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

与理论和实验科学一样，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、生物和金融等领域产生的大多数计算模型都涉及到复杂的微分方程组(DES)，这些微分方程组的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整体误差(近似解和精确解之间的差异)和缺陷(数值解未能满足方程的程度)。高质量的软件将调整计算，以便可以有效地获得数值解，并且使得对全局误差或缺陷的估计满足用户提供的容差。 我们的研究将集中在高效和准确地求解DES类型的数值软件上，这些DES被称为边值普通DES(BVODE)和抛物线部分DES(PDE)，其特点是自适应控制高效计算的全局误差和/或缺陷的估计。我们工作的一个主要目标是为依赖于时间和二维空间的抛物型偏微分方程组开发软件，该软件将使用时间和空间上的高精度方法来自适应地控制对空间和时间误差的准确和高效的计算估计。第二个主要目标是研究用于BVODE的采用混合全局错误/缺陷控制的软件，并开发新的算法和软件以扩展BVODE解算器目前可处理的问题类别，以包括例如具有周期性边界条件或具有延迟和提前项的问题。这项工作的意义在于，它将提高BVODE和PDE数值解软件的易用性、效率、健壮性和能力，从而改进可用于帮助计算科学家有效地处理其研究领域中出现的复杂计算模型的工具。

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.