Numerical Software for the Adaptive Error Controlled Solution of Ordinary and Partial Differential Equations

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Computational simulation of complex systems is now a central tool in all areas of scientific research. The complexity of these systems requires that numerical software be employed to obtain accurate solutions as efficiently as possible. ******Over my research career, my interests have focused on the development of numerical software for boundary value ordinary differential equations (BVODEs), one-dimensional time-dependent partial differential equations (1D PDEs), and two-dimensional time-dependent partial differential equations (2D PDEs). Our software had been applied in the solution of, e.g., tumor cell growth models, heart simulations, and in financial models. My research over the next several years will continue to focus on numerical software for BVODEs and PDEs:******BVODEs: Over the last 20 years I have developed error controlled numerical software packages based on special classes of Runge-Kutta (RK) methods; the most recent release of this package features options for global error control and defect control. (The defect is the amount by which the numerical solution fails to satisfy the BVODE.) Our new work will consider several improvements and extensions: (i) improved defect control schemes, (ii) new RK methods for stiff BVODEs, (iii) extensions to handle BVODEs with periodic boundary conditions, integral conditions, and/or delay and advance terms.******1D PDEs: Over the last 12 years I have developed a family of B-spline collocation software packages that provide both spatial and temporal error control. The most recent member of this family (which uses Backward Differential methods for time stepping) provides a Fortran 95 interface and features more efficient error estimation and control schemes. Our new work will consider extensions to improve the capabilities of this software: (i) new error estimation and control schemes within the RK time-stepping version of this software, (ii) automatic collocation order and error control scheme selection, (iii) handling of larger PDE systems, improved performance on problems with challenging initial conditions, and treatment of problems where termination depends on a solution-dependent condition.******2D PDEs: Our recent work has seen the development of prototype software based on extending the 1D algorithms to 2D. The current solver has a temporal error control capability but not a spatial error control capability. New work will focus on the development of spatial error estimation schemes and adaptive spatial error control.******The value of this work will to be to provide more robust and efficient software packages, able to handle more general problems classes, thus contributing to the software tools available to computational scientists, allowing them to consider more complex scientific investigations.