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Abstracts at Joint The 4th TWSIAM Annual Meeting
Tailored finite point method for solving
One-dimensional Burgers’ equation
Chih-Ching Tsai
Department of Applied Mathematics, National Chung-Hsing University
Taichung, Taiwan
[email protected]
Joint work with: Yin-Tzer Shih and Yu-Tuan Lin
Burgers’ equation is a well-known mathematical model describing a number of
physical phenomena that combine both nonlinear propagation and diffusive effects in
fluid mechanics and gas dynamics. It is also related to a variety of applications such
as traffic flow problems, continuous stochastic processes and the acoustic
transmission in viscous medium. Many numerical methods for solving Burgers’
equation have been presented for decades. Aside from solving the original Burgers’
equation, an alternative approach is to exploit the Hopf-Cole transformation (HCT),
which turns the nonlinear Burgers’ equation into a heat equation, and then to compute
the approximation of the transformed equation, e.g. [1]. The tailored finite point
method (TFPM) was first proposed by Han et al. [2] for solving a singular
perturbation problem. Later on, this method has been successfully applied to solve
convection-diffusion equation [3], first order wave equation [4], etc. The essence of
TFPM is to select basis functions that are particularly tailored to the property of the
solutions. In this work, we developed two TFPM schemes for solving viscous
Burgers’ problem. The first scheme utilizes the HCT while the second scheme
directly solves the equation. The selected basis functions for both the TFPM schemes
locally fit the properties of the solution in space and time simultaneously. Further, we
show that the first scheme is unconditionally stable and the second one is
conditionally stable. The numerical results demonstrate the efficiency and accuracy of
the proposed schemes.
Keywords: Tailored finite point method, Burgers’ equation, Hopf-Cole transformationstability analysis.
References
[1] A. Asaithambi, Numerical solution of the Burgers’ equation by automatic
differentiation,
Appl. Math. Comput. 216, 2700-2708, 2010.
[2] H. Han, Z. Huang and R.B. Kellogg, A tailored finite point method for a singular
perturbation problem on an unbounded domain, J. Sci. Comput., 36, 243-261, 2008.
[3] Y.T. Shih, R.B. Kellogg and Y. Chang, Characteristic tailored finite point method
for convection dominated convection-diffusion-reaction problems, J. Sci. Comput.,
47, 198-215, 2011.
[4] Z. Huang and X. Yang, Tailored finite point method for first order wave equation,
J. Sci. Comput., 49, 351-366, 2011.
Joint The 4th TWSIAM Annual Meeting