On the method of modified equations. V: Asymptotic analysis of and direct-correction and asymptotic successive-correction techniques for the implicit midpoint method

Abstract

The equivalent, second equivalent and (simply) modified equations for the implicit midpoint rule are shown to be asymptotically equivalent in the sense that an asymptotic analysis of these equations with the time step size as small parameter yields exactly the same results; for linear problems with constant coefficients, they are also equivalent to the original finite difference scheme. Straightforward (regular), multiple scales and summed-up asymptotic techniques are used for the analysis of the implicit midpoint rule difference method, and the accuracy of the resulting asymptotic expansion is assessed for several first-order, non-linear, autonomous ordinary differential equations. It is shown that, when the resulting asymptotic expansion is uniformly valid, the asymptotic method yields very accurate results if the solution of the leading order equation is smooth and does not blow up. The modified equation is also studied as a method for the development of new numerical schemes based on both direct-correction and asymptotic successive-correction techniques applied to the three kinds of modified equations, the linear stability of these techniques is analyzed, and their results are compared with those of Runge–Kutta schemes for several autonomous and non-autonomous, first-order, ordinary differential equations.