A Nonlinear Flux Split Method for Hyperbolic Conservation Laws

Abstract

We present and analyze the performance of a nonlinear, upwind flux split method for approximating solutions of hyperbolic conservation laws. The method is based on a new version of the single-state-approximate Riemann solver devised by Harten, Lax, and van Leer (HLL) and implemented by Einfeldt. It makes use of two-sided local characteristic variables to reduce the dissipation of HLL by introducing the flavor of HLL into the Steger-Warming flux vector splitting scheme. We use the characteristic decomposition and the method-of-lines approach to construct high-order versions of the first-order scheme and demonstrate their efficiency and robustness in several numerical tests.

Equilibrium real gas computations using Marquina's scheme

“…When an eigenvalue changes sign, Marquina's ux function characteristic component is constructed from the local-Lax-Friedrichs scalar numerical ux, which introduces additional dissipation to enforce an entropy condition in the case of a rarefaction wave (when p (Q i )¡0¡ p (Q i+1 )), or in other circumstances such as shock re ection (see References [2,13]). The ux splitting and the Jacobian evaluation in the ÿrst-order scheme are based on the physical data on the left and right sides of the cell wall.…”

The water hammer problem and its numerical analysis

“…We construct a numerical Lax-Wendroff scheme for (4.2) which has a rigorous error analysis compared to the engineering ODE model proposed by [35,68] with modelling based on the method of characteristics. This new dynamical equation set (4.2) also works for gas or petroleum pipelines industry [6,17,69], and can also refer to blood flowing through artificial valves in the human heart [23,42,46,54].…”