Discrete-error transport equation for error estimation in CFD

With computational fluid dynamics (CFD) becoming more accepted and more widely used in industry for design and analysis, there is increasing demand for not just more accurate solutions, but also error bounds on the solutions. One major source of error is from the grid or mesh. A number of methods have been developed to quantify errors in solutions of partial differential equations (PDEs) that arise from poor-quality or insufficiently fine grids/meshes. For PDEs of interest to CFD, it has been shown that the error at one location could be generated elsewhere and then transported there, and thus is not a function of the local mesh quality and the local solution. So, a transport equation for error is needed to understand the generation and evolution of errors. Error transport equations have been developed for finite-element methods but not for finite-difference (FD) and finite-volume (FV) methods.; In this study, a method is developed for deriving error-transport equations for estimating grid-induced errors in solutions obtained by using FD and FV methods. The error-transport equations derived are discrete in that they depend only on the FD or FV equations and are independent of the PDEs that the FD or FV equations are intended to represent. The usefulness of the DETEs developed was evaluated through test problems based on four one-dimensional (1-D) and two two-dimensional (2-D) PDES. The four 1-D PDEs are the advection-diffusion equation, the wave equation, the inviscid Burgers equation, and the steady Burgers equation. The two 2-D PDEs are the 2-D advection-diffusion equation and the system of Euler equations. For PDEs that are not linear, linearization procedures were proposed and examined. For all test problems based on 1-D PDEs, the residual is modeled by the leading term of the remainder in the modified equation for the FD or FV equation. The residual was also modeled by using functional relationship suggested by data mining, where actual residuals generated by the numerical experiments were fitted by using least-square minimization. For all test problems, grid-independent solutions were generated to assess how well the residuals are modeled and how well grid-induced errors are predicted by the DETEs.; Results obtained show that if the actual residuals are used, then the DETEs can predict the grid-induced errors perfectly. This is true for all test problems evaluated, including those based on PDEs that are nonlinear and have time derivatives and for test problems with weak solutions. Results obtained also show that the leading terms of the modified equation is useful in modeling the residual if the grid spacing or cell size is sufficiently small so that the leading terms are bounded, a condition that is often not satisfied in practice. The usefulness of data mining in constructing residuals show the power-law to produce better fit than local linear least square of smoothness, resolution, aspect ratio and solution gradient. However, a more extensive database is needed before this approach can be expected to yield a more generally applicable models for the residual. The usefulness of Euler DETE in predicting grid-induced errors in the Navier-Stokes solutions was also examined. Results obtained show that error predicted by Euler DETE matches very well with the actual error for the high-Reynolds-number Navier-Stokes solutions.