Efficient Alternative Finite Difference WENO Schemes for Hyperbolic Conservation Laws

Abstract

Higher order finite difference Weighted Essentially Non-Oscillatory (FD-WENO) schemes for conservation laws are extremely popular because, for multidimensional problems, they offer high order accuracy at a fraction of the cost of finite volume WENO or DG schemes. Such schemes come in two formulations. The very popular classical FD-WENO method (Shu and Osher J Comput Phys 83: 32–78, 1989) relies on two reconstruction steps applied to two split fluxes. However, the method cannot accommodate different types of Riemann solvers and cannot preserve free stream boundary conditions on curvilinear meshes. This limits its utility. The alternative FD-WENO (AFD-WENO) method can overcome these deficiencies, however, much less work has been done on this method. The reasons are three-fold. First, it is difficult for the casual reader to understand the intricate logic that requires higher order derivatives of the fluxes to be evaluated at zone boundaries. The analytical methods for deriving the update equation for AFD-WENO schemes are somewhat recondite. To overcome that difficulty, we provide an easily accessible script that is based on a computer algebra system in Appendix A of this paper. Second, the method relies on interpolation rather than reconstruction, and WENO interpolation formulae have not been documented in the literature as thoroughly as WENO reconstruction formulae. In this paper, we explicitly provide all necessary WENO interpolation formulae that are needed for implementing the AFD-WENO up to the ninth order. The third reason is that the AFD-WENO requires higher order derivatives of the fluxes to be available at zone boundaries. Since those derivatives are usually obtained by finite differencing the zone-centered fluxes, they become susceptible to a Gibbs phenomenon when the solution is non-smooth. The inclusion of those fluxes is also crucially important for preserving the order property when the solution is smooth. This has limited the utility of the AFD-WENO in the past even though the method per se has many desirable features. Some efforts to mitigate the effect of finite differencing of the fluxes have been tried, but so far they have been done on a case by case basis for the PDE being considered. In this paper we find a general-purpose strategy that is based on a different type of the WENO interpolation. This new WENO interpolation takes the first derivatives of the fluxes at zone centers as its inputs and returns the requisite non-linearly hybridized higher order derivatives of flux-like terms at the zone boundaries as its output. With these three advances, we find that the AFD-WENO becomes a robust and general-purpose solution strategy for large classes of conservation laws. It allows any Riemann solver to be used. The AFD-WENO has a computational complexity that is entirely comparable to the classical FD-WENO, because it relies on two interpolation steps which cost the same as the two reconstruction steps in the classical FD-WENO. We apply the method to several stringent test problems drawn from Euler flow, relativistic hydrodynamics (RHD), and ten-moment equations. The method meets its design accuracy for smooth flow and can handle stringent problems in one and multiple dimensions.

Appendices

Appendix A Mathematica Script for the Derivation of a Fifth Order AFD-WENO Scheme

We present the Mathematica script for understanding the formulae in Shu and Osher [55]. The script is extensible to all orders.

Appendix B Flattener Function for the Ten-Moment Rarefied Gas Flow Model

The flattening algorithm is employed in simulations to identify regions of strong shocks within the computational domain. The algorithm aims to improve the simulation accuracy by reducing numerical artifacts near discontinuities. The flattener functions rely on comparing the divergence of the velocity field to a characteristic speed associated with the specific problem being solved. For the Euler flows, such functions have been previously defined by Colella and Woodward [21] and Balsara [5]. For the Relativistic Magneto-hydrodynamics equations, a flattener has been presented in Balsara and Kim [9]. Along the same lines, we give a flattener function for the ten-moment model. The method begins by calculating the divergence of the velocity, $(\nabla \cdot {\mathbf{v}})_{i,j}$, and sound-like speed $c_{s;i,j}$ (some approximation of the characteristic speed), within a specific zone $(i,j)$. In the two-dimensional cartesian mesh, the quantities are defined by

$$(\nabla \cdot {\mathbf{v}})_{i,j} = \frac{{v_{x;i + 1,j} - v_{x;i - 1,j} }}{\Delta x} + \frac{{v_{y;i,j + 1} - v_{y;i,j - 1} }}{\Delta y},$$

$c_{s;i,j} = \sqrt {\frac{{P_{i,j} }}{{\rho_{i,j} }},} \quad {\text{where}}\quad P_{i,j} = \sqrt {p_{xx;i,j} p_{yy;i,j} - p_{xy;i,j}^{2} }.$

To detect a shock, the undivided divergence of the velocity within a zone must be compared to the minimum sound-like speed in the zone $(i,j)$ and its immediate neighbors. The minimum sound-like speed from all the neighbors is obtained as follows:

$$c_{s;i,j}^{{\text{min-nbr}}} = \min (c_{s;i - 1,j - 1} ,c_{s;i - 1,j} ,c_{s;i - 1,j + 1} ,c_{s;i,j - 1} ,c_{s;i,j} ,c_{s;i,j + 1} ,c_{s;i + 1,j - 1} ,c_{s;i + 1,j} ,c_{s;i + 1,j + 1} ).$$

In each zone with an extent of $\Delta x$ and $\Delta y$, the flattener function is defined as

$$\eta_{i,j} = {\text{min}}\left[ {1,{\text{max}}\left[ {0,\frac{{|\left( {\nabla \cdot {\mathbf{v}}} \right)_{i,j} |{\text{ max}}(\Delta x,\Delta y)}}{{\kappa c_{s;i,j}^{{\text{min-nbr}}} }} - 1} \right]} \right]{.}$$

The parameter $\kappa$ is set to 0.3, which has been found to work well across different orders and problem types. The flattener function does not modify the reconstruction when the flow is smooth or consists of rarefactions, and in that case, $\eta_{i,j} = 0.$ However, the flattener function gradually increases from $\eta_{i,j} = 0$ to $\eta_{i,j} = 1$ when strong shocks are present.

The inclusion of pressure variation in the flattener algorithm allows for a more comprehensive stabilization of the flow simulation. It ensures that not only the zones already influenced by shocks but also the zones on edge receive appropriate flattening treatment. This improvement helps maintain numerical stability and accuracy throughout the simulation, particularly in regions where shocks are forming or propagating. In the x-direction, the flattener can be extended to the neighboring cell if the following conditions are satisfied:

$$\begin{gathered} {\text{if }}\left( {\left( {\eta_{i,j} > 0} \right){\text{ and }}\left( {\eta_{i + 1,j} = 0} \right){\text{ and }}\left( {P_{i,j} > P_{i + 1,j} } \right)} \right),{\text{ then }}\eta_{i + 1,j} = \eta_{i,j}; \hfill \\ {\text{if }}\left( {\left( {\eta_{i,j} > 0} \right){\text{ and }}\left( {\eta_{i - 1,j} = 0} \right){\text{ and }}\left( {P_{i,j} > P_{i - 1,j} } \right)} \right),{\text{ then }}\eta_{i - 1,j} = \eta_{i,j} . \hfill \\ \end{gathered}$$

In situations involving multi-dimensional problems, the above strategy can be applied to each of the principal directions of the mesh.