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.

许多应用都可以用正的守恒常微分方程来描述，保证数值解的正性和守恒性是非常必要的。标准方法如Runge-Kutta(RK)方法保持保守性，但通常不能保证解分量的正性。这必须通过额外且昂贵的后处理来完成。一类既保证守恒性又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，几个出版物出现了这些有希望的计划。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，将统一阶次分析领域的现有理论，并弥合关于稳定性、时间适应性和稠密输出公式的理论空白。所有Patankar类型的方法都是基于用所谓的Patankar技巧对显式RK方法的修改。通过形式上将它们视为扰动RK格式，统一的阶次分析将成为可能，并有助于不同Patankar型方法的比较。该项目的主要目标是首次开发帕坦卡尔类型方法的稳定性分析。尽管MPRK格式在数值计算中被证明是非常稳定的，但到目前为止，关于这方面的理论研究还很缺乏。缺乏稳定性理论的一个主要原因是迭代的非线性相关性，甚至当该方法应用于线性系统时也会出现这种情况。该项目将同时关注当地和全球的稳定。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。这一分析将允许推导出保证稳定性的帕坦卡权重条件。Patankar类型方法使用低阶方法来确定所需的Patankar权重。这些又可以用来估计局部误差和自适应地选择时间步长。目前，还没有已知的自适应Patankar类型的方法在低公差下具有竞争力。利用新的稳定性分析方法，可以开发出高效的自适应Patankar型方法。最后，给出了Patankar型方法的稠密输出公式(DOF)，该公式可用于生成任意次数的适当阶近似。这里的一个新特点是，自由度还保证在任意时刻的正性和保守性。

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.