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

Abstract: 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.