Numerical solution of two dimensional scalar conservation laws using compact implicit numerical schemes on Cartesian meshes

TL;DR

This paper introduces a novel compact implicit finite volume scheme for 2D scalar conservation laws, enabling fast algebraic solutions and non-oscillatory results for large time steps.

Contribution

It develops a second-order accurate parametric scheme with ENO/WENO variants and a provably non-oscillatory limiting technique for large time steps.

Findings

Effective second-order accuracy demonstrated in numerical experiments.

Non-oscillatory solutions achieved for large time steps.

Applicable to both linear and nonlinear conservation laws.

Abstract

This paper deals with the numerical solution of conservation laws in the two dimensional case using a novel compact implicit time discretization that enables applications of fast algebraic solvers. We present details for the second order accurate parametric scheme based on the finite volume method including simple variants of ENO (Essentially Non-Oscillatory) and WENO (Weighted Essentially Non-Oscillatory) approximations. To avoid oscillatory numerical solutions for large time steps, we propose limiting in time which is provable non-oscillatory in the case of linear advection equation with variable velocity. We present numerical experiments for representative linear and nonlinear problems.