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

常微分方程自适应误差控制解的数值软件

• 批准号: RGPIN-2017-05811
• 负责人: Muir, Paul
• 金额: $ 1.89万
• 依托单位: Saint Mary's University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=681607
• 关键词: Numerical; Software; Adaptive; Error; Controlled

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.

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采用数值软件来尽可能有效地获得准确的解决方案。 ******在我的研究生涯中，我的兴趣集中在开发边值常微分方程 (BVODE)、一维时间相关偏微分方程 (1D PDE) 和二维时间数值软件相关偏微分方程（2D PDE）。 我们的软件已应用于肿瘤细胞生长模型、心脏模拟和金融模型等解决方案中。在接下来的几年里，我的研究将继续集中在 BVODE 和 PDE 的数值软件上：******BVODE：在过去的 20 年里，我开发了基于特殊类 Runge-Kutta (RK) 的误差控制数值软件包。 ） 方法;该软件包的最新版本具有全局错误控制和缺陷控制选项。 （缺陷是数值解无法满足 BVODE 的量。）我们的新工作将考虑多项改进和扩展：(i) 改进的缺陷控制方案，(ii) 用于刚性 BVODE 的新 RK 方法，(iii) 扩展处理具有周期性边界条件、积分条件和/或延迟和超前项的 BVODE。******一维偏微分方程：在过去 12 年中，我开发了一系列 B 样条搭配软件包，提供空间和超前条件。时间误差 控制。该系列的最新成员（使用向后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错误估计和控制方案，（ii）自动搭配顺序和错误控制方案选择，（iii）处理较大的 PDE 系统的性能，提高了具有挑战性的初始条件问题的性能，以及处理终止取决于解决方案相关条件的问题。******2D PDE：我们最近的工作已经看到了基于扩展的原型软件的开发将 1D 算法转换为 2D。当前的求解器具有时间误差控制能力，但不具有空间误差控制能力。新工作将侧重于空间误差估计方案和自适应空间误差控制的开发。******这项工作的价值将是提供更强大和更高效的软件包，能够处理更一般的问题类别，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。