Department of Mathematics
Construction and Implementation of General Linear Methods
for Ordinary Differential Equations
Zdzislaw Jackiewicz
Arizona State University
In the first part of this lecture we will give the overview of different approaches to the construction of diagonally implicit multistage integration methods for both nonstiff and stiff differential systems of ordinary differential equations. The identification of high order methods with appropriate stability properties requires the solution of large systems of nonlinear equations for the coefficients of the methods. For low orders these systems can be generated and solved by symbolic manipulation packages. For high orders the approach to the construction of such methods is based on the computation of the coefficients of the stability function by a variant of the Fourier series method and then solving the resulting large systems of polynomial equations of high degree by least squares minimization. Using these approaches both explicit and implicit methods were constructed up to the order eight with good stability properties (Runge-Kutta stability for explicit methods, A-stability and L-stability for implicit methods). In the second part of this talk we will address different issues related to the implementation of general linear methods. They include selection of initial stepsize and starting values, computation of Nordsieck representation, efficient and reliable estimation of the local discretization errors for nonstiff and stiff equations, step size ond order changing strategies, construction of continuous interpolants, and updating vector of external approximations to the solution. Experiments with variable step variable order experimental Matlab codes for both nonstiff and stiff differential systems on interesting test problems will be presented and compared with appropriate codes from Matlab ODE suite. These experiments demonstrate the high potential of diagonally implicit multistage integration methods, especially for stiff systems of differential equations.
All interested persons are welcome.
The talk will be accessible to upper class students.