Computation of one‐dimensional compressible flows including shocks

Abstract

Abstract The solutions of problems involving compressible flows have different functional forms in different regions, separated not only by discontinuities of the fluid itself but also by shock fronts and by characteristics starting from the origins of shock fronts and from points where such characteristics meet shocks. To obtain accuracy in numerical integration, small intervals and formulas using differences of low order are convenient immediately after crossing these lines, but much larger intervals, with differences of higher order, are sufficient elsewhere. The purpose of this paper is to explain a method of exploiting the economy made possible in this way. If the equations for the one‐dimensional case are transformed to characteristic variables, they form a ‘canonical hyperbolic system’. Adams' method for ordinary differential equations extends to a variable interval by using divided differences of the derivative, and it then becomes practical to use it for starting and across discontinuities. Further, it applies directly to canonical hyperbolic systems. Numerical solution is then very direct over regions determined by two characteristics and is convenient where the flow crosses already existing shocks. Finally, the origin of a new shock in any interval can be detected and arrangements can be made for it.