Applications of the Unsteady Error Transport Equation on Unstructured Meshes

Abstract

A numerical study of using the error transport equation, an auxiliary problem to a set of model equations, is performed to obtain higher-order accurate error estimates and corrections for applications in unsteady compressible flow. In many such applications, a functional of the solution is often examined as a proxy for the accuracy of the solution itself. Several measures of time-dependent functionals are used for test cases that have a periodic steady-state solution. The approach is verified by examining unsteady functionals for diffusion and advection model problems, and then the error transport method is applied to the von Kármán vortex shedding test case, a difficult problem to establish accuracy properties on its own. It was found that scalar measures of the unsteady functionals not only give the order of accuracy that would be expected from a primal discretization only but also the expected higher-order accuracy if the functionals were computed using the solutions corrected by this accurate error estimate, all without the otherwise necessary requirement of discretizing to higher order in both time and space for the primal problem. The results are consistent with previous studies on solution error using simple test cases with manufactured solutions.