A composite integration scheme for the numerical solution of systems of ordinary differential equations

AbstractA generalization of a composite linear multistep method [2] is developed and applied to the approximate integration of systems of ordinary differential equations. The proposed scheme is second-order accurate and L-stable. An algorithm, based on the integration formula derived in this paper, is applied to approximate the solutions of a number of standard test problems. The numerical results indicate that the method is competitive with other fixed-order methods particularly in terms of computational overhead and could provide the basis for efficient temporal integration in the semidiscretization of time dependent partial differential equations

Control of step size and order in extrapolation codes

AbstractExtrapolation of the semi-implicit midpoint rule is an effective way to solve stiff initial value problems for a system of ordinary differential equations. The theory of the control of step size and order is advanced by investigating questions not taken up before, providing additional justification for some algorithms, and proposing an alternative to the information theory approach of Deuflhard. An experimental code SIMP implementing the algorithms proposed is shown to be as good as, and in some respects better than, the research code METAN1 of Bader and Deuflhard

Evaluation of state of the art numerical integration schema

Tese de mestrado integrado. Engenharia Química. Faculdade de Engenharia. Universidade do Porto. 201

Diagonally Implicit Runge-Kutta Methods for Ordinary Differential Equations. A Review

A review of diagonally implicit Runge-Kutta (DIRK) methods applied to rst-order ordinary di erential equations (ODEs) is undertaken. The goal of this review is to summarize the characteristics, assess the potential, and then design several nearly optimal, general purpose, DIRK-type methods. Over 20 important aspects of DIRKtype methods are reviewed. A design study is then conducted on DIRK-type methods having from two to seven implicit stages. From this, 15 schemes are selected for general purpose application. Testing of the 15 chosen methods is done on three singular perturbation problems. Based on the review of method characteristics, these methods focus on having a stage order of two, sti accuracy, L-stability, high quality embedded and dense-output methods, small magnitudes of the algebraic stability matrix eigenvalues, small values of aii, and small or vanishing values of the internal stability function for large eigenvalues of the Jacobian. Among the 15 new methods, ESDIRK4(3)6L[2]SA is recommended as a good default method for solving sti problems at moderate error tolerances