Towards higher order discretization error estimation by error transport using unstructured finite-volume methods for unsteady problems

A numerical estimation of discretization error for solutions to unsteady laminar compressible flow equations is performed using the error transport equation (ETE) on unstructured meshes. This method is an extension to our previous work on steady problems, where it was found that solving the ETE can be more efficient and robust than solving the higher order primal problem. Computing the time-dependent ETE source term accurately is critical to the accuracy of the discretization error estimate, and several methods of doing so are considered. It was found that computing the ETE source term directly by a finite-difference approximation in time gives accurate error estimates, which we show is equivalent to an accurate corrected solution. A truncation error analysis was performed for the ETE to determine the expected accuracy of the error estimate, where a term that mixes the space and time discretization was observed. Although more stringent requirements for error estimation are needed when using unstructured meshes, constant time steps can be used and the best schemes we found were still able to attain an estimate of the discretization error that is higher order accurate in space and time, without discretizing both to higher order. Furthermore, unlike unsteady adjoints, the ETE requires only one other auxiliary equation to be solved, agnostic to the choice and number of output functionals, and co-advancing with the primal problem requires the storage of only local solutions in time, reducing memory requirements.