Transversal effects of high order numerical schemes for compressible fluid flows

Finite volume schemes for the two-dimensional (2D) wave system are taken to demonstrate the role of the genuine dimensionality of Lax-Wendroff flow solvers for compressible fluid flows. When the finite volume schemes are applied, the transversal variation relative to the computational cell interfaces is neglected, and only the normal numerical flux is used, thanks to the Gauss-Green formula. In order to offset such defects, the Lax-Wendroff flow solvers or the generalized Riemann problem (GRP) solvers are adopted by substituting the time evolution of flows into the spatial variation. The numerical results show that even with the same convergence rate, the error by the GRP2D solver is almost one ninth of that by the multistage Runge-Kutta (RK) method.