Implementation of Singly Diagonally Implicit Runge-Kutta Methods with Constant Step Sizes

Runge–Kutta methods can be used for solving ordinary differential equations of the form y0 = f(t, y) with initial condition y(t0) = y0 and where f : R x R^m -> R^m. The idea is to find a method that is efficient to implement. But it is also important for the method to be of high order and be stable. Diagonally Implicit RK-methods reduces an sm x sm matrix to s systems of m x m linear equations. Singly Diagonally Implicit RK-methods have only a single eigenvalue, which results in a reduction to only one LU-decomposition per time step. Combining the two methods, we get Singly Diagonally Implicit RK-methods.Many problems in engineering science, natural science and even social and life sciences can be better understood by mathematical simulations. There are often several methods that can solve the same problem with. When choosing which method to use there are properties that needs to be considered, for instance time efficency and accuracy. These properties depend on the problem to solve and what criterias there are on the solution. In this thesis only a specific class of methods are considered, for solving differential equations. These methods are called Runge–Kutta methods. Further on, some properties are described and also problems that can appear when implementing the method. A specific subclass of Runge–Kutta methods, namely Singly Diagonally Implicit Runge–Kutta methods, is constructed such that these common implementation issues are not a problem. The benefits from implementing and using these methods are described further and some tests are made to verify these statements