Computer Methods in Applied Mechanics and Engineering 100 (1992) 45-62 North-Holland CMA 262 A new approach to algorithms for convection problems which are based on exact transport + projection Claes Johnson Department of Mathematics, Chalmers University of Technology, 412 96 G6teborg, Sweden Received 19 December 1990 We present a new approach to algorithms for convective problems which are based on 'exact transport + projection', such as the characteristic Galerkin or modified method of characteristics. We show that in model cases such algorithms also arise by applying the streamline diffusion finite element method on space-time meshes oriented along characteristics. We exhibit some of the advantages of the latter approach concerning generality, design of artificial viscosity and precision of analysis. 1. Introduction A basic principle for the design of numerical methods for time-dependent convection- dominated flow problems may be stated as 'exact transport + projection'. Methods based on this principle have roughly the following form, where {tn} is an increasing sequence of discrete time levels and {Vn} is a corresponding sequencc~ of piecewise polynomial spaces on meshes (Exact transport) Given the approximate solution U~ E V~ at time t. solve the flow equations exactly on the time interval (to,, tn+i) with initial data U~ to give the solution Un+t at time t.+t. (1.1a) (Projection) Compute U. +I - P. +t U~+l where P. +i is a projection into V. +t. (1.1b) Different methods of this form may differ in the choice of piecewise polynomial spaces V n (degree of polynomials, continuous or discontinuous polynomials) and in the projections pn. We recall that the basic Godunov method for compressible flow from the 1950s has the form (1.1) with Vn consisting of piecewise constants, Pn the L2-projection and where the exact solution phase involves solving local Riemann problems. Generalizations of the Godunov method using discontinuous piecewise linears or parabolics and modified projections involving 'flux-correction' have been developed in the last decades and are used extensively in Correspondence to: Dr. Claes Johnson, Department of Mathematics, Chalmers University of Technology, 412 96 G6teborg, Sweden. 0045-7825/92/$05.00 © 1992 Elsevier Science Publishers B.V. All fights reserved