Grid and Solution Adaptation via Direct Optimization Methods
At present all numerical schemes based on some form of differencing approach are plagued by some lack of accuracy when compared to the exact solution. This lack of accuracy can be attributed to the presence of truncation error in the numerical method. Traditionally the error can be reduced by increasing the number of mesh points in the discrete domain or by implementing a higher order numerical scheme. In recent times the approach has taken a more intelligent direction where adaptation or distribution of the mesh points is affected in such a way to reduce the error. However, grid adaptation with all its progress over the past few decades still has not been able to completely address the issue as to what constitutes a best grid. To address this issue, direct optimization approach is used, where the solution and the grid adapts such that an optimum and correct solution is obtained. For some numerical schemes the truncation error associated with the scheme can be easily separated to form a modified equation, while for others the procedure can prove tedious and too laborious. Nevertheless, the kernel of this study is to find some way to improve the accuracy of a numerical solution via optimization where the movement of the grid points is predicated on minimizing the difference between the exact and numerical form of the partial differential equation, thus delivering a more correct solution. A best mesh for a given problem will reduce the variation between the discrete form of the pde and its exact form and in the process deliver a more correct solution to the problem. The study will also illustrate that the best mesh obtained for a given problem may not be consistent with conventional wisdom. In grid generation in most cases a good mesh is aesthetically pleasing to the eye, however this study will show that a best mesh could just as well be a dirty mesh. For numerical schemes in which the modified equation can be obtained without severe complication the study will show that by minimizing the leading truncation error terms in a difference scheme by adaptation the numerical order of the scheme is increased. At present the study is confined to the two dimensional Laplace problem discretized by the generalized and non-uniform formulation, while the one dimensional problem is the linearized viscous Burgers' problem discretized by the first order Roe's finite volume method. The exact solution for both the methods exist for a complete comparison with the numerical results. The study strives to answer two important questions regarding grid adaptation: (i) The best grid may not be unique for all types of problems, but if there is a best grid how does one attain it? (ii) If a best grid exists, is it worth the computational effort to obtain it? The efficiency of the present method is strongly influenced by the choice of the optimization method and how the control vector is set up over the solution domain. This study includes details of the work done on this facet of the overall work.
Dissertation Abstracts International
Mahajan, Ajay Mohan, "Grid and Solution Adaptation via Direct Optimization Methods" (2006). Mechanical Engineering Faculty Research. 529.