FINITE-DIFFERENCE METHOD-OF-CHARACTERISTICS SCHEME FOR THE EQUATIONS OF GASDYNAMICS

A finite-difference method for the characteristic equations of gasdynamics is developed. The method is based on a directed discretization of the characteristic compatibility conditions. The choice of wave propagation directions is arbitrary, and there is no need to fit interpolating functions to the initial data. The proposed method is not conservative and therefore must be used with a shock-fitting procedure in regions wherever the flow is discontinuous. Numerical results are presented for sample two-dimensional problems. UMERICAL schemes based on the method of character- istics have been applied successfully to a large number of problems in multidimensio nal gasdynamics. Schemes of this type are attractive because they are typically robust, in no small part because of their direct relationship to the physics of nonlinear wave propagation. The review paper by Roe 1 out- lines a number of characteristi cs-based methods, including algorithms for both the conservative and nonconservative forms of the governing equations. In this paper, we present a nonconservative discretization for the characteristic compati- bility conditions. It is important to keep in mind that al- gorithms of this type must, in general, incorporate a capability to track discontinuities in the flowfield explicitly to ensure that the Rankine-Hugoniot jump conditions are satisfied in these regions. A number of conservative and nonconservative schemes for multidimensional flow are constructed under the assumption that the multidimensional problem can be split into quasi-one- dimensional pieces in which waves travel normal to the mesh lines (this is true of approximate Riemann solvers,2'3 flux vector splitting schemes, 4>5 and also of the nonconservative SCM scheme6). How this assumption affects the quality of the numerical results is not entirely clear, especially if the compar- ison is made keeping the total computational effort fixed. Nevertheless, there is evidence that numerical methods that account for the multidimensional nature of the problem can produce high-resolution solutions. One family of methods of this type has evolved from Colella's conservative unsplit Godunov method:7 multidimensional characteristics informa- tion is propagated to the faces of the computational cell to compute the numerical fluxes required in the solution of the integral form of the conservation equations. Perhaps even more sophisticated is Roe's model,8 for which, unfortunately, little data is available. Several recent papers report research and opinions on this subject.8'9'10 Many nonconservative, truly multidimensional schemes, based on the method of characteristi cs for multidimensional flow, have evolved from the method of Butler.11 According to this method, a number of characteristi c compatibility condi- tions are combined to yield a set of equations in which the spatial derivatives of the dependent variables are evaluated along the projections onto space of the stream- and wave-type characteristic rays. The advantage in this approach is clear: the "upwind" directions are defined by the characteristic rays.