A space-time smooth artificial viscosity method with wavelet noise indicator and shock collision scheme, Part 1: the 1-D case

TL;DR

This paper introduces an advanced high-order numerical scheme for 1D shock wave simulations that effectively handles shock-wall collisions, removes high-frequency noise, and addresses wall heating issues using a space-time smooth artificial viscosity method with wavelet noise detection.

Contribution

The paper extends the C-method with a new collision indicator and wavelet-based noise removal, improving shock simulation accuracy and robustness in 1D gas dynamics.

Findings

Enhanced shock-wall collision handling and bounce-back simulation.

Effective removal of high-frequency numerical noise.

Improved accuracy and stability in strong discontinuity scenarios.

Abstract

In this first part of two papers, we extend the C-method developed in [40] for adding localized, space-time smooth artificial viscosity to nonlinear systems of conservation laws that propagate shock waves, rarefaction waves, and contact discontinuities in one space dimension. For gas dynamics, the C-method couples the Euler equations to a scalar reaction-diffusion equation, whose solution $C$ serves as a space-time smooth artificial viscosity indicator. The purpose of this paper is the development of a high-order numerical algorithm for shock-wall collision and bounce-back. Specifically, we generalize the original C-method by adding a new collision indicator, which naturally activates during shock-wall collision. Additionally, we implement a new high-frequency wavelet-based noise detector together with an efficient and localized noise removal algorithm. To test the methodology, we use a highly simplified WENO-based discretization scheme. We show that our scheme improves the order of accuracy of our WENO algorithm, handles extremely strong discontinuities (ranging up to nine orders of magnitude), allows for shock collision and bounce back, and removes high frequency noise. The causes of the well-known "wall heating" phenomenon are discussed, and we demonstrate that this particular pathology can be effectively treated in the framework of the C-method. This method is generalized to two space dimensions in the second part of this work [41].