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科学和工程环境中。该提案的目标集中于开发用于此类方程数值求解的新算法和软件。主要研究途径有两个：（a）边值常微分方程组（BVODE）的数值解：我们的研究将集中在BVODE软件的改进上，例如误差控制策略的改进（特别是数值解残差的控制），问题敏感性评估（即调节）和全局误差估计的改进，以及当前可以处理的问题类别的扩展，例如高阶导数系统，更一般（耦合）边界条件，以及具有延迟和超前项的微分方程（其中导数的评估取决于问题域中多个点的解信息），（b）一维中时间相关偏微分方程（PDE）系统的数值解：我们的研究将集中于对 PDE 软件的改进，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来简化软件的使用，并推广我们的技术以允许处理具有两个空间维度的偏微分方程。这项工作的意义在于提高 BVODE 和 PDE 数值求解软件的易用性、效率、鲁棒性和能力，从而提高应用专家解决与更复杂的数学模型相关的更复杂的微分方程组的能力。

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.