An order-adaptive compact approximation Taylor method for systems of conservation laws

TL;DR

This paper introduces the Adaptive Compact Approximation Taylor (ACAT) schemes, high-order shock-capturing methods that adaptively switch between first order and high order based on local smoothness, improving accuracy in smooth regions and robustness near discontinuities.

Contribution

The paper develops a novel family of ACAT schemes that combine adaptive stencils with smoothness indicators, extending CAT methods to nonlinear conservation laws with demonstrated effectiveness.

Findings

High-order accuracy in smooth regions

Effective shock capturing near discontinuities

Successful tests on Euler gas dynamics equations

Abstract

We present a new family of high-order shock-capturing finite difference numerical methods for systems of conservation laws. These methods, called Adaptive Compact Approximation Taylor (ACAT) schemes, use centered $(2p + 1)$-point stencils, where $p$ may take values in $\{1, 2, \dots, P\}$ according to a new family of smoothness indicators in the stencils. The methods are based on a combination of a robust first order scheme and the Compact Approximate Taylor (CAT) methods of order $2p$-order, $p=1,2,\dots, P$ so that they are first order accurate near discontinuities and have order $2p$ in smooth regions, where $(2p +1)$ is the size of the biggest stencil in which large gradients are not detected. CAT methods, introduced in \cite{CP2019}, are an extension to nonlinear problems of the Lax-Wendroff methods in which the Cauchy-Kovalesky (CK) procedure is circumvented following the strategy introduced in \cite{ZBM2017} that allows one to compute time derivatives in a recursive way using high-order centered differentiation formulas combined with Taylor expansions in time. The expression of ACAT methods for 1D and 2D systems of balance laws are given and the performance is tested in a number of test cases for several linear and nonlinear systems of conservation laws, including Euler equations for gas dynamics.