Piecewise-linearized methods for initial-value problems
Piecewise-linearized methods for the solution of initial value problems in ordinary differential equations are developed by approximating the right-hand sides of the equations by means of a Taylor polynomial of degree one. The resulting approximation can be integrated analytically to obtain the solution in each interval, and yields the exact solution for linear problems. Three adaptive methods, based on the norm of the Jacobian matrix, maintaining constant the value of the approximation errors made by the linearization, and Richardson extrapolation are considered. Numerical experiments with some nonstiff first- and second-order equations suggest that the accuracy of the piecewise-linearized methods is superior to that of the explicit, modified, or second-order Euler method, or the trapezoidal rule, but less than that of the fourth-order Runge-Kutta method.
scientific article; zbMATH DE number 1006498
scientific article