With increasing engineering demands, there need high order accurate schemes embedded with precise physical information in order to capture delicate small-scale structures with correct ``physics". Generally speaking, there are two families of high order methods popularly used in practice: One is the family of line methods, relying on the Runge-Kutta iteration to achieve the temporal discretization; the other is the family of one-stage Lax-Wendroff (L-W) type methods, the numerical realization of the Cauchy-Kowalevski approach for the corresponding partial differential equations (PDEs).
In the context of compressible fluid flows, the building block of the line methods is the Riemann solution (conservative quantities), which is labeled as ``1". Each step in the Runge-Kutta iteration just has first order accuracy. In order to design a fourth order accuracy scheme in time, for example, one needs four stages labeled as ``1¤1¤1¤1=4", besides the treatment of spatial discretization, which naturally spans computational stencils and decreases computational efficiency. Additionally, the spatial-temporal decoupling hampers to contain as sufficient physical information as possible, such as the thermodynamics. In contrast, the one-stage L-W type methods are more compact and contain all information, however, it is very complicated and hard to be used for the compressible fluid flows due to the high nonlinearity of underlying problems.
In recent years, the pair, the primitive variables and their dynamics labeled as ``2", e.g., the velocity and the acceleration, is taken as the building black to devise numerical schemes. In particular, a family of two-stage fourth order accurate schemes, labeled as ``2¤2=4", are designed for the computation of compressible fluid flows. The direct use of dynamics reflects the temporal-spatial coupling and entropy stability. The resulting schemes are compact, robust, and efficient. In this talk I will introduce how and why high order accurate schemes should be so designed, as sharp contrast to line methods and one-stage L-W methods.