Implementation of Finite Difference Weighted Compact Nonlinear Schemes with the Two-Stage Fourth-Order Accurate Temporal Discretization (Communications in Computational Physics, 2020)
In this paper, we present a new two-stage fourth-order finite differenceweighted compact nonlinear scheme (WCNS) for hyperbolic conservation laws withspecial application to compressible Euler equations. To construct this algorithm, apartfrom the traditional WCNS for the spatial derivative, it was necessary to first constructa linear compact/explicit scheme utilizing time derivative of flux at midpoints, which,in turn, was solved by a generalized Riemann solver. Combining these two schemes,the fourth-order time accuracy was achieved using only the two-stage time-steppingtechnique. The final algorithm was numerically tested for various one-dimensionaland two-dimensional cases. The results demonstrated that the proposed algorithmhad an essentially similar performance as that based on the fourth-order Runge-Kuttamethod, while it required 25 percent less computational cost for one-dimensionalcases, which is expected to decline further for multidimensional cases.