Numerical Methods Of The Generalized Borgers-Fisher Equation

The generalized Burgers-Fisher equation is an important non-linear equation involving the effects of non-linear advection, linear diffusion and non-linear logistic reaction. It has been wildly applied in physics, chemistry, biology and so on. There has been much work done in recent years, including the seek of different analytical solutions and some numerical solvers. We cite, among others, deduction of an analytical solution in the form of tanh function given by Wazwaz, the Adomian decomposition method introduced by Ismail, the pseudo-spectral method by Javidi, and the finite difference methods by Mickens and Chen.In the existing papers there is no particular attention paid to the problem associated to the boundary layer when the diffusion coefficient is small. In this paper we propose a new effecient method to solve the Generalized Burgers-Fisher equation. The main work of this paper consists: Firstly, we introduced a spectral element method in space and high order BD/AB schemes in time to discretize the Burgers-Fisher equation. Analysis and numerical validation are provided to demonstrate the stability and convergence of the method. Secondly, in order to overcome the difficulty caused by small diffusion coefficient, a spectral element viscosity vanishing method is introduced to stabilize the computation. Some numerical tests are carried out to confirm the efficiency of this approach.