The need to numerically solve initial value problems for large stiff systems of ordinary differential equations (ODEs) arises in an overwhelming majority of scientific and engineering fields. Traditionally implicit solvers have been used to overcome the stability restrictions on the time step and improve computational efficiency compared to explicit schemes. Recently, however, exponential integrators emerged as an efficient alternative to commonly used techniques. While several of such integrators have been proposed, significant research efforts are needed to construct, analyze and optimize these integrators. The PI proposed a class of new exponential propagation iterative schemes of Runge-Kutta type (EPIRK) which are designed to maximize computational efficiency for solving very large stiff systems. The research sponsored by this grant will produce new EPIRK methods designed to have good scalability on parallel high-performance computing platforms. The study of the new integrators will produce methodologies for construction of efficient exponential schemes and explore properties of these methods. An important part of the project is the development of adaptive versions of the algorithms and implementation of new integrators as a general-use software package for both serial and parallel platforms. The software will be used to study several application problems in plasma physics and biomodeling and will be made widely available.
In a variety of scientific and engineering applications researchers want to predict behavior of complex systems which evolve on a wide range of temporal and spatial scales. Such systems, for instance, arise in many geo-engineering applications such as storing greenhouse gases, extraction of oil and gas from highly porous and fractured media or managing groundwater resources. Another example of a multiscale problem is modeling magnetic reconnection, one of the most fundamental processes in astrophysical and laboratory plasmas that governs such important phenomena as solar flares, magnetic substorms in the Earth?s magnetosphere and dynamics of magnetic fusion experiments. Computer modeling has become an essential tool in studying such systems. However, in order to be able to simulate the behavior of these systems on a computer advanced mathematical tools that offer exceptional efficiency on high-performance computing platforms have to be developed. The numerical techniques that will result from this project will allow prediction of the behavior of a wide range of complex systems of scientific and engineering interest over the parameter regimes inaccessible to standard methods.