A variable high-order shock-capturing finite difference method with GP-WENO

TL;DR

This paper introduces a novel high-order shock-capturing finite difference scheme using Gaussian Process interpolation and a WENO-like nonlinear shock handling strategy, improving accuracy and non-oscillatory behavior for hyperbolic equations.

Contribution

It extends GP high-order methods to finite difference schemes with a new nonlinear smoothness indicator based on Gaussian likelihood, enhancing shock capturing and accuracy.

Findings

Achieves high-order accuracy in smooth regions.

Effectively captures shocks without oscillations.

Improves upon traditional WENO smoothness indicators.

Abstract

We present a new finite difference shock-capturing scheme for hyperbolic equations on static uniform grids. The method provides selectable high-order accuracy by employing a kernel-based Gaussian Process (GP) data prediction method which is an extension of the GP high-order method originally introduced in a finite volume framework by the same authors. The method interpolates Riemann states to high order, replacing the conventional polynomial interpolations with polynomial-free GP-based interpolations. For shocks and discontinuities, this GP interpolation scheme uses a nonlinear shock handling strategy similar to Weighted Essentially Non-oscillatory (WENO), with a novelty consisting in the fact that nonlinear smoothness indicators are formulated in terms of the Gaussian likelihood of the local stencil data, replacing the conventional $L_2$-type smoothness indicators of the original WENO method. We demonstrate that these GP-based smoothness indicators play a key role in the new algorithm, providing significant improvements in delivering high -- and selectable -- order accuracy in smooth flows, while successfully delivering non-oscillatory solution behavior in discontinuous flows.