A simple ODE solver based on 2-stage Radau IIA

AbstractBy simplifying the Newton process needed to solve the nonlinear equations associated with the 2-stage Radau IIA method, we come up with an efficient solver that needs only one LU-decomposition of the dimension of the problem per time step

Test set for initial value problem solvers

The CWI test set for IVP solvers presents a collection of Initial Value Problems to test solvers for implicit differential equations. This test set can both decrease the effort for the code developer to test his software in a reliable way, and cross the bridge between the application field and numerical mathematics. This document contains the descriptive part of the test set. It describes the test problems and their origin, and reports on the behavior of a few state-of-the-art solvers on these problems. The latest version of this document and the software part of the test set is available via the world wide web at url{http://www.cwi.nl/cwi/projects/IVPtestset/. The software part serves as a platform on which one can test the performance of a solver on a particular test problem oneself. Instructions how to use this software are in this paper as well. The idea to develop this test set was discussed at the workshop ODE to NODE, held in Geiranger, Norway, 19--22 June~1995

Parallel linear system solvers for Runge-Kutta methods

If the nonlinear systems arising in implicit Runge-Kutta methods like the Radau IIA methods are iterated by (modified) Newton, then we have to solve linear systems whose matrix of coefficients is of the form $I-A otimes hJ$ with A the Runge-Kutta matrix and J an approximation to the Jacobian of the righthand side function of the system of differential equations. For larger systems of differential equations, the solution of these linear systems by a direct linear solver is very costly, mainly because of the LU-decompositions. We try to reduce these costs by solving the linear systems by a second (inner) iteration process. This inner iteration process is such that each inner iteration again requires the solution of a linear system. However, the matrix of coefficients in these new linear systems is of the form $I-B otimes hJ$ where B is similar to a diagonal matrix with positive diagonal entries. Hence, after performing a similarity transformation, the linear systems are decoupled into s subsystems, so that the costs of the LU-decomposition are reduced to the costs of s LU-decompositions of dimension d. Since these LU-decompositions can be computed in parallel, the effective LU-costs on a parallel computer system are reduced by a factor $s^3$. It will be shown that matrices B can be constructed such that the inner iterations converge whenever A and J have their eigenvalues in the positive and nonpositive halfplane, respectively. The theoretical results will be illustrated by a few numerical examples. A parallel implementation on the four-processor Cray-C98/4256 shows a speed-up ranging from at least 2.4 until at least 3.1 with respect to RADAU5 applied in one-processor mode

Parallel Störmer-Cowell methods for high-precision orbit computations

Many orbit problems in celestial mechanics are described by (nonstiff) initial-value problems (IVPs) for second-order ordinary differential equations of the form y' = {bf f (y). The most successful integration methods are based on high-order Runge-Kutta-Nyström formulas. However, these methods were designed for sequential is paper, we consider high-order parallel methods that are not based on Runge-Kutta-Nyström formulas, but which fit into the class of general linear methods. In each step, these methods compute blocks of k approximate solution values (or stage values) at k different points using the whole previous block of solution values. The k stage values can be computed in parallel, so that on a k-processor computer system such methods effectively perform as a one-value method. The block methods considered in this paper are such that each equation defining a stage value resembles a linear multistep equation of the familiar Störmer-Cowell type. For k = 4 and k = 5 we constructed explicit PSC methods with stage order q = k and step point order p = k+1 and implicit PSC methods with q = k+1 and p = k+2. For k = 6 we can construct explicit PSC methods with q = k and p = k+2 and implicit PSC methods with q = k+1 and p = k+3. It turns out that for k = 5 the abscissae of the stage values can be chosen such that only k-1 stage values in each block have to be computed, so that the number of computational stages, and hence the number of processors and the number of starting values needed, reduces to k* = k-1. The numerical examples reported in this paper show that the effective number of righthand side evaluations required by a variable stepsize implementation of the 10th-order PSC method is 4 up to 30 times less than required by the Runge-Kutta-Nyström code DOPRIN (which is considered as one of the most efficient sequential codes for second-order ODEs). Furthermore, a comparison with the 12th-order parallel code PIRKN reveals that the PSC code is, in spite of its lower order, at least equally efficient, and in most cases more efficient than PIRKN