Adaptive Solution of Partial Differential Equations on Parallel Computers Using An Equational Language

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The goal of this project is to develop a software environment where scientists and engineers need not know intricate numerical and programming details in order to efficiently solve computational problems involving partial differential equations on parallel computers. The basic premise is that numerical software consists of two parts: a core which is invariant for a group of related methods designed for different architectures and an architecturally dependent part. Traditional languages tend to cloud common features of the software and interweave the two parts. This project aims at building a new language based on the assertive programming paradigm and at searching for unified principles for designing efficient parallel procedures for solving systems of partial differential equations. In the assertive programming paradigm, computations are specified as sets of assertions about properties of the solution, and not as detailed procedural implementations. Architecture and implementation language-dependent procedures are automatically generated from the assertive description. Assertive programming for parallel scientific processing is supported by equational languages in which assertions are expressed as algebraic equations. In this research, an Equational Programming Language (EPL) system is being built to (i) provide the tools for users to specify parallel numerical algorithms in an architecture-independent way and (ii) develop tools for automatic generation of architecturally dependent parts of those numerical algorithms. Adaptive methods for partial differential equations use local information about the computed solution and its discretization error to automatically refine meshes, redistribute meshes, and/or vary the numerical method in different parts of the problem domain. The project continues the investigation of parallel adaptive techniques for two- and three- dimensional partial differential systems. Particular studies include dynamic scheduling and load balancing techniques based on using local error estimates to predict the work remaining to solve a problem, parallel iterative techniques for algebraic systems, and parallel algorithms for finite quadtree and octree structured meshes. Newly designed procedures will be implemented using the EPL system.