Preventing numerical oscillations in the flux-split based finite difference method for compressible flows with discontinuities

Abstract

In simulating compressible flows with contact discontinuities or material interfaces, numerical pressure and velocity oscillations can be induced by point-wise flux vector splitting (FVS) or component-wise nonlinear difference discretization of convection terms. The current analysis showed that the oscillations are due to the incompatibility of the point-wise splitting of eigenvalues in FVS and the inconsistency of component-wise nonlinear difference discretization among equations of mass, momentum, energy, and even fluid composition for multi-material flows. Two practical principles are proposed to prevent these oscillations: (i) convective fluxes must be split by a global FVS, such as the global Lax-Friedrichs FVS, and (ii) consistent discretization between different equations must be guaranteed. The latter, however, is not compatible with component-wise nonlinear difference discretization. Therefore, a consistent discretization method that uses only one set of common weights is proposed for nonlinear weighted essentially non-oscillatory (WENO) schemes. One possible procedure to determine the common weights is presented that provided good results. The analysis and methods stated above are appropriate for both single-(e.g., contact discontinuity) and multi-material (e.g., material interface) discontinuities. For the latter, however, the additional fluid composition equation should be split and discretized consistently for compatibility with the other equations. Numerical tests including several contact discontinuities and multi-material flows confirmed the effectiveness, robustness, and low computation cost of the proposed method.

Preventing numerical oscillations in the flux‐split based finite difference method for compressible flows with discontinuities, II

“…In [21], as the ideal-gas EOS was considered, in order to make a clear comparison between different FVS methods for the simplest example of a problem: a single-material stationary contact discontinuity discretized by the first-order upwind difference scheme, we presented a unified form of the StegerWarming FVS and Lax-Friedrichs type FVS as…”

“…In [21], we reported that the general condition for Equation (5), coupled with the first-order upwind scheme to avoid velocity a nd pressure oscillations for a stationary contact discontinuity, is that…”

“…In [21], we reported that even with the ideal-gas EOS, numerical oscillations can be caused by component-wise nonlinear difference discretization among equations. In this section, we present the following theoretical derivation:…”

A parameter‐free ε‐adaptive algorithm for improving weighted compact nonlinear schemes

“…Initial conditions for a contact discontinuity are as follows: where γ = 1.4 is the specific heat ratio. Computational time is t end = 0.4, with 201 grid points and CFL = 0.1.…”

“…Computational results of a square wave indicate that this method helps to reduce error in smooth areas while maintaining the same ENO property near discontinuities as classical WENO‐Z. He et al modified this adaptive algorithm and free parameters for computations of Euler equations, yielding less numerical oscillation near a contact discontinuity. However, these adaptive algorithms include four free empirical parameters to be specified manually, making them difficult to be popularized.…”