Numerical error estimation for nonlinear hyperbolic PDEs via nonlinear error transport

Abstract

The estimation of discretization error in numerical simulations is a key component in the development of uncertainty quantification. In particular, there exists a need for reliable, robust estimators for finite volume and finite difference discretizations of hyperbolic partial differential equations. The approach espoused here, often called the error transport approach in the literature, is to solve an auxiliary error equation concurrently with the primal governing equation to obtain a point-wise (cell-wise) estimate of the discretization error. One-dimensional, nonlinear, time-dependent problems are considered. In contrast to previous work, fully nonlinear error equations are advanced, and potential benefits are identified. A systematic approach to approximate the local residual for both method-of-lines and space-time discretizations is developed. Behavior of the error estimates on problems that include weak solutions demonstrates that this nonlinear error evolution provides useful error estimates.

Accuracy of Discretization Error Estimation by the Error Transport Equation on Unstructured Meshes - Nonlinear Systems of Equations

“…As a corollary, there are some combinations of (p, q, r) that are clearly more computationally expensive but less accurate, for instance the (2, 4, 6) scheme compared to the (2,4,4) scheme. It addition to the difference in accuracy, it was observed that the superconvergent (p < q = r) schemes produce differences in exact error and error estimate that are smoother, while the other schemes have difference in error that are dominated by high frequency behavior.…”

“…Results for a typical (2,4,4) scheme applied here is shown in Figure 5. Furthermore, using p < q = r for a randomly perturbed grid gives O(h r ) convergence for the error estimate, which agrees with the results obtained previously for scalar problems, with the (2, 4, 4) case shown in Figure 6a and the (2, 6, 6) case shown in Figure 6b.…”

“…Banks et al 2 solved the error equation in a structured mesh context, and found that there are many schemes for which combinations of lower order solutions of the primal equation, the error equation, and sufficiently accurate residual evaluations led to a higher order error estimate. In the work of Hay and Visonneau, 3 some considerations were made for unstructured meshes, although obtaining higher-order estimates of discretization error was not a focus.…”

Truncation and Discretization Error for Diffusion Schemes on Unstructured Meshes

“…For the curved, stretched, and perturbed classes of meshes it was found that the truncation error is in general zeroth order accurate. By incorporating the jump term coecient α, it was found that for the specic uniform mesh, a particular choice of 4 3 gives a fourth order truncation error, and this motivated the search of optimal jump term coecients that minimize truncation error and discretization error for the vertex-centered unstructured mesh case.…”

“…It can be shown 3 that for the special case of a linear dierential operator L, the truncation error τ can be used to calculate the discretization error ε, as L(ε) = τ, (1) known as the error transport equation. For a general nonlinear dierential operator, Banks et al 4 noted that the error transport equation takes another form dierent from simply replacing L by N in Eq. (1).…”