Discretization Error Estimation Using the Unsteady Error Transport Equations

Abstract

The focus of this work is on discretization error estimation for time-dependent simulations. Based on previous work on steady-state problems, the unsteady Error Transport Equations (ETE) are used to generate local discretization error estimates for a finite volume CFD code SENSEI. For steady-state problems, the ETE only need to be solved once after the solution has converged, whereas the unsteady ETE need to be co-advanced with the primal solve. All the test cases chosen in this work have known analytical solutions so that order of accuracy test can be performed and the accuracy of the error estimates can be unambiguously determined. The 2D convected vortex is used as the test case for inviscid flow. A Cross-Term Sinusoidal (CTS) manufactured solution for the laminar Navier-Stokes equations is used as the test case for viscous flow. Order of accuracy of the corrected solution is used to assess the quality of the error estimate. When iterative correction is not applied, higher-order convergence rate has been observed for the 2D convected vortex test case. For the 2D CTS manufactured solution higher-order convergence rate can also be observed but not for the finest grid levels. The current implementation of iterative correction is less stable than the primal solve but can improve the discretization error estimate in general. After iterative correction, the discretization error estimate of the unsteady ETE is higher-order for all grid levels for the 2D CTS manufactured solution.