A note on a diagonally implicit Runge-Kutta-Nyström method

AbstractIt is shown that it is possible to obtain fourth-order accurate diagonally implicit Runge-Kutta-Nyström methods with only 2 stages. The scheme with the largest interval of periodicity, i.e. (0, 12), is given. Furthermore, the requirement of P-stability decreases the order to 2

Parallel-iterated Runge-Kutta methods for stiff ordinary differential equations

AbstractFor the numerical integration of a stiff ordinary differential equation, fully implicit Runge-Kutta methods offer nice properties, like a high classical order and high stage order as well as an excellent stability behaviour. However, such methods need the solution of a set of highly coupled equations for the stage values and this is a considerable computational task. This paper discusses an iteration scheme to tackle this problem. By means of a suitable choice of the iteration parameters, the implicit relations for the stage values, as they occur in each iteration, can be uncoupled so that they can be solved in parallel. The resulting scheme can be cast into the class of Diagonally Implicit Runge-Kutta (DIRK) methods and, similar to these methods, requires only one LU factorization per step (per processor). The stability as well as the computational efficiency of the process strongly depends on the particular choice of the iteration parameters and on the number of iterations performed. We discuss several choices to obtain good stability and fast convergence. Based on these approaches, we wrote two codes possessing local error control and stepsize variation. We have implemented both codes on an ALLIANT FX/4 machine (four parallel vector processors and shared memory) and measured their speedup factors for a number of test problems. Furthermore, the performance of these codes is compared with the performance of the best stiff ODE codes for sequential computers, like SIMPLE, LSODE and RADAU5

An implicit-explicit Runge-Kutta-Chebyshev scheme for diffusion-reaction equations

An implicit-explicit (IMEX) extension of the explicit Runge-Kutta-Chebyshev (RKC) scheme designed for parabolic PDEs is proposed for diffusion-reaction problems with severely stiff reaction terms. The IMEX scheme treats these reaction terms implicitly and diffusion terms explicitly. Within the setting of linear stability theory, the new IMEX scheme is unconditionally stable for reaction terms having a Jacobian matrix with a real spectrum. For diffusion terms the stability characteristics remain unchanged. A numerical comparison for a stiff, nonlinear radiation-diffusion problem between an RKC solver, an IMEX-RKC solver and the popular implicit BDF solver VODPK using the Krylov solver GMRES illustrates the excellent performance of the new scheme

On stabilized integration for time-dependent PDEs

An integration method is discussed which has been designed totreat parabolic and hyperbolic terms explicitly and stiff reaction terms implicitly. The method is a special two-step form of the one-step IMEX (Implicit-Explicit) RKC (Runge-Kutta-Chebyshev) method. The special two-step form is introduced with the aim of getting a non-zero imaginary stability boundary which is zero for the one-step method. Having a non-zero imaginary stability boundary allows, for example, the integration of pure advection equations space-discretized with centered schemes, the integration of damped or viscous wave equations, the integration of coupled sound and heat flow equations, etc. For our class of methods it also simplifies the choice of temporal step sizes satisfying the Von Neumann stability criterion, by embedding a thin long rectangle inside the stability region. Embedding rectangles or other tractable domains with this purpose is an idea of Wesselin