Semi-implicit high resolution numerical scheme for conservation laws

Abstract

We present a novel semi-implicit scheme for numerical solutions of time-dependent conservation laws. The core idea of the presented method consists of exploiting and approximating mixed partial derivatives of the solution that occur naturally when deriving higher-order accurate schemes. Such an approach is introduced in the context of the Lax-Wendroff (or Cauchy-Kowalevski) procedure when the time derivatives are not completely replaced by space derivatives using the PDE, but some mixed derivatives are allowed. If approximated in a suitable way, one obtains algebraic systems that have a more convenient structure than the systems derived by standard fully implicit schemes. We derive high-resolution TVD form of the semi-implicit scheme for some representative hyperbolic equations in one-dimensional case including related numerical experiments.