Estimating Grid-Induced Errors in CFD by Discrete-Error-Transport Equations

Abstract

This paper is dedicated to the memory of Derlon Chu (Nov. 20, 1959-June 19, 2003)-a colleague and a friend who has contributed much to make CFD impact the design of automotive components. Derlon Chu spearheaded the formation of the "Partnership on CFD Codes and Models for the Automotive Industry" to address error issues in CFD. This paper presents and evaluates a method for estimating grid-induced errors in CFD solutions that recognizes error at one location in the flow domain may not be generated there, but rather generated elsewhere and then transported there. This paper derives a system of discrete error-transport equations (DETEs) to compute the evolution of grid-induced errors in finite-volume solutions of the Euler equations for compressible flows in two dimensions. The finite-volume method to which the DETEs were derived is one which can be applied to structured or unstructured meshes with cells that can be triangular, rectangular, or other polygons. Results for a test problem involving an oblique shock wave show that if the residuals in the DETEs are modeled accurately, then the DETEs can predict grid-induced errors accurately.

Numerical error estimation for nonlinear hyperbolic PDEs via nonlinear error transport

“…In contrast and in a completely analogous way, one can develop a discrete error equation directly by beginning with the difference equation solved exactly by the discrete solution and introducing a discrete form of (2). In this case, the error equation is driven by the truncation error instead of the residual, but the evaluation of the source term is still an important issue [11,[13][14][15]18]. Such methods have been referred to as "Discrete Error Transport Equation" or "Approximate Operator Residual" methods.…”

“…We assume thatũ i is a discrete approximation ofũ that satisfies (18). Thus, following the discussion in Section 3.1, the primal discretization (18) provides a discrete expression of the first term in (21).…”

“…The source of error is the residual of the approximate solution, and, in the nonlinear case, the error evolves by a different differential operator than the solution. In the linear error transport approach [10][11][12][13][14][15][16][17][18], the differential operator on the left-hand side of (4) is linearized aboutũ , which is a reasonable assumption so long as |e(x, t)| |u(x, t)| . In practice, and in some very important cases, this is not valid, and such a linearization becomes questionable.…”

“…The linear error transport approach has been applied to the steadystate [12,[16][17][18] and time-dependent [10] Euler and Navier-Stokes systems that describe fluid flow. We are particularly interested in the time-dependent, nonlinear hyperbolic case because of its admission of weak solutions, but we wish to first restrict our consideration to a scalar equation for clarity.…”

“…In all cases in the error transport literature, the error equations are linearized, and the focus is typically on the approximation used to evaluate the local residual, that is, the local source (sink) of error that drives an error transport equation. Linear error transport techniques have been applied to linear and nonlinear models of advection [10,11] and advection-diffusion [11][12][13][14][15][16] equations, subsonic and supersonic inviscid flow [10,17,18], and subsonic viscous flow [12,16]. Most investigations have considered steady-state solutions [13][14][15][16][17][18], and only a few examples of timedependent applications [10,11] exist.…”