Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Impact of Numerical Fluxes on High Order Semidiscrete WENO-DeC Finite Volume Schemes

Lorenzo Micalizzi, Eleuterio F. Toro

TL;DR

This work analyzes how numerical flux selection influences the performance of high-order semidiscrete FV schemes that combine WENO spatial reconstruction with Deferred Correction time stepping up to order $7$. By testing eight fluxes (LxF, FORCE, Rus, HLL, CU, LDCU, HLLC, Ex.RS) on 1D and 2D Euler problems, it demonstrates that complete upwind fluxes (HLLC, Ex.RS) deliver superior accuracy and robustness, especially for slow/intermediate waves and long-time evolutions, while centered fluxes are more diffusive and often unstable in 2D at low order. Increasing the space-time order markedly improves accuracy across fluxes, but cannot fully compensate for inherently diffusive fluxes; in many cases a suitably chosen complete upwind flux with moderate order achieves results comparable to higher-order schemes with poorer fluxes. The findings provide practical guidance for selecting fluxes in high-order FV methods and motivate future work on even higher-order WENO–DeC and flux designs.

Abstract

The numerical flux determines the performance of numerical methods for solving hyperbolic partial differential equations (PDEs). In this work, we compare a selection of 8 numerical fluxes in the framework of nonlinear semidiscrete finite volume (FV) schemes, based on Weighted Essentially Non-Oscillatory (WENO) spatial reconstruction and Deferred Correction (DeC) time discretization. The methodology is implemented and systematically assessed for order of accuracy in space and time up to seven. The numerical fluxes selected in the present study represent the two existing classes of fluxes, namely centred and upwind. Centred fluxes do not explicitly use wave propagation information, while, upwind fluxes do so from the solution of the Riemann problem via a wave model containing $A$ waves. Upwind fluxes include two subclasses: complete and incomplete fluxes. For complete upwind fluxes, $A=E$, where $E$ is the number of characteristic fields in the exact problem. For incomplete upwind ones, $A<E$. Our study is conducted for the one- and two-dimensional Euler equations, for which we consider the following numerical fluxes: Lax-Friedrichs (LxF), First-Order Centred (FORCE), Rusanov (Rus), Harten-Lax-van Leer (HLL), Central-Upwind (CU), Low-Dissipation Central-Upwind (LDCU), HLLC, and the flux computed through the exact Riemann solver (Ex.RS). We find that the numerical flux has an effect on the performance of the methods. The magnitude of the effect depends on the type of numerical flux and on the order of accuracy of the scheme. It also depends on the type of problem; that is, whether the solution is smooth or discontinuous, whether discontinuities are linear or nonlinear, whether linear discontinuities are fast- or slowly-moving, and whether the solution is evolved for short or long time.

Impact of Numerical Fluxes on High Order Semidiscrete WENO-DeC Finite Volume Schemes

TL;DR

This work analyzes how numerical flux selection influences the performance of high-order semidiscrete FV schemes that combine WENO spatial reconstruction with Deferred Correction time stepping up to order . By testing eight fluxes (LxF, FORCE, Rus, HLL, CU, LDCU, HLLC, Ex.RS) on 1D and 2D Euler problems, it demonstrates that complete upwind fluxes (HLLC, Ex.RS) deliver superior accuracy and robustness, especially for slow/intermediate waves and long-time evolutions, while centered fluxes are more diffusive and often unstable in 2D at low order. Increasing the space-time order markedly improves accuracy across fluxes, but cannot fully compensate for inherently diffusive fluxes; in many cases a suitably chosen complete upwind flux with moderate order achieves results comparable to higher-order schemes with poorer fluxes. The findings provide practical guidance for selecting fluxes in high-order FV methods and motivate future work on even higher-order WENO–DeC and flux designs.

