The construction of difference schemes for hyperbolic equations based on characteristic relations

Abstract

THERE have recently been intensive developments in various numerical methods for solving multi-dimensional problems for partial differential equations [1–9]. In particular, schemes using the characteristic equations of gas dynamics have found wide application [4–8]. Numerical methods, using the characteristics of hyperbolic equations, have definite advantages over ordinary finite-difference methods, in particular, in taking into account the physical nature of the problem, weak variation along the characteristics of some complexes of the required functions, and the possibility of predicting the instant when discontinuities will occur, for instance, suspended jumps. These advantages stand out very sharply in the solution of problems with two independent variables. But when the number of independent variables and the amount of information to be processed is increased a fixed choice of nodes becomes desirable, as in the network method. In this respect the numerical schemes of [6–8] for analyzing supersonic space flows of gas, for which a fixed network and the use of some form of characteristic relations are usual, are most effective. Obviously the name “method of characteristics” must be retained only for numerical schemes in which characteristics curves or surfaces are used in the process of computation. Then, numerical schemes, in which characteristic reations (conditions of compatibility) are used only for the derivation of difference equations at fixed nodes, can be considered as variants of the network method. This is even more advisable because for schemes of such a kind the problem of stability arises even in the case where the Courant-Friedrich-Levy (CFL) condition is satisfied, i.e. the region of dependence for the differential equation lies inside that of the difference equations [10–12]. In multi-dimensional cases there is a wide range in the choice of numerical methods based on characteristic relations, but obviously it is reasonable to have some fairly general and simple method of constructing explicit difference schemes for multi-dimensional problems based on characteristics of the same type as that put forward in [13] for a hyperbolic system in two variables (this method is described in [1, 14]). In the present paper it is proposed to construct such difference schemes for multi-dimensional quasilinear equations of a rather general form, using the lines of intersection of coordinate and characteristic surfaces and a fixed uniform network. To find the required functions at the points of contact of these lines, drawn from the points considered, with the preceding time layer, where the solution is already known, we can use linear or quadratic interpolation. Schemes I and II (Section 1) thus obtained, and also their modifications are investigated by an example of very simple equations with constant coefficients in Section 2. Difference schemes I and II obtained in Section 3 for the equations of gas dynamics are carried out numerically for a problem on the supersonic space flow round bodies (Section 4). The results given there indicate the possibility of applying Scheme I to the calculation of discontinuous solutions.