Adaptive hr-mesh refinement for the numerical solution of advection-diffusion equations

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DMS Award AbstractAward #: 0209313PI: Weiming CaoInstitution: University of Texas, San AntonioProgram: Computational MathematicsProgram Manager: Catherine MavriplisTitle: Adaptive hr-mesh refinement for the numerical solution of advection-diffusion equationsA fundamental issue in the numerical solutions of partial differential equations (PDEs) is adaptive mesh refinement. Local refinements (the h-methods) and moving mesh methods (the r-methods) are two basic types of approaches. Each has its features. The h-methods are robust and reliable. The r-methods can be highly efficient and effective for time dependent problems. The goal of this project is to develop a comprehensive combined h- and r-refinement method to achieve the advantages of both approaches. This combined method is especially useful in solving advection-diffusion problems, with which the numerical diffusion and advection speed can be significantly reduced. Major issues to be addressed in this research include: the error estimates used to guide the hr-refinement, stable and accurate time integration methods used in conjunction with the spatial hr-refinement; the conservative numerical schemes based on the hr-refined meshes; the efficient solution of the linear algebraic equations arising from the discretization; the implementation and software development. While the analysis and algorithms will be established in the context of general time dependent advection-diffusion problems, we shall focus on two specific applications: the contamination and remediation of ground water systems in environmental science, and the motion and deformation of white blood cells in biophysics.With the advances of modern computer technology, numerical experiments have now become one of the two primary tools in science, industry, and engineering (the other is physical experiments). Numerical simulation of many important processes involves the solution of advection-diffusion problems. Examples include optimal design of aircrafts and automobiles, forecast of global weather change, remediation of surface and ground water contamination, and innovation of biomedical devices. This project aims to develop a robust, reliable, and highly efficient mesh refinement method used for solving the advection-diffusion problems. Major issuesin the design, analysis, and implementation of this method will be addressed. In particular, it will be applied directly to solve environmental and biophysical problems. We anticipate that successful completion of this project will not only enrich the mathematical theory of adaptive solution of partial differential equations, but also generate highly efficient and accurate numerical algorithms, which will find applications in a variety of industrial and engineering areas that are essential to our national economy and security.Date: May 22nd, 2002