Local principle satisfying high order total variation diminishing approximation for non-sonic data extrema

TL;DR

This paper develops higher-order TVD schemes that maintain high accuracy at non-sonic extrema without oscillations, using a local maximum principle framework and hybrid shock detection.

Contribution

It introduces a novel approach to construct higher than second order TVD schemes that preserve accuracy at non-sonic extrema without oscillations, based on non-conservative reformulation and LMP analysis.

Findings

Hybrid schemes achieve TVD with second or higher order convergence.

Schemes preserve high accuracy at non-sonic extrema.

Numerical results confirm stability and accuracy improvements.

Abstract

The main contribution of this work is to construct higher than second order accurate total variation diminishing (TVD) schemes which can preserve high accuracy at non-sonic extrema with out induced local oscillations. It is done in the framework of local maximum principle (LMP) and non-conservative formulation. The representative uniformly second order accurate schemes are converted in to their non-conservative form using the ratio of consecutive gradient. These resulting schemes are analyzed for their non-linear LMP/TVD stability bounds using the local maximum principle. Based on the bounds, second order accurate hybrid numerical schemes are constructed using a shock detector. Numerical results are presented to show that such hybrid schemes yield TVD approximation with second or higher order convergence rate for smooth solution with extrema.