Parallel Nonlinear Elimination Methods and Software for Partial Differential Equations

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Nonlinear Partial Differential Equations (PDEs) are the basic mathematical description for a wide variety of important application areas. In particular, this project will consider PDEs that arise in fluid dynamics, biology, and radiation diffusion. Because of their complexity, these equations can only be solved numerically by computers, and because of their particular properties (shock waves, sharp fronts, and local singularities) they are difficult to solve even then. This project will design, analyze, and implement software for a class of iterative methods to numerically solve nonlinear PDEs. The software will be provided in two forms - Matlab codes and a package interoperating with the PETSc library - for other researchers to apply the methods.Technically, the project will study a class of nonlinear elimination algorithms for solving algebraic nonlinear equations with unbalanced nonlinearities. The elimination methods avoid traditional methods' slow convergence when local singularities appear by identifying "misscaled" nonlinear components and replacing them with a function of the remaining more uniformly scaled components. The family of algorithms thus devised will obtain parallelism from domain decomposition, scalability (with respect to problem size) from multilevel methods, and robustness (against local singularities) from incomplete elimination. The methods will be tested on three important classes of applications: transonic compressible flows (CFD), electric wave problems in the heart (computational biology), and Marshak wave problems (radiation transport). The proposed algorithm and software development will have a great impact on the three applications, and will also have substantial influence on other areas of computational science where large nonlinear equations need to be solved.