Grant: Discovery Grants Program - Individual

Along with theoretical and experimental science, computational modeling is now viewed as a third fundamental branch of scientific inquiry. A majority of computational models, arising in such areas as chemistry, physics, biology, and finance, involve complex systems of differential equations (DEs), for which approximate solutions are obtained through the use of sophisticated numerical software. Because the numerical solutions are approximate, it is essential that the software also assess the quality of the computed solution. Two popular types of assessment are the global error (the difference between the approximate and exact solutions) and the defect (the amount by which the numerical solution fails to satisfy the equations). Good quality software will adapt a computation so that the numerical solution can be obtained efficiently and so that an estimate of the global error or defect satisfies a user-provided tolerance. Our research will focus on numerical software for the efficient and accurate solution of types of DEs, known as boundary value ordinary DEs (BVODEs) and parabolic partial DEs (PDEs), that features adaptive control of efficiently computed estimates of the global error and/or defect. One major goal of our work is to develop software, for parabolic PDEs that depend on time and two spatial dimensions, that will employ high accuracy methods in time and space to adaptively control accurate and efficiently computed estimates of the spatial and temporal errors. A second major goal is to investigate software for BVODEs that employs hybrid global error/defect control and to develop new algorithms and software to extend the problem class currently treatable by BVODE solvers to include, e.g., problems with periodic boundary conditions or with delay and advance terms. The significance of this work is that it will improve the ease-of-use, efficiency, robustness, and capability of software for the numerical solution of BVODEs and PDEs, thereby improving the tools available to help computational scientists efficiently treat sophisticated computational models arising in their areas of investigation.