Diagnonally implicit Runge Kutta methods

Differential equations are equations that are frequently encountered in the modeling of real world problems in many branches of science, especially in engineering, physics, mathematics, chemistry and biology. Therefore, it becomes very important to find analytical or numerical solutions of these bottom equations. With the analytical solution of a differential equation, the sought function is obtained in such a way that it gives a value at every point of the solution region. This is not always possible. The necessity of using numerical methods is inevitable, especially in cases where the differential equation is nonlinear, analytical solution cannot be found or it is quite difficult. The numerical solution is based on the fact that the function f is calculated at discrete points, not continuous. The use of initial values to find the integration constants that emerge in the analytical solution of differential equations is also necessary to find numerical solutions. In this thesis, which is prepared as a master's thesis, the Runge-Kutta methods, which is one of the numerical solution methods of initial value problems, will be emphasized. Runge-Kutta methods, which have a very high stability and approach, will be examined in detail according to their open and closed situations, their advantages and disadvantages and differences will be revealed, and the claim will be supported with numerical examples and simulations. The primary aim of this thesis will be the diagonal closed Runge-Kutta methods, which is one of the closed methods that do not have many studies in the literature. These methods are especially useful and effective in numerical solutions of Stiff differential equations, which are special differential equations.