A complete characterization of the convergence factor can be very useful when analyzing the asymptotic convergence of an iterative method. We will here establish a formula for the convergence factor of the method called residual inverse iteration, which is a method for nonlinear eigenvalue problems and a generalization of the well known inverse iteration. The formula for the convergence factor ...

We propose a new Eulerian-Lagrangian Runge-Kutta finite volume method for numerically solving convection and convection-diffusion equations. semi-Lagrangian methods have grown in popularity mostly due to their ability allow large time steps. Our proposed scheme is formulated by integrating the PDE on space-time region partitioned approximations of characteristics determined from Rankine-Hugonio...

This paper will concentrate on contributions of CWI to the development of parallel Runge-Kutta (RK) methods. We shall describe two approaches to construct such methods. In both approaches, a conventional implicit RK method is used as a corrector equation whose solution is approximated by an iterative method. In the first approach, the iteration method uses a fixed number of iterations without s...

The Interaction Picture (IP) method is a valuable alternative to Split-step methods for solving certain types of partial differential equations such as the nonlinear Schrödinger equation or the Gross-Pitaevskii equation. Although very similar to the Symmetric Split-step (SS) method in its inner computational structure, the IP method results from a change of unknown and therefore do not involve ...

To integrate large systems of ordinary differential equations (ODEs) with disparate timescales, we present a multirate method with error control that is based on embedded, explicit Runge-Kutta (RK) formulas. The order of accuracy of such methods depends on interpolating certain solution components with a polynomial of sufficiently high degree. By analyzing the method applied to a simple test eq...

This paper investigates the performance of two explicit pseudo two-step Runge-Kutta methods of order 5 and 8 for rst-order nonstii ODEs on a parallel shared memory computer. For expensive right hand sides the parallel implementation gives a speedup of 3{4 with respect to the sequential one. Furthermore we compare the codes with the two eecient nonstii codes DOPRI5 and DOP853. For problems, wher...

Discontinuous Galerkin (DG) spatial discretizations are often used in a method-of-lines approach with explicit strong-stability-preserving (SSP) Runge– Kutta (RK) time steppers for the numerical solution of hyperbolic conservation laws. The time steps that are employed in this type of approach must satisfy Courant–Friedrichs–Lewy (CFL) stability constraints that are dependent on both the region...

A personal view of the historical development of software based on explicit Runge-Kutta formulas is presented, and some future developments are suggested.

Locally refined meshes severely impede the efficiency of explicit Runge-Kutta (RK) methods for the simulation of time-dependent wave phenomena. By taking smaller time-steps precisely where the smallest elements are located, local time-stepping (LTS) methods overcome the bottleneck caused by the stringent stability constraint of but a few small elements in the mesh. Starting from classical or lo...