An Overview of Rosenbrock-Krylov Methods for the Numerical Solution of Ordinary Differential Equations

Sammanfattning: This thesis offers an overview of the relatively new family of Rosenbrock-Krylov numerical methods for ODEs. These methods are a further development of Rosenbrock methods, using a lower-dimension approximation of the Jacobian. The thesis gives the mathematical background to Rosenbrock-Krylov methods by first presenting Runge-Kutta methods and Rosenbrock methods, followed by the concept of Krylov spaces and the Arnoldi Iteration. Rosenbrock-Krylov methods are described in relation to these concepts. The implementation of the three mentioned families of methods is chronicled in the thesis, and their numerical performance on some problems is studied. Rosenbrock-Krylov methods show promising results for accuracy of the solution for fixed step sizes. Regarding raw computing performance, the implemention of Rosenbrock-Krylov methods made for this thesis is considerably faster than an ordinary Rosenbrock method for larger problems. As such, Rosenbrock-Krylov methods appear to be an appealing alternative to current methods.