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Compact Runge-Kutta Flux Reconstruction for Hyperbolic Conservation Laws with admissibility preservation

Arpit Babbar, Qifan Chen

TL;DR

This work extends the compact Runge-Kutta concept to the Flux Reconstruction framework, recasting cRK as a time-averaged flux approximation to enable a single numerical flux per time step. It introduces three dissipative models (D-CSX, D1, D2) and two central flux strategies (AE, EA) within the FR setting, and proves equivalence to Lax-Wendroff-type schemes under linearity assumptions. A subcell-based blending limiter (Gauss-Legendre points with MUSCL-Hancock reconstruction) and a flux limiter for admissibility in means ensure robustness for nonsmooth solutions, including those with source terms. Numerical experiments on scalar equations, Euler flows, and the ten-moment model demonstrate high-order accuracy, strong shock robustness, and admissibility preservation, with competitive wall-clock performance relative to LWFR. The framework shows promise for efficient, high-order, physically admissible simulations in complex hyperbolic systems and sets the stage for extensions to curvilinear grids and implicit-explicit time stepping.

Abstract

Compact Runge-Kutta (cRK) Discontinuous Galerkin (DG) methods, recently introduced in [Q. Chen, Z. Sun, and Y. Xing, SIAM J. Sci. Comput. SIAM J. Sci. Comput., 46: A1327-A1351, 2024], are a variant of RKDG methods for solving hyperbolic conservation laws and are characterized by their compact stencil including only immediate neighboring finite elements. This article proposes a cRK Flux Reconstruction (FR) method by interpreting cRK as a procedure to approximate time-averaged fluxes, which requires computing only a single numerical flux for each time step and further reduces data communication. The numerical flux is carefully constructed to maintain the same Courant-Friedrichs-Lewy (CFL) numbers as cRKDG methods and achieve optimal accuracy uniformly across all polynomial degrees, even for problems with sonic points. A subcell-based blending limiter is then applied for problems with nonsmooth solutions, which uses Gauss-Legendre solution points and performs MUSCL-Hancock reconstruction on subcells to mitigate the additional dissipation errors. Additionally, to achieve a fully admissibility-preserving cRKFR scheme, a flux limiter is applied to the time-averaged numerical flux to ensure admissibility preservation in the means, combined with a positivity-preserving scaling limiter. The method is further extended to handle source terms by incorporating their contributions as additional time averages. Numerical experiments including Euler equations and the ten-moment problem are provided to validate the claims regarding the method's accuracy, robustness, and admissibility preservation.

Compact Runge-Kutta Flux Reconstruction for Hyperbolic Conservation Laws with admissibility preservation

TL;DR

This work extends the compact Runge-Kutta concept to the Flux Reconstruction framework, recasting cRK as a time-averaged flux approximation to enable a single numerical flux per time step. It introduces three dissipative models (D-CSX, D1, D2) and two central flux strategies (AE, EA) within the FR setting, and proves equivalence to Lax-Wendroff-type schemes under linearity assumptions. A subcell-based blending limiter (Gauss-Legendre points with MUSCL-Hancock reconstruction) and a flux limiter for admissibility in means ensure robustness for nonsmooth solutions, including those with source terms. Numerical experiments on scalar equations, Euler flows, and the ten-moment model demonstrate high-order accuracy, strong shock robustness, and admissibility preservation, with competitive wall-clock performance relative to LWFR. The framework shows promise for efficient, high-order, physically admissible simulations in complex hyperbolic systems and sets the stage for extensions to curvilinear grids and implicit-explicit time stepping.

