Adaptive interval wavelet precise integration method for partial differential equations. (English) Zbl 1144.65325

Mei, Shu-Li; Lu, Qi-Shao; Zhang, Sen-Wen; Jin, Li

Appl. Math. Mech., Engl. Ed. 26, No. 3, 364-371 (2005).

Summary: The quasi-Shannon interval wavelet is constructed based on the interpolation wavelet theory, and an adaptive precise integration method, which is based on extrapolation method is presented for nonlinear ordinary differential equations (ODEs). And then, an adaptive interval wavelet precise integration method (AIWPIM) for nonlinear partial differential equations (PDEs) is proposed. The numerical results show that the computational precision of AIWPIM is higher than that of the method constructed by combining the wavelet and the 4th Runge-Kutta method, and the computational amounts of these two methods are almost equal. For convenience, the Burgers equation is taken as an example in introducing this method, which is also valid for more general cases.

MSC:

65T60	Numerical methods for wavelets
65L05	Numerical methods for initial value problems involving ordinary differential equations
35F30	Boundary value problems for nonlinear first-order PDEs

Keywords:

precise integration method; extrapolation method; Burgers equation; interval wavelet

References:

[1]	Zhong Wanxie. Precise computation for transient analysis[J].Computational Structural Mechanics and Applications, 1995,12(1):1–6 (in Chinese). · Zbl 0846.76064
[2]	Wei G W. Quasi wavelets and quasi interpolating wavelets[J].Chemical Physics Letters, 1998,296 (6):215–222. · doi:10.1016/S0009-2614(98)01061-6
[3]	Silvia Bertoluzza. Adaptive wavelet collocation method for the solution of burgers equation[J].Transport Theory and Statistical Physics, 1996,25(3/5):339–352. · Zbl 0868.65071 · doi:10.1080/00411459608220705
[4]	Wan Decheng, Wei Guowei. The study of quasi wavelets based numerical method applied to Burgers equations[J].Applied Mathematics and Mechanics (English Edition), 2000,21(10):1099–1110. · Zbl 1003.76070 · doi:10.1007/BF02458986
[5]	Yan Guangwu. Study of Burgers equation using a lattice Bolizmann method[J].Acta Mechanica Sinica, 1999,31(2):143–151 (in Chinese).
[6]	Zhang Xunan, Jiang Jiesheng. On precise time-integration method for nonlinear dynamics equations [J].Chinese Journal of Applied Mechanics, 2000,17(4):164–168 (in Chinese).