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Multi-Derivative Runge-Kutta Flux Reconstruction for hyperbolic conservation laws

Arpit Babbar, Praveen Chandrashekar

Abstract

We extend the fourth order, two stage Multi-Derivative Runge Kutta (MDRK) scheme to the Flux Reconstruction (FR) framework by writing both stages in terms of a time averaged flux and then using the approximate Lax-Wendroff procedure to compute the time averaged flux. Numerical flux is carefully constructed to enhance Fourier CFL stability and accuracy. A subcell based blending limiter is developed for the MDRK scheme which ensures that the limited scheme is provably admissibility preserving. Along with being admissibility preserving, the blending scheme is constructed to minimize dissipation errors by using Gauss-Legendre solution points and performing MUSCL-Hancock reconstruction on subcells. The accuracy enhancement of the blending scheme is numerically verified on compressible Euler's equations, with test cases involving shocks and small-scale structures.

Multi-Derivative Runge-Kutta Flux Reconstruction for hyperbolic conservation laws

Abstract

We extend the fourth order, two stage Multi-Derivative Runge Kutta (MDRK) scheme to the Flux Reconstruction (FR) framework by writing both stages in terms of a time averaged flux and then using the approximate Lax-Wendroff procedure to compute the time averaged flux. Numerical flux is carefully constructed to enhance Fourier CFL stability and accuracy. A subcell based blending limiter is developed for the MDRK scheme which ensures that the limited scheme is provably admissibility preserving. Along with being admissibility preserving, the blending scheme is constructed to minimize dissipation errors by using Gauss-Legendre solution points and performing MUSCL-Hancock reconstruction on subcells. The accuracy enhancement of the blending scheme is numerically verified on compressible Euler's equations, with test cases involving shocks and small-scale structures.
Paper Structure (56 sections, 1 theorem, 134 equations, 17 figures, 2 tables)

This paper contains 56 sections, 1 theorem, 134 equations, 17 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Conservation law and solution space
  3. Runge-Kutta FR
  4. Step 1.
  5. Step 2.
  6. Step 3.
  7. Correction functions.
  8. Multi-derivative Runge-Kutta FR scheme
  9. Conservation property
  10. Reconstruction of the time average flux
  11. Step 1.
  12. Step 2.
  13. Step 3.
  14. Step 4.
  15. Approximate Lax-Wendroff procedure
  16. ...and 41 more sections

Key Result

Theorem 1

Consider the MDRK scheme with blending eq:mdrk.blended.scheme where low and high order schemes use the same numerical fluxes $\boldsymbol{F}_{e + \frac{1}{2}}, \boldsymbol{F}^{\ast}_{e + \frac{1}{2}}$eq:blended.numfluxes at every element interface. Then the following can be said about admissibility

Figures (17)

  • 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 $N = 3$.
  • Figure 3: Error convergence for constant linear advection equation comparing MDRK and RK - (a) GL points with Radau correction, (b) GLL points with $g_2$ correction
  • Figure 4: Error convergence for variable linear advection equation with $a (x) = x^2$; (a) AE scheme, (b) EA scheme, (c) AE versus EA
  • Figure 5: Comparing AE and EA schemes using D2 dissipation for 1-D Burgers' equation at $t = 2$. (a) GL points with Radau correction, (b) GLL points with $g_2$ correction
  • ...and 12 more figures

Theorems & Definitions (9)

  • Remark 1
  • Remark 2
  • Definition 1
  • Definition 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Theorem 1
  • Remark 6