First-order greedy invariant-domain preserving approximation for hyperbolic problems: scalar conservation laws, and p-system

TL;DR

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Abstract

The paper focuses on first-order invariant-domain preserving approximations of hyperbolic systems. We propose a new way to estimate the artificial viscosity that has to be added to make explicit, conservative, consistent numerical methods invariant-domain preserving and entropy inequality compliant. Instead of computing an upper bound on the maximum wave speed in Riemann problems, we estimate a minimum wave speed in the said Riemann problems such that the approximation satisfies predefined invariant-domain properties and predefined entropy inequalities. This technique eliminates non-essential fast waves from the construction of the artificial viscosity, while preserving pre-assigned invariant-domain properties and entropy inequalities.

Figures (6)

• Figure 1: Riemann problem and Riemann fan.
• Figure 2: Approximation of a scalar conservation equation with piecewise linear flux: (a) viscosity solely based on $\lambda_{12}$ given in \ref{['def_lambda_12']}; (b) viscosity based on an entropy inequality, Eq. \ref{['Lambda_Scalar_Entrop_Ineq']} with the choice $k_i=\frac{1}{2} ({\mathsf U}_i^{\min,n}+{\mathsf U}_i^{\max,n}).$
• Figure 3: 1D two-sonic point problem computed with the square entropy $\eta(v)=\frac{1}{2}v^2$. The "entropy stable" method does not converge to the entropy solution.
• Figure 4: 2D KPP problem with ${\mathbb P}_1$ elements on nonuniform Delaunay mesh (118850 grid points) at $t=1$, $\text{CFL}=0.5$, computed with three different strategies: (a) $\lambda_{\max}= \lambda_{12}$ using \ref{['def_lambda_12']}; (b) wave speed $\lambda^{\text{grdy}}$ computed with \ref{['Lambda_Scalar_Entrop_Ineq']} using $k_i= \theta {\mathsf U}_i^{\min,n}+(1-\theta){\mathsf U}_i^{\max,n}$ with $\theta=\frac{1}{2}$; (c) wave speed $\lambda^{\text{grdy}}$ computed with \ref{['Lambda_Scalar_Entrop_Ineq']} using $k_i= \theta_i {\mathsf U}_i^{\min,n}+(1-\theta_i){\mathsf U}_i^{\max,n}$ where $\theta_i\in(0,1)$ is a uniformly random number changing for every $i\in{\mathcal{V}}$. Only the solution in the right panel is the correct entropy solution.
• Figure 5: Approximation of the $u$ component in the $p$ system, $t=0.5$. Left: comparisons between the methods using ${\widehat{\lambda}}_{\max}$, $\lambda_{\max}$, and $\lambda^{\textup{gdy}}$ with $101$ grid points. Right: Three refinements: $101$ grid points (top), $401$ grid points (middle), $1600$ grid points (bottom).

Theorems & Definitions (30)

• Definition 2.1: Invariant domain
• Lemma 2.2: Invariance of the auxiliary states
• Proof 1
• Remark 2.3: Literature
• Theorem 2.4: Local invariance
• Proof 2
• Remark 2.5: $\lambda_\epsilon$ and $\lambda_{\max}^{\mathcal{V}}$
• Remark 2.6: Key observation
• Remark 2.7: Literature
• Definition 3.1: Quasiconcavity