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: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=711565
• 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中心工具。这些系统的复杂性要求使用数值软件来尽可能有效地获得准确的解。 在我的研究生涯中，我的兴趣主要集中在边值常微分方程组(BVODE)、一维含时偏微分方程(1D PDE)和二维含时偏微分方程(2D PDE)的数值软件开发上。我们的软件已经应用于肿瘤细胞生长模型、心脏模拟和金融模型的求解。在接下来的几年里，我的研究将继续专注于BVODE和PDE的数值软件： BVODE：在过去的20年里，我开发了基于特殊类别龙格-库塔(Runge-Kutta，RK)方法的误差控制数值软件包；该软件包的最新版本具有用于全局误差控制和缺陷控制的选项。(缺陷是数值解未能满足BVODE的量。)我们的新工作将考虑几个改进和扩展：(I)改进的缺陷控制方案，(Ii)用于刚性BVODE的新的RK方法，(Iii)扩展以处理具有周期边界条件、积分条件和/或延迟和提前项的BVODE。 1DPDES：在过去的12年里，我开发了一系列B-Spline配置软件包，提供空间和时间误差控制。这一系列的最新成员(使用反向差分法进行时间推进)提供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能力：(I)该软件的RK时间推进版本中的新的误差估计和控制方案，(Ii)自动配置顺序和误差控制方案选择，(Iii)处理较大的PDE系统，改善具有挑战性初始条件的问题的性能，以及处理终止依赖于解决方案的条件的问题。 2DPDES：我们最近的工作看到了基于将1D算法扩展到2D的原型软件的开发。当前解算器具有时间误差控制能力，但不具有空间误差控制能力。新的工作将集中在开发空间误差估计方案和自适应空间误差控制。 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，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调查。