On Efficiently Obtaining Higher Order Accurate Discretization Error Estimates for Unstructured Finite Volume Methods Using the Error Transport Equation

A numerical estimation of discretization error for steady compressible flow solutions is performed using the error transport equation (ETE). There is a deficiency in the literature for obtaining efficient, higher order accurate error estimates for finite volume discretizations using nonsmooth unstructured meshes. We demonstrate that to guarantee sharp, higher order accurate error estimates, one must discretize the ETE to a higher order than the primal problem, a requirement not necessary for uniform meshes. Linearizing the ETE can limit the added cost, rendering the overall computational time competitive, while retaining accuracy in the error estimate. For the Navier–Stokes equations, when the primal solution is corrected using this error estimate, for the same level of solution accuracy the overall computational time is more than two times faster compared to solving the higher order primal problem. In addition, our scheme has robustness advantages, because we solve the primal problem only to lower order.