On differentiable local bounds preserving stabilization for Euler equations

TL;DR

This paper introduces a differentiable local bounds preserving stabilization method for Euler equations, combining shock detection and artificial diffusion, which enhances robustness and convergence in finite element simulations of shocks.

Contribution

It develops a novel differentiable stabilization scheme with a continuation method, improving nonlinear convergence and computational efficiency for Euler equations with complex shocks.

Findings

Improved robustness and convergence for steady Euler problems.

Reduced computational cost for transient simulations.

Sharp, well-resolved shock capturing demonstrated.

Abstract

This work presents the design of nonlinear stabilization techniques for the finite element discretization of Euler equations in both steady and transient form. Implicit time integration is used in the case of the transient form. A differentiable local bounds preserving method has been developed, which combines a Rusanov artificial diffusion operator and a differentiable shock detector. Nonlinear stabilization schemes are usually stiff and highly nonlinear. This issue is mitigated by the differentiability properties of the proposed method. Moreover, in order to further improve the nonlinear convergence, we also propose a continuation method for a subset of the stabilization parameters. The resulting method has been successfully applied to steady and transient problems with complex shock patterns. Numerical experiments show that it is able to provide sharp and well resolved shocks. The importance of the differentiability is assessed by comparing the new scheme with its non-differentiable counterpart. Numerical experiments suggest that, for up to moderate nonlinear tolerances, the method exhibits improved robustness and nonlinear convergence behavior for steady problems. In the case of transient problem, we also observe a reduction in the computational cost.