In this article, up to tenth-order finite difference schemes are proposed to solve the generalized BurgersHuxley equation. The schemes based on high-order differences are presented using Taylor series expansion. To establish the numerical solutions of the corresponding equation, the high-order schemes in space and a fourth-order Runge-Kutta scheme in time have been combined. Numerical experiments have been conducted to demonstrate the high-order accuracy of the current algorithms with relatively minimal computational effort. The results showed that use of the present approaches in the simulation is very applicable for the solution of the generalized Burgers-Huxley equation. The current results are also seen to be more accurate than some results given in the literature. The proposed algorithms are seen to be very good alternatives to existing approaches for such physical applications.

Up to tenth-order finite difference schemes are proposed in this paper to solve one-dimensional advection-diffusion equation. The schemes based on high-order differences are presented using Taylor series expansion. To obtain the solutions, up to tenth-order finite difference schemes in space and a fourth-order Runge-Kutta scheme in time have been combined. The methods are implemented to solve two problems having exact solutions. Numerical experiments have been conducted to demonstrate the efficiency and high-order accuracy of the current methods. The techniques are seen to be very accurate in solving the advection-diffusion equation for 5 Pe ≤. The produced results are also seen to be more accurate than some available results given in the literature.

In this paper, we attempt to display effective numerical simulations of shock wave propagating represented by the Burgers equations known as a significant mathematical model for turbulence. A high order hybrid approximation based on seventh order weighted essentially non-oscillatory finite difference together with the sixth order finite difference scheme implemented for spatial discretization is presented and applied without any transformation or linearization to the Burgers equation and its modified form. Then, the produced system of first order ordinary differential equations is solved by the MacCormack method. The efficiency, accuracy and applicability of the proposed technique are analyzed by considering three test problems for several values of viscosity that can be caused by the steep shock behavior. The performance of the method is measured by some error norms. The results are in good agreement with the results reported previously, and moreover, the suggested approximation relatively comes to the forefront in terms of its low cost and easy implementation.

SUMMARYUp to tenth-order finite difference (FD) schemes are proposed in this paper to solve the generalized Burgers-Fisher equation. The schemes based on high-order differences are presented using Taylor series expansion. To obtain the solutions, up to tenth-order FD schemes in space and fourth-order Runge-Kutta scheme in time have been combined. Numerical experiments have been conducted to demonstrate the efficiency and high-order accuracy of the present methods. The produced results are also seen to be more accurate than some available results given in the literature. Comparisons showed that there is very good agreement between the numerical solutions and the exact solutions in terms of accuracy. The present methods are seen to be very good alternatives to some existing techniques for such realistic problems.

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