Substantial extension and unification of the theory of Patankar-type schemes by means of unified order analysis, first-time investigation of stability, time-step adaptation and dense-output formulas.

通过统一阶次分析、首次稳定性研究、时间步自适应和密集输出公式，对Patankar型方案理论进行了实质性扩展和统一。

• 批准号: 466355003
• 负责人: Professor Dr. Andreas Meister
• 金额: --
• 依托单位: Arbeitsgruppe Analysis und angewandte Mathematik
• 依托单位国家: 德国
• 项目类别: Research Grants
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/466355003?language=en
• 关键词: Substantial; extension; unification; theory; Patankar

Many applications can be described by positive and conservative ordinary differential equations and it is highly desirable to guarantee the positivity and conservativity also for the numerical solution. Standard methods such as Runge-Kutta (RK) methods preserve conservativity, but in general cannot guarantee positivity of the solution components. This has to be done by additional and costly postprocessing. A class of methods which guarantee not only conservativity but also unconditional positivity are the Patankar-type methods. This class is divided into BBKS and MPRK schemes and in the last three years several publications appeared to these promising schemes. In particular, since MPRK methods have proven excellent for the solution of stiff problems.In the proposed project, existing theory in the field of order analysis will be unified and theoretical gaps regarding stability, time adaptation and dense output formulas will be closed. All Patankar-type methods are based on the modification of explicit RK methods with the so-called Patankar trick. By formally considering them as perturbed RK schemes, a unified order analysis will be possible and facilitate the comparison of the different Patankar-type methods. The main goal of the project is to develop for the first time a stability analysis for Patankar-type methods. Although MPRK schemes in particular have been shown to be very stable in numerical calculations, theoretical investigations of this have been lacking up to now. A major reason for the lack of a stability theory is the nonlinear dependence of the iterates, which even occur when the methods are applied to linear systems. The project will be concerned with both local and global stability. For this purpose, the theory of nonlinear dynamical systems with several unknowns and parameters will be applied. This analysis will allow to derive conditions on the Patankar weights which guarantee stability. Patankar type methods use lower order methods to determine the required Patankar weights. These, in turn, can be used to estimate local error and select the time step size adaptively. Currently, there are no known adaptive Patankar-type methods that are competitive at low tolerances. Using the new stability analysis, efficient adaptive Patankar-type methods can be developed. Finally, dense output formulas (DOF) for Patankar-type methods are developed, which can be used to generate approximations of appropriate order for arbitrary times. A new feature here is that the DOF also guarantee positivity and conservativity at arbitrary times.

许多应用都可以用正的和守恒的常微分方程来描述，因此保证数值解的正性和守恒性是非常必要的。标准方法，如龙格-库塔（RK）方法保持保守性，但一般不能保证解决方案的组件的积极性。这必须通过额外的和昂贵的后处理来完成。一类不仅保证保守性而且保证无条件正性的方法是Patankar型方法。这一类被分为BBKS和MPRK计划，在过去的三年里，一些出版物出现了这些有前途的计划。特别是，由于MPRK方法已被证明是解决刚性问题的优秀方法，在拟议的项目中，阶次分析领域的现有理论将得到统一，有关稳定性，时间适应性和密集输出公式的理论差距将得到弥补。所有的Patankar型方法都是基于修改显式RK方法与所谓的Patankar技巧。通过将它们形式化地视为摄动RK格式，统一的阶次分析将成为可能，并便于不同Patankar型方法的比较。该项目的主要目标是首次为Patankar型方法开发稳定性分析。虽然MPRK格式在数值计算中特别稳定，但迄今为止还缺乏理论研究。缺乏稳定性理论的一个主要原因是迭代的非线性依赖性，甚至当方法应用于线性系统时也会出现这种情况。该项目将关注地方和全球稳定。为此，将应用具有多个未知数和参数的非线性动力系统理论。该分析将允许导出保证稳定性的Patankar权重的条件。Patankar类型方法使用低阶方法来确定所需的Patankar权重。这些，反过来，可以用来估计局部误差和选择的时间步长自适应。目前，还没有已知的自适应帕坦卡型方法在低公差下具有竞争力。使用新的稳定性分析，有效的自适应Patankar型方法可以开发。最后，稠密输出公式（DOF）的Patankar型方法，它可以用来产生适当的顺序为任意时间的近似。这里的一个新特性是自由度还保证在任意时间的正性和保守性。