High resolution compact implicit numerical scheme for conservation laws

TL;DR

This paper introduces a new implicit numerical scheme for conservation laws that leverages mixed spatial-temporal derivatives to achieve high resolution and compact algebraic systems, demonstrated through 1D hyperbolic equations.

Contribution

The paper presents a novel implicit scheme that approximates mixed derivatives to improve structure and resolution in conservation law simulations.

Findings

Produces algebraic systems with more convenient structure

Achieves high resolution TVD solutions for 1D hyperbolic equations

Demonstrates effectiveness through numerical experiments

Abstract

We present a novel implicit scheme for the numerical solution of time-dependent conservation laws. The core idea of the presented method is to exploit and approximate the mixed spatial-temporal derivative of the solution that occurs naturally when deriving some second order accurate schemes in time. Such an approach is introduced in the context of the Lax-Wendroff (or Cauchy-Kowalevski) procedure when the second time derivative is not completely replaced by space derivatives using the PDE, but the mixed derivative is kept. If approximated in a suitable way, the resulting compact implicit scheme produces algebraic systems that have a more convenient structure than the systems derived by fully implicit schemes. We derive a high resolution TVD form of the implicit scheme for some representative hyperbolic equations in the one-dimensional case, including illustrative numerical experiments.