Software for Error Controlled Numerical Solution of Boundary Value Ordinary Differential Equations and Parabolic Partial Differential Equations

沿着理论和实验科学，计算建模现在被视为科学探究的第三个基本分支。在化学、物理、生物和金融等领域中出现的大多数计算模型都涉及复杂的微分方程（DE）系统，其近似解是通过使用复杂的数值软件获得的。 由于数值解是近似的，因此软件还必须评估计算解的质量。两种流行的评估类型是全局误差（近似解和精确解之间的差异）和缺陷（数值解不满足方程的量）。高质量的软件将调整计算，使得可以有效地获得数值解，并且使得全局误差或缺陷的估计满足用户提供的公差。 我们的研究将集中在数值软件的有效和准确的解决方案类型的DE，被称为边界值普通DE（BVODE）和抛物偏DE（PDE），其功能自适应控制的有效计算的估计的全局误差和/或缺陷。我们的工作的一个主要目标是开发软件，抛物偏微分方程，依赖于时间和两个空间维度，这将采用高精度的方法在时间和空间自适应控制准确和有效地计算估计的空间和时间误差。第二个主要目标是研究采用混合全局误差/缺陷控制的BVODE软件，并开发新的算法和软件来扩展目前可由BVODE求解器处理的问题类别，例如，具有周期边界条件或延迟和提前项的问题。这项工作的意义在于，它将提高易用性，效率，鲁棒性和能力的软件的数值解的BVODEs和偏微分方程，从而改善工具，可帮助计算科学家有效地处理复杂的计算模型在他们的调查领域。

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.