The work deals with two major topics concerning the numerical analysis of Runge-Kutta-like (RK-like) methods, namely their stability and order of convergence. RK-like methods differ from additive RK methods in that their coefficients are allowed to depend on the solution and the step size. As a result of this, we also refer to them as non-standard additive RK (NSARK) methods. The first major part of this thesis is dedicated to providing a tool for deriving order conditions for NSARK methods. The proposed approach may yield implicit order conditions, which can be rewritten in explicit form using the NB-series of the stages. The obtained explicit order conditions can be further reduced using Gr\"obner bases computations. With the presented approach, it was possible for the first time to obtain conditions for the construction of 3rd and 4th order GeCo as well as 4th order MPRK schemes. Moreover, a new fourth order MPRK method is constructed using our theory and the order of convergence is validated numerically. The second major part is concerned with the stability of nonlinear time integrators preserving at least one linear invariant. We discuss how the given approach generalizes the notion of A-stability. We can prove that investigating the Jacobian of the generating map is sufficient to understand the stability of the nonlinear method in a neighborhood of the steady state. This approach allows for the first time the investigation of several modified Patankar. In the case of MPRK schemes, we compute a general stability function in a way that can be easily adapted to the case of PDRS. Finally, the approach from the theory of dynamical systems is used to derive a necessary condition for avoiding unrealistic oscillations of the numerical approximation.

翻译：本文围绕龙格-库塔型（RK-like）方法数值分析的两大核心问题展开研究，即其稳定性与收敛阶。RK-like方法与加法型RK方法的区别在于，其系数允许依赖于解和步长。因此，我们也将这类方法称为非标准加法型RK（NSARK）方法。论文的第一部分致力于为NSARK方法提供导出阶条件的工具。所提出的方法可能产生隐式阶条件，通过各阶段的NB级数可将其改写为显式形式。利用格罗布纳基计算可进一步简化所获得的显式阶条件。借助该研究方法，我们首次获得了构造三阶和四阶GeCo格式以及四阶MPRK格式的条件。此外，基于我们的理论构建了一种新的四阶MPRK方法，并通过数值实验验证了其收敛阶。第二部分主要研究至少保持一个线性不变量的非线性时间积分器的稳定性。我们讨论了所提出方法如何推广A-稳定性的概念，并证明分析生成映射的雅可比矩阵足以理解非线性方法在稳态邻域内的稳定性。该研究方法首次允许对多种改进型Patankar格式进行稳定性分析。针对MPRK格式，我们以可便捷适配至PDRS情形的方式计算了通用稳定性函数。最后，利用动力系统理论的方法推导了避免数值近似产生非物理振荡的必要条件。