$2\odot 2=4$: Temporal-Spatial Coupling and Beyond in Computational Fluid Dynamics (CFD)

Abstract: With increasing engineering demands, there need high order accurate schemes embedded with precise physical information in order to capture delicate small scale structures and strong waves with correct "physics". There are two families of high order methods: One is the method of line, relying on the Runge-Kutta (R-K) time-stepping. The building block is the Riemann solution labeled as the solution element "1". Each step in R-K just has first order accuracy. In order to derive a fourth order accuracy scheme in time, one needs four stages labeled as "$1\odot 1\odot 1\odot 1=4$". The other is the one-stage Lax-Wendroff (L-W) type method, which is more compact but is complicated to design numerical fluxes and hard to use when applied to highly nonlinear problems. In recent years, the pair of solution element and dynamics, labeled as "$2$", are taken as the building black. The direct adoption of the dynamics implies the inherent temporal-spatial coupling. With this type of building blocks, a family of two-stage fourth order accurate schemes, labeled as "$2\odot 2=4$", are designed for the computation of compressible fluid flows. The resulting schemes are compact, robust and efficient. This paper contributes to elucidate how and why high order accurate schemes should be so designed. To some extent, the "$2\odot 2=4$" algorithm extracts the advantages of the method of line and one-stage L-W method. As a core part, the pair "$2$" is expounded and L-W solver is revisited. The generalized Riemann problem (GRP) solver, as the discontinuous and nonlinear version of L-W flow solver, and the gas kinetic scheme (GKS) solver, the microscopic L-W solver, are all reviewed. The compact Hermite-type data reconstruction and high order approximation of boundary conditions are proposed. Besides, the computational performance and prospective discussions are presented.