Abstract

Compact Runge-Kutta (cRK) Discontinuous Galerkin (DG) methods, recently introduced in [Q. Chen, Z. Sun, and Y. Xing, SIAM J. Sci. Comput. SIAM J. Sci. Comput., 46: A1327-A1351, 2024], are a variant of RKDG methods for solving hyperbolic conservation laws and are characterized by their compact stencil including only immediate neighboring finite elements. This article proposes a cRK Flux Reconstruction (FR) method by interpreting cRK as a procedure to approximate time-averaged fluxes, which requires computing only a single numerical flux for each time step and further reduces data communication. The numerical flux is carefully constructed to maintain the same Courant-Friedrichs-Lewy (CFL) numbers as cRKDG methods and achieve optimal accuracy uniformly across all polynomial degrees, even for problems with sonic points. A subcell-based blending limiter is then applied for problems with nonsmooth solutions, which uses Gauss-Legendre solution points and performs MUSCL-Hancock reconstruction on subcells to mitigate the additional dissipation errors. Additionally, to achieve a fully admissibility-preserving cRKFR scheme, a flux limiter is applied to the time-averaged numerical flux to ensure admissibility preservation in the means, combined with a positivity-preserving scaling limiter. The method is further extended to handle source terms by incorporating their contributions as additional time averages. Numerical experiments including Euler equations and the ten-moment problem are provided to validate the claims regarding the method's accuracy, robustness, and admissibility preservation.

Paper Structure

This paper contains 47 sections, 1 theorem, 105 equations, 19 figures, 1 table.

Table of Contents

  1. Introduction
  2. Runge-Kutta Flux Reconstruction
  3. Compact Runge-Kutta Flux Reconstruction
  4. cRK22.
  5. cRK33.
  6. cRK44.
  7. Time average numerical flux computation
  8. Dissipative part of the numerical flux
  9. D-CSX (Chen, Sun, Xing chen2024) dissipation.
  10. D1 dissipation.
  11. D2 dissipation.
  12. Central part of the numerical flux
  13. Conservation property and admissibility preservation
  14. High order boundary treatment
  15. Periodic boundaries.
  16. ...and 32 more sections

Key Result

Theorem 1

Consider the cRKFR blending scheme eq:blended.scheme where low and high order schemes use the same numerical flux $\boldsymbol{F}_{e+\frac{1}{2}}$ at every element interface. Then we obtain the following results on admissibility preserving in means (Definition defn:admissibility.preserving.means) of

Figures (19)

  • Figure 1: (a) Piecewise polynomial solution at time $t_n$, and (b) discontinuous and continuous flux. The figure has been taken from babbar2022.
  • Figure 2: Subcells used by lower order scheme for degree $N = 3$ using (a) Gauss-Legendre (GL) solution points (GL), (b) Gauss-Legendre-Lobatto (GLL) solution points.
  • Figure 3: $L^2$ error versus number of degrees of freedom for degrees $N=1,2,3$ for 1-D linear advection equation with periodic boundary conditions at $t = 2$ comparing D1 and D2 dissipation schemes using (a) Gauss-Legendre (GL) solution points with Radau correction functions, (b) Gauss-Legendre-Lobatto (GLL) solution points with $g_2$ correction functions.
  • Figure 4: $L^2$ error versus number of degrees of freedom for degrees $N=1,2,3$ for variable advection equation with flux $f(x,u) = x^2 u$ and non-periodic boundary conditions using Gauss-Legendre (GL) solution points with Radau correction functions. (a) Comparison of the cRKFR scheme using the AE flux \ref{['eq:extrapolate']} with the standard RKFR scheme, (b) comparison of the cRKFR scheme using the EA flux \ref{['eq:evaluate']} with the standard RKFR scheme, (c) comparison of the cRKFR schemes using AE\ref{['eq:extrapolate']} and EA\ref{['eq:evaluate']} fluxes.
  • Figure 5: Comparing AE and EA schemes for 1-D Burgers' equation at $t = 2$ using Gauss-Legendre (GL) solution points with (a) Radau correction function with Gauss-Legendre (GL) solution points, (b) $g_2$ correction function with Gauss-Legendre-Lobatto (GLL) solution points.
  • ...and 14 more figures

Theorems & Definitions (3)

  • Definition 1
  • Definition 2
  • Theorem 1