Piecewise-linearized methods for initial-value problems

Abstract

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 incurred by the linearization of the right-hand sides of the ordinary differential equations, and Richardson's extrapolation are developed. Numerical experiments with some nonstiff, first- and second-order, ordinary differential equations, indicate that the accuracy of piecewise-linearized methods is, in general, superior to those of the explicit, modified, second-order accurate Euler method and the implicit trapezoidal rule, but lower than that of the explicit, fourth-order accurate Runge-Kutta technique. It is also shown that piecewise-linearized methods do not exhibit computational (i.e., spurious) modes for the relaxation oscillations of the van der Pol oscillator, and, for those systems of equations which satisfy certain conservation principles, conserve more accurately the invariants than the trapezoidal rule. An error bound for piecewise-linearized methods is provided for ordinary differential equations whose right-hand-sides satisfy certain Lipschitz conditions.