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High-order accurate entropy stable schemes for compressible Euler equations with van der Waals equation of state on adaptive moving meshes

Shangting Li, Huazhong Tang

TL;DR

This work develops high-order entropy-stable schemes for the three-dimensional compressible Euler equations with the van der Waals equation of state on adaptive moving meshes. It constructs explicit two-point entropy-conservative fluxes in curvilinear coordinates, enforces discrete geometric conservation laws, and then augments these EC fluxes with a WENO-based dissipation to obtain ES schemes that satisfy a semi-discrete entropy inequality. The moving mesh is generated via Winslow-type mesh equations with a monitor function incorporating the fundamental derivative $G$ to resolve non-classical wave structures, and time integration is performed with SSP-RK methods on MPI-parallel hardware. A wide range of 1D, 2D, and 3D tests demonstrate fifth-order accuracy on moving meshes, superior resolution of non-classical waves, and substantial efficiency gains over uniform-mesh runs. These contributions advance robust, high-fidelity simulations of dense-gas flows near critical points where non-classical phenomena arise.

Abstract

This paper develops the high-order entropy stable (ES) finite difference schemes for multi-dimensional compressible Euler equations with the van der Waals equation of state (EOS) on adaptive moving meshes. Semi-discrete schemes are first nontrivially constructed built on the newly derived high-order entropy conservative (EC) fluxes in curvilinear coordinates and scaled eigenvector matrices as well as the multi-resolution WENO reconstruction, and then the fully-discrete schemes are given by using the high-order explicit strong-stability-preserving Runge-Kutta time discretizations.The high-order EC fluxes in curvilinear coordinates are derived by using the discrete geometric conservation laws and the linear combination of the two-point symmetric EC fluxes, while the two-point EC fluxes are delicately selected by using their sufficient condition, the thermodynamic entropy and the technically selected parameter vector.The adaptive moving meshes are iteratively generated by solving the mesh redistribution equations, in which the fundamental derivative related to the occurrence of non-classical waves is involved to produce high-quality mesh. Several numerical tests on the parallel computer system with the MPI programming are conducted to validate the accuracy, the ability to capture the classical and non-classical waves, and the high efficiency of our schemes in comparison with their counterparts on the uniform mesh.

High-order accurate entropy stable schemes for compressible Euler equations with van der Waals equation of state on adaptive moving meshes

TL;DR

This work develops high-order entropy-stable schemes for the three-dimensional compressible Euler equations with the van der Waals equation of state on adaptive moving meshes. It constructs explicit two-point entropy-conservative fluxes in curvilinear coordinates, enforces discrete geometric conservation laws, and then augments these EC fluxes with a WENO-based dissipation to obtain ES schemes that satisfy a semi-discrete entropy inequality. The moving mesh is generated via Winslow-type mesh equations with a monitor function incorporating the fundamental derivative to resolve non-classical wave structures, and time integration is performed with SSP-RK methods on MPI-parallel hardware. A wide range of 1D, 2D, and 3D tests demonstrate fifth-order accuracy on moving meshes, superior resolution of non-classical waves, and substantial efficiency gains over uniform-mesh runs. These contributions advance robust, high-fidelity simulations of dense-gas flows near critical points where non-classical phenomena arise.

Abstract

This paper develops the high-order entropy stable (ES) finite difference schemes for multi-dimensional compressible Euler equations with the van der Waals equation of state (EOS) on adaptive moving meshes. Semi-discrete schemes are first nontrivially constructed built on the newly derived high-order entropy conservative (EC) fluxes in curvilinear coordinates and scaled eigenvector matrices as well as the multi-resolution WENO reconstruction, and then the fully-discrete schemes are given by using the high-order explicit strong-stability-preserving Runge-Kutta time discretizations.The high-order EC fluxes in curvilinear coordinates are derived by using the discrete geometric conservation laws and the linear combination of the two-point symmetric EC fluxes, while the two-point EC fluxes are delicately selected by using their sufficient condition, the thermodynamic entropy and the technically selected parameter vector.The adaptive moving meshes are iteratively generated by solving the mesh redistribution equations, in which the fundamental derivative related to the occurrence of non-classical waves is involved to produce high-quality mesh. Several numerical tests on the parallel computer system with the MPI programming are conducted to validate the accuracy, the ability to capture the classical and non-classical waves, and the high efficiency of our schemes in comparison with their counterparts on the uniform mesh.
Paper Structure (10 sections, 1 theorem, 89 equations, 16 figures, 5 tables)

This paper contains 10 sections, 1 theorem, 89 equations, 16 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Entropy conditions
  3. Numerical schemes
  4. Two-point EC fluxes in curvilinear coordinates
  5. Discrete GCLs
  6. ES schemes
  7. Adaptive moving mesh strategy
  8. Numerical results
  9. Conclusion
  10. Different forms of linear equations to obtain two-points EC flux

Key Result

Theorem 3.1

Assuming that $\bm{z}_l, \bm{z}_r$ satisfy the constraints in Remark rem-eq:constraint and $\{\!\!\{ \bm{z} \}\!\!\}$ satisfies the linear equations eq:linear_U-eq:linear_F have a unique solution.

Figures (16)

  • Figure 2.1: The curves of the pressure $p = {8T}/{(3\nu-1)} - {3}/{\nu^2}$ with respect to specific volume $\nu=1/\rho$ at different temperature $T$.
  • Figure 5.1: Example \ref{['ex:1Dsmooth']}: $\ell^1$- and $\ell^\infty$-errors in $\rho$ with respect to $N_1$ at $t = 1.0$.
  • Figure 5.2: The numerical results of RP1 at $t = 0.15Z_c^{-1/2}.$
  • Figure 5.3: The numerical results of RP2 at $t = 0.45Z_c^{-1/2}.$
  • Figure 5.4: The numerical results of RP3 at $t = 0.2Z_c^{-1/2}.$
  • ...and 11 more figures

Theorems & Definitions (17)

  • Remark 2.1
  • Remark 2.2
  • Definition 2.1: Entropy function
  • Theorem 3.1
  • proof
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Example 5.1: 1D Smooth Sine Wave
  • ...and 7 more