Moving Mesh Methods for Numerical Solution of Time Dependent Partial Differential Equations in Two and Three Spatial Dimensions

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9626107 Huang The investigator develops a moving mesh method for the numerical solution of two- and three-dimensional time-dependent PDEs (partial differential equations) that extends the MMPDE (Moving Mesh PDE) approach introduced by the investigator and his coworkers for PDEs in one spatial dimension. With this approach, an MMPDE is formulated explicitly based upon the heat flow equation and the theory of harmonic maps. The MMPDE is employed to move the mesh, to concentrate nodes in regions where the physical solution has large variations, and to control certain mesh qualities, in particular the skewness. Techniques are investigated to discretize and solve efficiently the extended system consisting of the underlying physical PDE and the MMPDE. A number of related issues are considered - e.g., the computation of a reference mesh for an arbitrary connected domain, the use of flow directional control, the study of preconditioning techniques, the examination of the space-time finite element method, and the investigation of robustness and reliability of the method. Moreover, the method is applied to problems in airfoil analysis and wing design, moving boundary problems, and problems involving blow-up solutions. The new algorithms and computational techniques are simultaneously implemented in mathematical software that could find widespread usage among scientists and engineers. This project is concerned with the development of new computational methods that are essential to enhance the ability of scientists and engineers to solve large scale computational problems that are crucial to our economy, environment, and security. The research is focused on development of so-called adaptive numerical techniques, where the special features of the particular problem being solved are adapted to. More specifically, the places (mesh points) where the solution to the scientific problem is being approximated are moved with time to adapt to the changes in the solu tion. Mesh adaptation has recently played an indispensable role in the numerical solution of the problems, as, supercomputers notwithstanding, such problems can generally not otherwise be solved satisfactorily. This is because in many problems of science and engineering, there is a small portion of the physical domain where large changes in the solution occur over very small separations in the mesh. Numerical solution of these problems using fixed uniform meshes is formidable because millions of mesh points are required to resolve the physical phenomena. On the other hand, use of adaptive mesh methods can significantly reduce the number of mesh points and thus economies can be gained. Unlike other adaptive mesh methods, the moving mesh method under study changes the mesh smoothly in time to adapt to the key solution features. It is suitable for parallel computing. The moving mesh method should be very useful in the numerical simulation of many industrial manufacturing problems. A particular area is airfoil analysis and wing design. For example, in the airfoil and wing design the airfoil or wing shape is computed when a pressure distribution is specified. It will save computing resources tremendously if the flow solution and the computational mesh, and hence the airfoil shape, can be updated simultaneously, as these methods do. This advantage holds for a variety of other applications as well, such as the problem of studying how and why industrial and house fires start. The investigator believes that a `moving mesh' approach is the best way to reduce computing time and improve accuracy, and that the actual technology transfer for these methods is close at hand.