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.

许多应用可以通过正面和保守的普通微分方程来描述，并且非常需要保证数值解决方案的积极性和保守性。诸如Runge-Kutta（RK）方法之类的标准方法可保留保守性，但通常不能保证溶液组件的阳性。这必须通过额外且昂贵的后处理来完成。一类方法不仅可以保证保守性，而且是无条件阳性的方法是patankar型方法。该课程分为BBK和MPRK方案，在过去的三年中，这些有前途的计划出现了一些出版物。特别是，由于MPRK方法已被证明是解决僵局问题的绝佳方法。在拟议的项目中，将在订单分析领域中的现有理论统一，并且有关稳定性，时间适应和密集输出公式的理论差距将关闭。所有Patankar型方法均基于使用所谓的Patankar Trick修改显式RK方法。通过正式将它们视为扰动的RK方案，将进行统一的订单分析，并促进不同patankar型方法的比较。该项目的主要目标是首次开发用于Patankar型方法的稳定性分析。尽管特别是在数值计算中已显示MPRK方案非常稳定，但是到目前为止，对此的理论研究一直缺乏。缺乏稳定理论的一个主要原因是迭代的非线性依赖性，甚至在将方法应用于线性系统时也会发生。该项目将与本地和全球稳定有关。为此，将应用具有几个未知数和参数的非线性动力学系统理论。该分析将允许在Patankar权重的情况下得出任何保证稳定性的条件。 patankar型方法使用较低阶的方法来确定所需的patankar权重。这些反过来又可用于估计本地错误，并自适应地选择时间步长。当前，尚无已知的自适应patankar型方法，它们在低公差下具有竞争力。使用新的稳定性分析，可以开发有效的自适应patankar型方法。最后，开发了用于patankar型方法的致密输出公式（DOF），可用于在任意时间生成适当顺序的近似值。这里的一个新功能是，DOF还可以保证在任意时期的积极性和保守性。