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：本文主要研究龙格-库塔类（RK-like）方法数值分析中的两大核心问题，即其收敛阶与稳定性。与加法型龙格-库塔方法不同，RK-like方法的系数允许依赖于解和步长，因此我们将其称为非标准加法型龙格-库塔（NSARK）方法。本文第一部分致力于为NSARK方法推导阶条件提供工具。所提出的方法可导出隐式阶条件，并通过子阶段的NB级数将其改写为显式形式。利用Gröbner基计算可进一步简化得到的显式阶条件。基于该理论，首次获得了构建三阶和四阶GeCo格式以及四阶MPRK格式的阶条件。此外，我们根据该理论构造了一种新的四阶MPRK方法，并通过数值实验验证了其收敛阶。第二部分关注至少保持一个线性不变量的非线性时间积分器的稳定性。我们讨论了该方法如何推广A-稳定性的概念，并证明通过研究生成映射的雅可比矩阵即可充分理解非线性方法在稳态附近的稳定性。该理论首次实现了对多种改进型Patankar格式的稳定性分析。针对MPRK格式，我们以易于推广至PDRS格式的方式计算了通用稳定性函数。最后，利用动力系统理论推导出避免数值近似出现非物理振荡的必要条件。