Investigation of Optimal Grid Distributions for Burgers' Equation

This paper explores the potential of numerical optimization methods to determine grid distributions that minimize truncation error. The technique is explored using 1D and 2D Burgers’ equations where an exact solution to a steady viscous shock is available. To accomplish the optimization the relationship between discretization error and truncation error is exploited. The truncation error is chosen to drive the optimization process because it serves as the local source for the discretization error and analytic expressions can be derived for a given equation set and the numerical scheme. A complete derivation of the truncation error terms for the 1D Burgers’ equation using second order accurate finite differences is presented. The importance of including discrete grid transformation metrics (i.e., derivatives of the grid distribution function) in the truncation error derivation is discussed. Methods for estimation of truncation error are also presented. In the case of the 2D Burgers’ equation a truncation error estimation method is applied to drive the optimization process. For the case considered, since an exact solution to the differential equation is available, the estimation method gives an exact evaluation of truncation error. To represent the grid distribution a spline method that preserves higher order continuous derivative connectivity is applied. The method is applied for both 1D and 2D cases. Several objective functions and spline applications are explored for the 1D problem and the best practices are applied to the 2D problem. For the 1D case a two order of magnitude reduction in discretization error was found when the optimal grid was compared to the initial uniform grid. For the 2D case improvement was not as substantial, not quite a full order of magnitude, due to the limited ability to morph the mesh under the constraints applied. I. Introduction ISCRETIZATION error is a direct result of solving a partial differential equation (PDE) with an approximate numerical technique. It is formally defined as the difference between the exact solution to the discretized PDE and the exact solution to the original governing PDE. Discretization error can be difficult to estimate and therefore it is often neglected. At best a grid resolution study is performed to make the case that the solution is converged with respect to grid size. This says nothing about the actual discretization error. To estimate discretization error from a grid resolution study the study must be carried out on at least two grids from the same family and the solution must be in the asymptotic range. To show that grids are asymptotic, at least three grids from the same family are required. Thus it is clear why discretization error is often ignored in complex CFD simulations. Unfortunately the discretization error and the lack of its characterization is a major factor contributing to the uncertainty in a given CFD prediction. In the current paper, past work by Roy (2010) that investigates discretization error, its estimation, and its relation to truncation error is expanded to drive optimal grid design. We argue that global grid refinement (over the entire domain) is non-trivial, especially for unstructured meshes, and results in refinement in regions where it is not needed adding additional computational expense for little gain. In this paper global refinement is not considered, but grid