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Coupled general Riemann problems for the Euler equations

Zhifang Du, Aleksey Sikstel

TL;DR

The paper develops a coupled generalized Riemann problem (GRP) framework for hyperbolic conservation laws, enabling a second-order-in-time, non-synchronous interface data approximation via coupled half-GRPs. The authors instantiate the framework for the one-dimensional Euler equations coupled by a gas generator, derive solvability conditions for the resulting coupled GRP, and provide a numerical validation using a high-order discontinuous Galerkin scheme. Key contributions include the formalization of coupled GRP/half-GRP methods, a solvability analysis for the gas-generator problem, and numerical evidence of correct convergence rates across multiple test cases. The approach facilitates decoupled, parallelizable solvers for coupled systems and offers a pathway to higher-order coupling via extended GRP formulations, with applicability to gas networks and similar interfaces in complex flows.

Abstract

We introduce a novel method for systems of conservation laws coupled at a sharp interface based on generalized Riemann problems. This method yields a piecewise-linear in time approximation of the solution at the interface, thus, descynchronising the solvers for the coupled systems. We apply this framework to a problem of compressible Euler equations coupled via a gas generator and prove its solvability. Finally, we conduct numerical experiments and show that our algorithm performs at correct convergence rates.

Coupled general Riemann problems for the Euler equations

TL;DR

The paper develops a coupled generalized Riemann problem (GRP) framework for hyperbolic conservation laws, enabling a second-order-in-time, non-synchronous interface data approximation via coupled half-GRPs. The authors instantiate the framework for the one-dimensional Euler equations coupled by a gas generator, derive solvability conditions for the resulting coupled GRP, and provide a numerical validation using a high-order discontinuous Galerkin scheme. Key contributions include the formalization of coupled GRP/half-GRP methods, a solvability analysis for the gas-generator problem, and numerical evidence of correct convergence rates across multiple test cases. The approach facilitates decoupled, parallelizable solvers for coupled systems and offers a pathway to higher-order coupling via extended GRP formulations, with applicability to gas networks and similar interfaces in complex flows.

Abstract

We introduce a novel method for systems of conservation laws coupled at a sharp interface based on generalized Riemann problems. This method yields a piecewise-linear in time approximation of the solution at the interface, thus, descynchronising the solvers for the coupled systems. We apply this framework to a problem of compressible Euler equations coupled via a gas generator and prove its solvability. Finally, we conduct numerical experiments and show that our algorithm performs at correct convergence rates.

Paper Structure

This paper contains 10 sections, 1 theorem, 42 equations, 4 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Coupled generalized half Riemann problems
  3. Coupled generalized Riemann problem for the Euler equations
  4. Euler equations coupled by a gas-generator
  5. Generalized Riemann problem for single Euler equations
  6. Coupled generalized Riemann problem for the gas-generator
  7. Numerical results
  8. Case 1: Linear Outtake
  9. Case 2: Nonlinear Outtake
  10. Case 3: Density bump passing a nonlinear outtake

Key Result

Theorem 1

The system eq:gcc-grp has a unique solution if the left-going 1-wave of the GGC is a rarefaction wave or a weak shock, regardless of the type the right-going 3-wave.

Figures (4)

  • Figure 1: General setup of two coupled hyperbolic conservation laws.
  • Figure 2: Time series of the momentum $q$ for Case 1 on level $L=5$.
  • Figure 3: Time series of the momentum $q$ for Case 2 on level $L=5$.
  • Figure 4: Time series of the momentum $q$ for Case 3 on level $L=5$.

Theorems & Definitions (6)

  • Definition 1
  • Definition 2
  • Remark 1
  • Theorem 1
  • proof
  • Remark 2