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Efficient optimization-based invariant-domain-preserving limiters in solving gas dynamics equations

Chen Liu, Dionysis Milesis, Chi-Wang Shu, Xiangxiong Zhang

TL;DR

This work presents a general framework for enforcing invariant-domain-preserving properties in high-order gas-dynamics solvers by formulating optimization-based limiters that postprocess cell averages. Central to the approach is an explicit projection onto the convex invariant-domain set $G^\varepsilon$, and the use of operator-splitting methods—Douglas-Rachford for $L^1$ and Davis-Yin for $L^2$—to solve the resulting convex programs efficiently. The authors derive an explicit cubic-root projection onto $G^\varepsilon$, enable two limiter variants ($L^2$ and $L^1$), and demonstrate their application to high-order discontinuous Galerkin schemes across one- and two-dimensional benchmarks, including Sedov blasts and high Mach-number jets. Comparisons show that the $L^2$ limiter is cheaper and can improve accuracy, while the $L^1$ limiter can reduce limiter activity in certain challenging problems, offering a flexible, general, and scalable strategy for invariant-domain-preserving high-order discretizations of compressible flows.

Abstract

We introduce effective splitting methods for implementing optimization-based limiters to enforce the invariant domain in gas dynamics in high order accurate numerical schemes. The key ingredients include an easy and efficient explicit formulation of the projection onto the invariant domain set, and also proper applications of the classical Douglas-Rachford splitting and its more recent extension Davis-Yin splitting. Such an optimization-based approach can be applied to many numerical schemes to construct high order accurate, globally conservative, and invariant-domain-preserving schemes for compressible flow equations. As a demonstration, we apply it to high order discontinuous Galerkin schemes and test it on demanding benchmarks to validate the robustness and performance of both $\ell^1$-norm minimization limiter and $\ell^2$-norm minimization limiter.

Efficient optimization-based invariant-domain-preserving limiters in solving gas dynamics equations

TL;DR

This work presents a general framework for enforcing invariant-domain-preserving properties in high-order gas-dynamics solvers by formulating optimization-based limiters that postprocess cell averages. Central to the approach is an explicit projection onto the convex invariant-domain set , and the use of operator-splitting methods—Douglas-Rachford for and Davis-Yin for —to solve the resulting convex programs efficiently. The authors derive an explicit cubic-root projection onto , enable two limiter variants ( and ), and demonstrate their application to high-order discontinuous Galerkin schemes across one- and two-dimensional benchmarks, including Sedov blasts and high Mach-number jets. Comparisons show that the limiter is cheaper and can improve accuracy, while the limiter can reduce limiter activity in certain challenging problems, offering a flexible, general, and scalable strategy for invariant-domain-preserving high-order discretizations of compressible flows.

Abstract

We introduce effective splitting methods for implementing optimization-based limiters to enforce the invariant domain in gas dynamics in high order accurate numerical schemes. The key ingredients include an easy and efficient explicit formulation of the projection onto the invariant domain set, and also proper applications of the classical Douglas-Rachford splitting and its more recent extension Davis-Yin splitting. Such an optimization-based approach can be applied to many numerical schemes to construct high order accurate, globally conservative, and invariant-domain-preserving schemes for compressible flow equations. As a demonstration, we apply it to high order discontinuous Galerkin schemes and test it on demanding benchmarks to validate the robustness and performance of both -norm minimization limiter and -norm minimization limiter.
Paper Structure (43 sections, 3 theorems, 94 equations, 5 figures, 2 tables)

This paper contains 43 sections, 3 theorems, 94 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Optimization-based invariant-domain-preserving limiters
  4. Operator splitting methods in convex optimization
  5. Two-operator Douglas-Rachford splitting
  6. Three-operator Davis-Yin splitting
  7. The main results of this paper
  8. Related work
  9. Contributions and organization of the paper
  10. Bound-preserving limiters for scalar conservation laws
  11. The problem setup and existsing results for the scalar case
  12. Davis-Yin splitting for $\ell^2$ minimization
  13. Douglas-Rachford splitting for $\ell^1$ minimization only for scalar variables
  14. Douglas-Rachford splitting combined with Davis-Yin splitting for for $\ell^1$ minimization
  15. An efficient implementation of invariant-domain-preserving limiters by splitting methods
  16. ...and 28 more sections

Key Result

Theorem 1

Let $\mathbfsf{X}^\ast$ denote the minimizer of eq:invariant_domain_limiter2 and let $\overline{\mathbfsf{U}^\mathrm{exact}}\in \mathds{R}^{N\times (2+d)}$ be a matrix storing the cell averages of an exact solution. If the DG solution has the same integrals over the whole domain as the exact solutio

Figures (5)

  • Figure 1: Top left: out-of-bound cell averages of data set $1000$ (generated in time step $1000$). Top right: the postprocessed result from the $\ell^2$ limiter with DYS method. Bottom: the postprocessed results from the $\ell^1$ limiter with direct method ($\mathtt{ClipAndAssuredSum}$) and DRS method. There is no difference between $\ell^2$ limited result and two $\ell^1$ limited results up to $\epsilon$.
  • Figure 2: Top row: DYS method solving $\ell^2$ model. Bottom row: DRS method nested with DYS of solving $\ell^1$ model. From left to right: number of iterations for data at different time steps, number of projections, and the convergence error in processing one particular data set at time step $1000$. Minimizers and fixed points, denoted by superscript star, are computed numerically by using sufficiently many iterations of splitting methods.
  • Figure 3: Top row: DYS method solving $\ell^2$ model. Bottom row: DRS method nested with DYS of solving $\ell^1$ model. From left to right: number of iterations for data at different time steps, total number of projections to admissible set, and the convergence error in processing the data set at time step $1000$. Minimizers and fixed points, denoted by superscript star, are computed numerically by taking sufficiently many iterations.
  • Figure 4: Sedov blast wave. Left: snapshot of density at $T = 1$. Plot of density: $50$ exponentially distributed contour lines of density from $0.001$ to $6$. Right: number of DYS and DRS iterations in processing out-of-bound data of $\ell^2$ and $\ell^1$ model in the second RK stage at time step $74669$, which is the only time step (during the whole time evolution) where the cell averages violate the invariant domain.
  • Figure 5: The Mach 2000 astrophysical jet. Top row: $\ell^2$-norm limiter solved by with DYS. Bottom row: $\ell^1$-norm limiter solved by DRS nested with DYS. Left: snapshot of density at $T=0.001$ with logarithmic color scale. Right: number of projections to admissible set is number of computations of proximal operators to the indicator function $\iota_{\Lambda_2}(\mathbfsf{X})$ in the optimization solvers during each step of the fourth-order RK time marching, which is a fair way to compare computational cost between two optimization-based limiters.

Theorems & Definitions (10)

  • Remark 1
  • Theorem 1
  • proof
  • Remark 2
  • Remark 3
  • Remark 4
  • Lemma 1
  • proof
  • Lemma 2
  • proof