Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Admissibility preserving subcell limiter for Lax-Wendroff flux reconstruction

Arpit Babbar, Sudarshan Kumar Kenettinkara, Praveen Chandrashekar

TL;DR

This work develops an admissibility-preserving subcell limiter for the Lax-Wendroff Flux Reconstruction (LWFR) scheme by blending LWFR with a lower-order scheme on GL-based subcells and enforcing admissibility in means through a carefully constructed blended flux. A MUSCL-Hancock-based higher-order blending is introduced on subcells to improve small-scale resolution, with a problem-independent slope limiting approach extended to non-cell-centered grids. Admissibility in means enables the use of a Zhang scaling limiter to ensure positivity of polynomial solutions, while flux corrections guarantee conservation. The method demonstrates robust positivity preservation and enhanced resolution on a wide range of 1-D and 2-D Euler tests, including shocks, vortices, and detonations, highlighting practical impact for high-order shock-capturing CFD. Overall, the approach achieves high-order accuracy away from discontinuities and robust, admissible behavior near shocks, with improved performance over prior subcell blending strategies.

Abstract

Lax-Wendroff Flux Reconstruction (LWFR) is a single-stage, high order, quadrature free method for solving hyperbolic conservation laws. We develop a subcell based limiter by blending LWFR with a lower order scheme, either first order finite volume or MUSCL-Hancock scheme. While the blending with a lower order scheme helps to control oscillations, it may not guarantee admissibility of discrete solution, e.g., positivity property of quantities like density and pressure. By exploiting the subcell structure and admissibility of lower order schemes, we devise a strategy to ensure that the blended scheme is admissibility preserving for the mean values and then use a scaling limiter to obtain admissibility of the polynomial solution. For MUSCL-Hancock scheme on non-cell-centered subcells, we develop a slope limiter, time step restrictions and suitable blending of higher order fluxes, that ensures admissibility of lower order updates and hence that of the cell averages. By using the MUSCL-Hancock scheme on subcells and Gauss-Legendre points in flux reconstruction, we improve small-scale resolution compared to the subcell-based RKDG blending scheme with first order finite volume method and Gauss-Legendre-Lobatto points. We demonstrate the performance of our scheme on compressible Euler's equations, showcasing its ability to handle shocks and preserve small-scale structures.

Admissibility preserving subcell limiter for Lax-Wendroff flux reconstruction

TL;DR

This work develops an admissibility-preserving subcell limiter for the Lax-Wendroff Flux Reconstruction (LWFR) scheme by blending LWFR with a lower-order scheme on GL-based subcells and enforcing admissibility in means through a carefully constructed blended flux. A MUSCL-Hancock-based higher-order blending is introduced on subcells to improve small-scale resolution, with a problem-independent slope limiting approach extended to non-cell-centered grids. Admissibility in means enables the use of a Zhang scaling limiter to ensure positivity of polynomial solutions, while flux corrections guarantee conservation. The method demonstrates robust positivity preservation and enhanced resolution on a wide range of 1-D and 2-D Euler tests, including shocks, vortices, and detonations, highlighting practical impact for high-order shock-capturing CFD. Overall, the approach achieves high-order accuracy away from discontinuities and robust, admissible behavior near shocks, with improved performance over prior subcell blending strategies.

Abstract

Lax-Wendroff Flux Reconstruction (LWFR) is a single-stage, high order, quadrature free method for solving hyperbolic conservation laws. We develop a subcell based limiter by blending LWFR with a lower order scheme, either first order finite volume or MUSCL-Hancock scheme. While the blending with a lower order scheme helps to control oscillations, it may not guarantee admissibility of discrete solution, e.g., positivity property of quantities like density and pressure. By exploiting the subcell structure and admissibility of lower order schemes, we devise a strategy to ensure that the blended scheme is admissibility preserving for the mean values and then use a scaling limiter to obtain admissibility of the polynomial solution. For MUSCL-Hancock scheme on non-cell-centered subcells, we develop a slope limiter, time step restrictions and suitable blending of higher order fluxes, that ensures admissibility of lower order updates and hence that of the cell averages. By using the MUSCL-Hancock scheme on subcells and Gauss-Legendre points in flux reconstruction, we improve small-scale resolution compared to the subcell-based RKDG blending scheme with first order finite volume method and Gauss-Legendre-Lobatto points. We demonstrate the performance of our scheme on compressible Euler's equations, showcasing its ability to handle shocks and preserve small-scale structures.
Paper Structure (37 sections, 13 theorems, 182 equations, 25 figures, 2 algorithms)

This paper contains 37 sections, 13 theorems, 182 equations, 25 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Lax-Wendroff FR scheme
  3. On controlling oscillations
  4. TVD limiter
  5. Blending scheme
  6. Smoothness indicator
  7. First order blending
  8. Higher order blending
  9. Flux limiter for admissibility preservation
  10. Some implementation details
  11. Numerical results
  12. 1-D Euler equations
  13. Shu-Osher problem
  14. Blast wave
  15. Sedov's blast wave
  16. ...and 22 more sections

Key Result

Theorem 1

Consider a conservation law of the form eq:con.law which preserves the admissibility set $\mathcal{U}_{\textrm{ad}}$eq:conv.pres.con.law. Let $\left\{u_j^n\right\}_{j}$ be the approximate solution at time level $n$ and assume that $u_j^n \in \mathcal{U}_{\textrm{ad}}$ for all $j$. Consider conservat Define $u_j^{*,\pm}$ by where Assume that the slope $\delta_j$ is chosen so that Then, under app

Figures (25)

  • Figure 1: (a) Piecewise polynomial solution at time $t_n$, and (b) discontinuous and continuous flux.
  • Figure 2: Subcells used by lower order scheme for degree $N=4$.
  • Figure 3: Logistic function used to map energy to a smoothness coefficient $\alpha \in [0,1]$ shown for various solution polynomial degrees $N$.
  • Figure 4: Shu-Osher problem, numerical solution with degree $N=4$ using first order (FO) and MUSCL-Hancock (MH) blending schemes, and TVB limited scheme (TVB-300) with parameter $M=300$. (a) Full and (b) zoomed density profiles of numerical solutions are shown up to time $t=1.8$ on a mesh of 400 cells.
  • Figure 5: Blast wave problem, numerical solution with degree $N=4$ using first order (FO) and MUSCL-Hancock (MH) blending schemes, and TVB limited scheme (TVB-300) with parameter $M=300$. (a) Density, (b) pressure profiles are shown at time $t=0.038$ on a mesh of 400 cells.
  • ...and 20 more figures

Theorems & Definitions (17)

  • Definition 1
  • Definition 2
  • Remark 1
  • Theorem 1
  • Theorem 2
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • ...and 7 more