Mathematical Sciences: Computational Error Estimation and Adaptive Error Control for Numerical Methods for Differential Equations

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Estep The investigator develops and implements accurate approximation methods for differential equations using a posteriori error estimates and adaptive error control. The main target is systems of reaction-diffusion equations. Such problems are important in practical terms because they occur as mathematical models in applied science and engineering, including applications in genetics, material science, chemistry, and biology, among others. The challenge is to compute accurate approximations of solutions that generically include multiple scales in their space and time behavior and whose behavior depends strongly on parameters prescribed as part of the model. Moreover, using computation as a scientific tool requires an estimate of the accuracy of the approximation. The approach to these problems is based on developing a posteriori error estimates that bound the error in terms of computable information obtained from the approximation once a computation is completed. The analysis takes into account both the difficulty of solving the differential equation over a small interval and the global accumulation of errors. In particular, the stability properties of the solution being approximated are measured by auxilary computations performed during the approximation. The result is robust and reliable computational error estimates. In addition, the investigator examines the dynamical properties of numerical schemes in the context of obtaining schemes with improved accuracy for a specified problem and obtaining more accurate error estimates for such schemes. The third component of the project is the development and implementation into code of adaptive error control algorithms based on the a posteriori error estimates. The ultimate goal of this project is the public release of a parallel code that can solve systems of reaction-diffusion equations in two and three dimensions reliably and efficiently. Mathematical models in applied science, including genetic s, material science, chemistry, and biology, are often expressed as nonlinear reaction-diffusion differential equations that contain source terms balanced against terms that diffuse energy. The goal of such modelling is to describe the physical situation in terms of the solution of the differential equation. However, the nonlinear nature of most models makes it impossible to solve the equations explicitly; consequently numerical approximation is an important tool in science. This approach has its own difficulties. The balance between reaction and diffusion is usually delicate and difficult to handle accurately. Moreover, solutions of such problems typically evolve on several scales, i.e. some interesting behavior occurs in very localized regions in space and time while other behavior evolves over long times or over larger regions in space. The use of a uniform numerical discretization for a real application results in huge computations that tax even the largest computers. The investigator aims to produce numerical schemes that adapt themselves to the localized behavior of the target solution so as to make the computations both as accurate as desired and as efficient as possible. Another benefit is that the estimate of the accuracy can then be reported, which increases the scientific level of numerical analysis. The mathematical approach is develop estimates of the error that use information obtained from the approximation, which can then be used to adapt the discretization, that is make the computations self-governing. The investigator also is implementing this theory in a code for parallel computers that can solve very general problems with minimum user input. The intent is to make the code publicly available, yielding a scientific tool that benefits the engineering and scientific infrastructure.