B-method approach to blow-up solutions of fourth-order semilinear parabolic equations
- 2 Downloads
B-method is a novel method developed by Beck et al. (SIAM J. Sci. Comput. 37(5), A2998–A3029, 2015), and has been shown theoretically to be very advantageous in time discretization of the second-order parabolic equations with blow-up solutions. In this paper, we extend the B-method to approximate the blow-up solution of a class of fourth-order parabolic equations, which plays very important role in many engineering applications. First, by following the systematic means of constructing numerical schemes based on the technique of variation of constants proposed by Beck et al., we give some B-method schemes for the fourth-order semilinear parabolic equations. Second, we perform a truncation error analysis to show when and why the B-method scheme is advantageous over its classical counterpart. Third, we take one of the constructed numerical schemes as an example to show the well-posedness using the technique of upper and lower solutions. Last, we carry out numerical experiments to approximate the blow-up solutions and illustrate the efficiency of our numerical schemes.
KeywordsFourth-order parabolic equation B-method Blow-up solution
We thank the anonymous referees for their constructive suggestions and useful comments, which have substantially improved our paper.
Yongkui Zou is supported by NSFC-11771179 and NSFC-91630201. Yingxiang Xu is supported by NSFC-11671074 and the Fundamental Research Funds for the Central Universities (No. 2412018ZD001)
- 1.Ball, J.M.: Finite time blow-up in nonlinear problems. In: Crandall, M.G. (ed.) Nonlinear Evolution Equations, pp 189–205. Academic Press, New York (1978)Google Scholar
- 11.Frank-Kamenetskii, D.A.: Towards temperature distributions in a reaction vessel and the stationary theory of thermal explosion. Doklady Acad. Nauk SSSR 18, 411–412 (1938)Google Scholar
- 16.Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, volume 31. Springer Science & Business Media (2006)Google Scholar