Abstract

The numerical flux determines the performance of numerical methods for solving hyperbolic partial differential equations (PDEs). In this work, we compare a selection of 8 numerical fluxes in the framework of nonlinear semidiscrete finite volume (FV) schemes, based on Weighted Essentially Non-Oscillatory (WENO) spatial reconstruction and Deferred Correction (DeC) time discretization. The methodology is implemented and systematically assessed for order of accuracy in space and time up to seven. The numerical fluxes selected in the present study represent the two existing classes of fluxes, namely centred and upwind. Centred fluxes do not explicitly use wave propagation information, while, upwind fluxes do so from the solution of the Riemann problem via a wave model containing waves. Upwind fluxes include two subclasses: complete and incomplete fluxes. For complete upwind fluxes, , where is the number of characteristic fields in the exact problem. For incomplete upwind ones, . Our study is conducted for the one- and two-dimensional Euler equations, for which we consider the following numerical fluxes: Lax-Friedrichs (LxF), First-Order Centred (FORCE), Rusanov (Rus), Harten-Lax-van Leer (HLL), Central-Upwind (CU), Low-Dissipation Central-Upwind (LDCU), HLLC, and the flux computed through the exact Riemann solver (Ex.RS). We find that the numerical flux has an effect on the performance of the methods. The magnitude of the effect depends on the type of numerical flux and on the order of accuracy of the scheme. It also depends on the type of problem; that is, whether the solution is smooth or discontinuous, whether discontinuities are linear or nonlinear, whether linear discontinuities are fast- or slowly-moving, and whether the solution is evolved for short or long time.

Paper Structure

This paper contains 27 sections, 1 theorem, 37 equations, 11 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Governing equations and Finite Volume method
  3. Hyperbolic systems of conservation laws
  4. Finite Volume method
  5. Space discretization
  6. Weighted Essentially Non--Oscillatory reconstruction
  7. One--dimensional case
  8. Multidimensional case
  9. Reconstruction of characteristic variables
  10. Numerical fluxes
  11. Time discretization
  12. Abgrall's Deferred Correction formalism
  13. bDeC for ordinary differential equations
  14. Numerical results
  15. One--dimensional Euler equations
  16. ...and 12 more sections

Key Result

Theorem 1

Let us assume that the operators $\mathcal{L}_{\Delta}^1,\mathcal{L}_{\Delta}^2: X \rightarrow Y$ fulfill the following properties: For a given $\underline{\boldsymbol{u\mkern-3mu}\mkern3mu}^{(0)}\in X$, let us consider the sequence of vectors $\left\lbrace\underline{\boldsymbol{u\mkern-3mu}\mkern3mu}^{(p)}\right\rbrace_{p\in\mathbb{N}}\subseteq X$ obtained by solving the following iterative proc

Figures (11)

  • Figure 1: One--dimensional Euler equations, Advection of smooth density: Convergence analysis and efficiency analysis for all numerical fluxes and orders
  • Figure 2: One--dimensional Euler equations, Riemann problem 1: Density profile obtained with all numerical fluxes for different orders with $C_{CFL}:=0.95$
  • Figure 3: One--dimensional Euler equations, Riemann problem 5: Density profile obtained with all numerical fluxes for different orders with $C_{CFL}:=0.1$
  • Figure 4: One--dimensional Euler equations, Riemann problem 5: Density profile obtained with all numerical fluxes for a first order FV setting with $C_{CFL}:=0.95$
  • Figure 5: One--dimensional Euler equations, Riemann problem 6: Density profile obtained with all numerical fluxes for different orders with $C_{CFL}:=0.95$ at final time $T_f=2$
  • ...and 6 more figures

Theorems & Definitions (8)

  • Remark 1: On the importance of nonlinearity
  • Remark 2: On source terms
  • Remark 3: On optimality and symmetry of stencils
  • Remark 4: On negative linear weights
  • Remark 5: On speed estimates
  • Theorem 1: DeC, Abgrall Decremi
  • proof
  • Remark 6: On the time step computation