Application of the Space–Time Conservation Element and Solution Element Method to One-Dimensional Convection–Diffusion Problems

Abstract

In the space–time conservation element and solution element (CE/SE) method, the independent marching variables used comprise not only the mesh values of the physical dependent variables but also, in contrast to a typical numerical method, the mesh values of the spatial derivatives of these physical variables. The use of the extra marching variables results from the need to construct the two-level, explicit and nondissipative schemes which are at the core of the CE/SE development. It also results from the need to minimize the stencil while maintaining accuracy. In this paper, using the 1D a–μ scheme as an example, the effect of this added complication on consistency, accuracy, and operation count is assessed. As part of this effort, an equivalent yet more efficient form of the a–μ scheme in which the independent marching variables are the local fluxes tied to each mesh point is introduced. Also, the intriguing relations that exist among the a–μ. Leapfrog, and DuFort–Frankel schemes are further explored. In addition, the redundancy of the Leapfrog, DuFort–Frankel, and Lax schemes and the remedy for this redundancy are discussed. This paper is concluded with the construction and evaluation of a CE/SE solver for the inviscid Burgers equation.