Direct modeling for computational fluid dynamics and the construction of high-order compact scheme for compressible flow simulations

Computational fluid dynamics is a direct modeling of physical laws in a discretized space. The basic physical laws include the mass, momentum and energy conservations, numerical scale-consistent transport process, and similar domain of dependence and influence in the physical and the numerical wave propagations. Therefore, a physically soundable numerical scheme should be a compact one which involves the closest neighboring cells within the domain of dependence for the solution update under a CFL number (similar to 1). In order to keep the compactness of the scheme and be high-order, subcell flow distributions or the equivalent degrees of freedom beyond the cell averaged flow variables are needed, such as the gradients of the flow variables inside each control volume. Under the above consideration, the direct use of Riemann solver as the evolution model doesn't provide such a mechanism. High-order dynamic evolution process should be used to update the time accurate cell-averaged flow variables and their gradients. The direct modeling of flow evolution under generalized initial condition will be developed. In order to provide reliable cell averaged flow variables and their gradients for the compact data reconstruction, the evolution process is constructed to provide possible discontinuous time-dependent flow variables across a cell interface. At the same time, the time accurate flux function at a cell interface can become a discontinuous function of time in its evolution process. Same as the spatial limiter in the conventional computational fluid dynamics methods, such as the total variation diminishing and weighted essentially non-oscillatory methods, the temporal limiter for the time accurate flux function is designed as well. The direct modeling method in this paper will update flow variables differently on both sides of a cell interface and nonlinearly limit the high-order time derivatives of the flux function in case of shock wave passing through a cell interface within a time step. The direct modeling method unifies the concepts of nonlinear limiters in spatial data reconstruction and temporal flux transport. The equivalent treatment of space and time makes the compact gas-kinetic scheme be super robust and accurate for the compressible flow simulation. At the same time, the 4th to 8th-order compact schemes can use a large CFL number (similar to 0.8) in flow computation, even for the hypersonic flow simulation. (c) 2023 Elsevier Inc. All rights reserved.