| For solving initial-boundary value problem in nonlinear diffusion dominated partial functional differential equations of Volterra type by method of line, the most commonly used time discretization methods are implicit Euler method and trapezoidal method for Volterra functional differential equations(VFDEs), and sometimes implicit midpoint method and the 2-stage Lobatto IIIC Runge-Kutta method for VFDEs recommended by Bellen and Zennaro in 2003 are also used. To avoid ”order failure” and ”stability failure” phenomena pointed out by Bellen and Zennaro,in this paper, for solving VFDEs, we always use Canonical Runge-Kutta methods(abbr. CaRK) rather than continuous Runge-Kutta methods(abbr. CRK), and for convenience we always use the symbols CaIE, CaTr, CaIM and CaLo2 to denote the aforementioned time discretization methods, respectively.The ?rst major work of this paper is to compare carefully the aforementioned time discretization methods by theoretical analysis and numerical experiments in Section 3, and we ?nd that among which CaIM method is most e?cient and excellent.This discovery provides important new scienti?c basis for the selection of time discretization methods. The importance of this discovery lies in that for solving strongly nonlinear problems of above mentioned type, it may lead to CaIM method gradually increased to the most widely used time discretization method.The second major work of this paper is that in Section 2 CaIM method is further proved to be optimally B-convergent of order 2 provided that the time stepsize is uniform. It is this creative achievement makes people realize that CaIM method may be superior to CaTr method since the later is only classical convergent of order 2rather than B-convergent.As extension of the above research work, we have also constructed a high accuracy numerical method in Section 4 for solving a class of fractional differential equations. |
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