The characteristic-based-split procedure: An efficient and accurate algorithm for fluid problems

In 1995 the two senior authors of the present paper introduced a new algori thm designed to replace the Taylor-Galerkin (or Lax-Wendroff) methods, used by them so far in the solution of compressible flow problems. The new algo rithm was applicable to a wide variety of situations, including fully incom pressible flows and shallow water equations, as well as supersonic and hype rsonic situations, and has proved to be always at least as accurate as othe r algorithms currently used. The algorithm is based on the solution of cons ervation equations of fluid mechanics to avoid any possibility of spurious solutions that may otherwise result. The main aspect of the procedure is to split the equations into two parts, (I) a part that is a set of simple sca lar equations of convective-diffusion type for which it is well known that the characteristic Galerkin procedure yields an optimal solution; and (2) t he part where the equations are self-adjoint and therefore discretized opti mally by the Galerkin procedure. It is possible to solve both the first and second parts of the system explicitly, retaining there the time step limit ations of the Taylor-Galerkin procedure. But it is also possible to use sem i-implicit processes where in the first part we use a much bigger time step generally governed by the Peclet number of the system while the second par t is solved implicitly and is unconditionally stable. It turns out that the characteristic-based-split (CBS) process allows equal interpolation to be used for all system variables without difficulties when the incompressible or nearly incompressible stage is reached. It is hoped that the paper will help to make the algorithm more widely available and understood by the prof ession and that its advantages can be widely realised. Copyright (C) 1999 J ohn Wiley & Sons, Ltd.