Monolithic convex limiting and implicit pseudo-time stepping for calculating steady-state solutions of the Euler equations

TL;DR

This paper introduces a monolithic convex limiting approach combined with implicit pseudo-time stepping to reliably compute steady-state solutions of the Euler equations while preserving physical invariants and preventing spurious oscillations.

Contribution

The work develops a novel IDP scheme with an efficient fixed-point solver and adaptive underrelaxation for steady Euler solutions, ensuring invariant domain preservation and improved convergence.

Findings

The proposed method guarantees invariant domain preservation for steady solutions.

The fixed-point iteration enhances computational efficiency and robustness.

Numerical tests demonstrate reliable convergence and physical accuracy.

Abstract

In this work, we use the monolithic convex limiting (MCL) methodology to enforce relevant inequality constraints in implicit finite element discretizations of the compressible Euler equations. In this context, preservation of invariant domains follows from positivity preservation for intermediate states of the density and internal energy. To avoid spurious oscillations, we additionally impose local maximum principles on intermediate states of the density, velocity components, and specific total energy. For the backward Euler time stepping, we show the invariant domain preserving (IDP) property of the fully discrete MCL scheme by constructing a fixed-point iteration that meets the requirements of a Krasnoselskii-type theorem. Our iterative solver for the nonlinear discrete problem employs a more efficient fixed-point iteration. The matrix of the associated linear system is a robust low-order Jacobian approximation that exploits the homogeneity property of the flux function. The limited antidiffusive terms are treated explicitly. We use positivity preservation as a stopping criterion for nonlinear iterations. The first iteration yields the solution of a linearized semi-implicit problem. This solution possesses the discrete conservation property but is generally not IDP. Further iterations are performed if any non-IDP states are detected. The existence of an IDP limit is guaranteed by our analysis. To facilitate convergence to steady-state solutions, we perform adaptive explicit underrelaxation at the end of each time step. The calculation of appropriate relaxation factors is based on an approximate minimization of nodal entropy residuals. The performance of proposed algorithms and alternative solution strategies is illustrated by the convergence history for standard two-dimensional test problems.