuracy in space, a nonlinear weighted essentially non-oscillatory reconstruction procedure is applied to the cell averages at the current time level. Second, the temporal evolution of the resulting reconstruction polynomials is computed locally inside each cell exploiting directly the full system of governing equations. In previous ADER schemes, this was achieved via the Cauchy-Kovalewski procedure, where the governing equation is repeatedly differentiated with respect to space and time to construct a Taylor series expansion of the local solution. As the CauchyKovalewski procedure is based on Taylor series expansions, it is not able to handle systems with stiff source terms since the Taylor series diverges for this case. Therefore, in this article, we present a new strategy that replaces the Cauchy-Kovalewski procedure for high order time interpolation: we present a special local space-time discontinuous Galerkin (DG) finite element scheme that is able to handle arbitrarily stiff source terms in a stable manner. The solution of this space-time DG method can be proven to have several important robustness properties in the presence of stiff source terms. This step is the only part of the entire algorithm which is locally implicit. The third and last step of the proposed ADER finite volume schemes consists of the standard explicit space-time integration over each control volume, using the local space-time DG solutions at the Gaussian integration points for the intercell fluxes and for the space-time integral over the source term. We will show numerical convergence studies for nonlinear systems in one space dimension with both non-stiff and with very stiff source terms up to sixth order of accuracy in space and time. The application of the new method to a large set of different test cases is shown, in particular the stiff scalar model problem of LeVeque and Yee [34], the Preprint submitted to Elsevier Science 31 January 2007 relaxation system of Jin and Xin [30] and the full compressible Euler equations with stiff friction source terms.