New Numerical Interface Scheme for the Kurganov-Tadmor second-order Method

TL;DR

This paper introduces a numerical interface scheme for the Kurganov-Tadmor method that maintains second-order accuracy and shock transmission across interfaces without sharing interior grid information.

Contribution

It develops a novel interface scheme that relaxes the KT method to first order at interfaces, preserving overall second-order accuracy and TVD properties.

Findings

Successfully transmits shocks across interfaces without deformation.

Maintains second-order convergence in 1D tests.

Effectively handles 2D Euler equations with complex shocks.

Abstract

In this paper, we develop a numerical scheme to handle interfaces across computational domains in multi-block schemes for the approximation of systems of conservation laws. We are interested in transmitting shock discontinuities without lowering the overall precision of the method. We want to accomplish this without using information from interior points of adjacent grids, that is, sharing only information from boundary points of those grids. To achieve this, we choose to work with the second-order Kurganov-Tadmor (KT) method at interior points, relaxing it to first order at interfaces. This allows us to keep second-order overall accuracy (in the relevant norm) and at the same time preserve the TVD property of the original scheme. After developing the method we performed several standard one and two-dimensional tests. Among them, we used the one-dimensional advection and Burgers equations to verify the second-order convergence of the method. We also tested the two-dimensional Euler equations with an implosion and a Gresho vortex\cite{liska2003}. In particular, in the two-dimensional implosion test we can see that regardless of the orientation of shocks with respect to the interface, they travel across them without appreciable deformation both in amplitude and front direction.