Mathematical Sciences: Domain Decomposition Algorithms for Linear Hyperbolic Equations

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The use of parallel computers for solving partial differential equations is important in areas such as fluid dynamics, transonic flow, reservoir simulation, weather prediction and structural analysis, where many of the problems of interest can not be solved without the use of supercomputers. One technique for applying parallel computers to the solution of these problems is known as domain decomposition where the domain of interest is subdivided into several smaller subdomains and the task of solving the partial differential equation on each subdomain problem is assigned to a different processor. The global solution is then pieced together from the solutions computed on the individual subdomains. Much of the current work in the application of domain decomposition techniques has been in the area of elliptic partial differential equations with very little attention being given to hyperbolic equations. The investigator proposes to use the methods of domain decomposition for the solution of linear hyperbolic equations. The idea of using overlapping domains is introduced in the context of linear hyperbolic equations to develop a domain decomposition algorithm which is shown to be well suited for parallel processors. The questions of communication costs and load balancing is presented. The investigator also propose to implement parallel algorithms usong domain decomposition techniques in an existing distributed computing environment. Domain decomposition algorithm are important for the areas of applications mentioned above.