The numerical solution of a generalized Burgers–Huxley equation through a conditionally bounded and symmetry-preserving method

• Published in: Computers & mathematics with applications (1987) 2011-06, Vol.61 (11), p.3330-3342
• Main Authors: Macías-Díaz, J.E., Ruiz-Ramírez, J., Villa, J.
• Format: Article
• Language: English
• Subjects: Approximation; Boundedness preservation; Burgers–Huxley equation; Computer simulation; Constants; Mathematical analysis; Mathematical models; Non-standard method; Partial differential equations; Positivity preservation; Preserves; Symmetry preservation; Tuning
• ISSN: 0898-1221; 1873-7668
• DOI: 10.1016/j.camwa.2011.04.022

In this article, we propose a non-standard, finite-difference scheme to approximate the solutions of a generalized Burgers–Huxley equation from fluid dynamics. Our numerical method preserves the skew-symmetry of the partial differential equation under study and, under some analytical constraints of the model constants and the computational parameters involved, it is capable of preserving the boundedness and the positivity of the solutions. In the linear regime, the scheme is consistent to first order in time (due partially to the inclusion of a tuning parameter in the approximation of a temporal derivative), and to second order in space. We compare the results of our computational technique against the exact solutions of some particular initial-boundary-value problems. Our simulations indicate that the method presented in this work approximates well the theoretical solutions and, moreover, that the method preserves the boundedness of solutions within the analytical constraints derived here. In the problem of approximating solitary-wave solutions of the model under consideration, we present numerical evidence on the existence of an optimum value of the tuning parameter of our technique, for which a minimum relative error is achieved. Finally, we linearly perturb a steady-state solution of the partial differential equation under investigation, and show that our simulations still converge to the same constant solution, establishing thus robustness of our method in this sense.