Estimation of Discretization Errors Using the Method of Nearby Problems

Abstract

The method of nearby problems is developed as an approach for estimating numerical errors due to insufficient mesh resolution. A key aspect of this approach is the generation of accurate, analytic curve fits to an underlying numerical solution. Accurate fits are demonstrated using fifth-order Hermite splines that provide for solution continuity up to the third derivative, which is recommended for second-order differential equations. This approach relies on the generation of a new problem (and corresponding exact solution) that is "nearby" the original problem of interest, and the nearness requirements are discussed. The method of nearby problems is demonstrated as an accurate discretization error estimator for steady-state Burgers's equation for a viscous shock wave at Reynolds numbers of 8 and 64. A key advantage of using the method of nearby problems as an error estimator is that it requires only one additional solution on the same mesh, as opposed to multiple mesh solutions required for extrapolationbased error estimators. Furthermore, the present results suggest that the method of nearby problems can produce better error estimates than other methods in the preasymptotic regime. The method of nearby problems is also shown to provide a useful framework for evaluating other discretization error estimators. This framework is demonstrated by the generation of exact solutions to problems nearby Burgers's equation as well as a form of Burgers's equation with a nonlinear viscosity variation. Nomenclature a i -f i = spline coefficients E = discrete L 2 norm error function f = general function h = global measure of cell or element size L = differential operator L ref = reference length of domain N = number of mesh points n = number of spline points p = order of accuracy r = grid refinement factor (r > 1) S = spline polynomial s = source term t = time coordinate, s u = local solution variable, m=s x = spatial coordinate, m = scaling constant = viscosity, m 2 =s = solution domain Subscripts exact = exact solution to differential equation i = spline zone number j = node number k = mesh level (1 finest mesh) Superscripts 0 = dimensionless variable, first derivative 00= second derivative 000 = third derivative = scaled variablẽ = numerical solution = estimated exact solution to differential equation

Discretization error estimation and exact solution generation using the method of nearby problems.

“…11 The results for steady-state Burgers equation with a Reynolds number of 64 are presented below in Figure 3. The numerical solution appears to be approximated quite well since the fifth-order Hermite spline fit with 65 spline points is visually indistinguishable from the underlying numerical solution.…”

“…Furthermore, the source terms become larger at the boundaries as the polynomial order is increased. When the fifth-order Hermite splines are implemented, 11 the magnitude of the source term is significantly smaller. The source term for the Reynolds number 8 case using 17 spline points is shown in Figure 11 along the entire domain.…”

On the generation of exact solutions for evaluating numerical schemes and estimating discretization error

“…MNP has been successfully demonstrated for one-dimensional problems [4]. Roy et al used MNP to estimate discretization errors in steady-state Burgers equation for viscous shocks.…”

“…Recently, an approach for estimating discretization error has been proposed by our group called the Method of Nearby Problems (MNP) [4]. MNP requires two numerical solutions on the same grid, thereby eliminating the problems associated with generating multiple grids that are all within the asymptotic grid convergence regime.…”

Estimation of grid-induced errors in computational fluid dynamics solutions using a discrete error transport equation

“…An approach similar to the DETE, called the method of nearby problems, has recently been introduced by Roy, et al [33][34][35]. Conceptually, the procedure of this method is to model the residual, then solve Eq.…”