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SUMMARY. The numerical scheme for the computation of a shock discontinuity developed by MacCormack has been extended to solve a number of differential equations

The numerical scheme for the computation of a shock discontinuity developed by MacCormack has been extended to solve a number of differential equations, including cases explicitly containing higher-order derivatives. Comparisons with previous results are made, if available, to illustrate the advantages of the present method. The question of convergence of the numerical calculation is discussed. In the paper author analyzes the stability of the scheme and conjectures that alternating between the order in which the backward/forward steps are applied will allow the full (one-dimensional) CFL limit on time step. This conjecture certainly has been verified again and again in numerous applications of the method, but has not been proved.

The authors suggest a positive correlation between the accuracy and efficiency of a numerical method, pointing out that an explicit scheme operating at its maximum allowable time step has all the data needed to advance the solution, with a minimum of extraneous data. The purpose of this paper is to demonstrate the advantages of MacCormack scheme used earlier for the shock wave/laminar boundary layer interaction problem to more general fluid dynamics problems. The split explicit MacCormack scheme is applied to the inviscid equations of compressible flow to solve for the supersonic flow past symmetric diamond-shaped airfoils and double compression corners using simple, non-orthogonal, sheared meshes. They achieve results in excellent agreement with the exact (inviscid) solutions for these problems, demonstrating a reduction in computational time of more than a factor of two, relative to the unsplit method. The split method allows both: 1) advancing the solution at the full one-dimensional CFL limit in each space dimension, and 2) advancing the solution in the direction of the smaller mesh spacing multiple time steps for each time step in the coarser direction, allowing a better matching of the numerical and physical domains of dependence.

The authors use three problems to illustrate three different points. The linear wave (advection) equation is used to show that the MacCormack explicit method reproduces the exact solution at a Courant number of unity due, the authors argue, to the alignment of the spacetime mesh with the solution for this value of Courant number. Second, the inviscid Burgers equation is used to show that, without corrective measures, the numerical scheme may capture (physically incorrect) expansion " shocks." Finally, the authors consider solutions of the Euler equations for several two-dimensional, supersonic flows, including flows past wedges, diamond airfoils, and a sphere. For these flows it is shown that the numerical error is reduced when the mesh is aligned with the shock position.