臺灣博碩士論文加值系統

We consider the frst order nonlinear ODE blow-up problem u′(t)=G(u). Nakagawa [6] proposed a scheme with adaptively-defned temporal increment for the computation of the blow-up solutions. Later, an algorithm using uniform time mesh was proposed by Cho [3] for the computation of the numerical blow-up time. Nevertheless, these schemes are of order 1 in time variable so that the convergence order of the numerical blow-up time is also of order 1. In this presentation, we would like to consider a question as to can we have a higher convergence order if we use a higher order scheme for both methods to compute the blow-up time.

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Numerical Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.1 Nakagawa’s idea . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.2 Cho’s idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3 Implicit schemes using Cho’s idea . . . . . . . . . . . . . . . . . . . 3
3.1 A full implicit scheme . . . . . . . . . . . . . . . . . . . . . . . . 4
3.2 A linearly implicit scheme . . . . . . . . . . . . . . . . . . . . . .6
4 Second order schemes . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.1 Adaptive temporal mesh . . . . . . . . . . . . . . . . . . . . . . . . 10
4.2 Uniform temporal mesh . . . . . . . . . . . . . . . . . . . . . . . . 14
5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

[1] Berger, M., Kohn, R. V.: A rescaling algorithm for the numerical calculation of blowing-up solutions. Commun. Pure Appl. Math. 41, 841–863 (1988).
[2] Cho, C.-H., Hamada, S., Okamoto, H.: On the fnite difference approximation for a parabolic blow-up problem. J.pan J. Indus. Appl. Math. 24(2), 131ȉ 160 (2007)
[3] Cho, C.-H.: On the computation of numerical blow-up time. Jpn. J. Indust. Appl. Math. 30, 331–349 (2013).
[4] Cho, C.-H.: A numerical algorithm for blow-up problems revisited. Numer. Algor. 75, 675–697 (2017).
[5] Huang, W.-Z., Ren, Y., Russell, R. D.: Moving mesh partial differential equations (MMPDES) based on the equidistribution principle. SIAM J. Numer. Anal. 31, 709–730 (1994).
[6] Nakagawa, T.: Blowing up of a fnite difference solution to ut = uxx + u2. Appl. Math. Optim. 2, 337–350 (1976).