On the method of modified equations. II: Numerical techniques based on the equivalent equation for the Euler forward difference method

摘要

New numerical successive-correction techniques for ordinary differential equations based on the Euler forward explicit method and the first modified or equivalent equation are presented. These techniques are similar to iterative updating deferred methods and are based on the application of asymptotic methods to modified equations which do not require initial conditions for the high order derivatives in the truncation terms and which yield stable numerical methods. It is shown that, depending on the discretization of the high order derivatives in the high order correction equations, different methods of as high order of consistency as required can be developed. In this paper, backward and centered formulas are used, but the resulting numerical methods are not self-starting. It is shown that, if the starting procedure is not adequate, the numerical order of the method can be smaller than the theoretical one. In order to avoid this loss of numerical order, a method for consistently starting the asymptotic successive-correction technique based on the use of fictitious times is presented and applied to autonomous and non-autonomous, ordinary differential equations, and compared with the results of second and fourth-order Runge–Kutta methods. It is shown that the fourth-order Runge–Kutta method is more accurate than the successive-correction techniques for large time steps due to the higher order derivatives in the successive-correction, but, for sufficiently small time steps, these techniques have almost the same accuracy as the fourth-order Runge–Kutta method.