Robustness and efficiency improvements and problem class extensions for numerical software for differential equations

常微分方程和偏微分方程出现在各种各样的科学和工程设置。本提案的目标集中在为这些方程的数值解开发新的算法和软件。有两种主要的研究途径：(a)边值常微分方程系统的数值解；我们的研究将集中在BVODE软件的改进，例如，误差控制策略的改进（特别是在数值解的残差控制方面），问题敏感性评估（即条件作用）和全局误差估计的改进，以及目前可以处理的问题类别的扩展，例如，高阶导数系统，更一般的（耦合）边界条件，具有延迟和提前项的微分方程（其中导数的求值取决于问题域中若干点的解信息），(b)一维空间中时变偏微分方程系统的数值解；我们的研究将集中在PDE软件的改进上，例如，更有效地估计空间全局误差，估计和控制全局时间误差，提高效率或稳定性的时间积分器，通过使用更强大的计算机语言简化软件的使用，以及推广我们的技术以允许处理两个空间维度的PDE。这项工作的意义在于提高BVODEs和pde数值解软件的易用性、效率、稳健性和能力，从而提高应用专家解决更复杂的微分方程系统与更复杂的数学模型的能力。

Ordinary and partial differential equations arise in a wide variety of scientific and engineering settings. The objectives of this proposal focus on the development of new algorithms and software for the numericalsolution of such equations. There are two main avenues of investigation:(a) The numerical solution of systems of boundary value ordinary differential equations (BVODEs): Our investigations will focus on improvements to BVODE software, e.g., improvements in error control strategies (specifically in the control of the residual of the numerical solution), improvements in assessment of problem sensitivity (i.e. conditioning) and global error estimation, and extension of the problem class that can currently be treated, e.g. higher order derivative systems, more general (coupled) boundary conditions, and differential equations with delay and advance terms (in which the evaluation of the derivative depends on solution information at several points in the problem domain),(b) The numerical solution of systems of time-dependent partial differential equations (PDEs) in one space dimension: Our investigations will focus on improvements to PDE software, e.g., more efficient estimation of the spatial global error, estimation and control of global temporal error, time integrators with improved efficiency or stability, simplification of the usage of the software through the use of a more powerful computer languages, and the generalization of our techniques to allow the treatment of PDEs with two spatial dimensions. The significance of this work will to be to increase the ease-of-use, efficiency, robustness and capability of software for the numerical solution of BVODEs and PDEs, thereby increasing the ability of applications experts to solve more complicated systems of differential equations associated with more sophisticated mathematical models.