Numerical Software for the Adaptive Error Controlled Solution of Ordinary and Partial Differential Equations

复杂系统的计算模拟现在是所有科学研究领域的核心工具。这些系统的复杂性要求使用数值软件尽可能高效地获得精确的解。******在我的研究生涯中，我的兴趣集中在边值常微分方程（BVODEs），一维时变偏微分方程（1D PDEs）和二维时变偏微分方程（2D PDEs）的数值软件的开发上。我们的软件已应用于解决方案，例如，肿瘤细胞生长模型，心脏模拟和金融模型。在接下来的几年里，我的研究将继续集中在BVODEs和PDEs的数值软件上：******BVODEs：在过去的20年里，我开发了基于特殊类龙格-库塔（RK）方法的误差控制数值软件包；这个包的最新版本提供了全局错误控制和缺陷控制的选项。（缺陷是数值解不能满足BVODE的量。）我们的新工作将考虑几个改进和扩展：(i)改进缺陷控制方案，（ii）刚性BVODEs的新RK方法，（iii）扩展处理具有周期边界条件，积分条件和/或延迟和提前条件的BVODEs。******1D PDEs：在过去的12年里，我开发了一系列b样条搭配软件包，提供空间和时间误差控制。该系列的最新成员（使用反向微分方法进行时间步进）提供了Fortran 95接口，并具有更有效的误差估计和控制方案。我们的新工作将考虑扩展以提高该软件的功能：(i)该软件的RK时间步进版本中的新误差估计和控制方案，（ii）自动搭配顺序和错误控制方案选择，（iii）处理更大的PDE系统，在具有挑战性初始条件的问题上提高性能，以及处理终止依赖于解依赖条件的问题。******2D PDEs：我们最近的工作是开发基于将1D算法扩展到2D的原型软件。目前的求解器具有时间误差控制能力，但不具备空间误差控制能力。新的工作将集中在空间误差估计方案和自适应空间误差控制的发展。******这项工作的价值在于提供更健壮和高效的软件包，能够处理更一般的问题类，从而为计算科学家提供可用的软件工具，使他们能够考虑更复杂的科学研究。

Computational simulation of complex systems is now a central tool in all areas of scientific research. The complexity of these systems requires that numerical software be employed to obtain accurate solutions as efficiently as possible. ******Over my research career, my interests have focused on the development of numerical software for boundary value ordinary differential equations (BVODEs), one-dimensional time-dependent partial differential equations (1D PDEs), and two-dimensional time-dependent partial differential equations (2D PDEs). Our software had been applied in the solution of, e.g., tumor cell growth models, heart simulations, and in financial models. My research over the next several years will continue to focus on numerical software for BVODEs and PDEs:******BVODEs: Over the last 20 years I have developed error controlled numerical software packages based on special classes of Runge-Kutta (RK) methods; the most recent release of this package features options for global error control and defect control. (The defect is the amount by which the numerical solution fails to satisfy the BVODE.) Our new work will consider several improvements and extensions: (i) improved defect control schemes, (ii) new RK methods for stiff BVODEs, (iii) extensions to handle BVODEs with periodic boundary conditions, integral conditions, and/or delay and advance terms.******1D PDEs: Over the last 12 years I have developed a family of B-spline collocation software packages that provide both spatial and temporal error control. The most recent member of this family (which uses Backward Differential methods for time stepping) provides a Fortran 95 interface and features more efficient error estimation and control schemes. Our new work will consider extensions to improve the capabilities of this software: (i) new error estimation and control schemes within the RK time-stepping version of this software, (ii) automatic collocation order and error control scheme selection, (iii) handling of larger PDE systems, improved performance on problems with challenging initial conditions, and treatment of problems where termination depends on a solution-dependent condition.******2D PDEs: Our recent work has seen the development of prototype software based on extending the 1D algorithms to 2D. The current solver has a temporal error control capability but not a spatial error control capability. New work will focus on the development of spatial error estimation schemes and adaptive spatial error control.******The value of this work will to be to provide more robust and efficient software packages, able to handle more general problems classes, thus contributing to the software tools available to computational scientists, allowing them to consider more complex scientific investigations.