The first goal of the project is to develop new domain decomposition algorithms and software for the numerical solution of some highly nonlinear, coupled systems of partial differential equations arising from multi-physics applications. For simple problems, implicit methods are relatively easy to develop from a given explicit or semi-implicit method, but for some multi-physics problems it is quite difficult to develop a fully implicit method that allows a high performance implementation. Most of the existing techniques are non-smooth and therefore difficult to solve with Newton type solver. In the project, some discretization techniques will be developed that involve high order non-smooth discretization for capturing the domain information, and low order locally smooth discretization for building the Jacobian system of the Newton iterations. The low order discretization serves as a nonlinear preconditioner that speeds up the convergence, but doesn't change the accuracy of the solution. The second goal of the project is to develop an efficient implementation of the proposed linear and nonlinear preconditioning approaches on high performance computers with a large number of processors.
Relatively mature technologies including algorithms and software are available for solving many types of single physics problems, but for coupled multi-physics problems, robust and scalable techniques are badly needed, especially for large scale parallel computers with accelerators. The proposed algorithms and software will have a great impact on several important application areas, such as the simulation of global atmospheric flows and the bio-fluids, and will also have substantial influence on other areas of computational sciences where large linear and nonlinear equations need to be solved. To broaden the impact of the research, the software will be made fully compatible with the widely used PETSc package. The research is a rich area in opportunities for both graduate and undergraduate students interested in high performance computing and general Computational Science and Engineering.
