On an exact numerical simulation of solitary-wave solutions of the Burgers-Huxley equation through Cardano's method
This article deals with one-dimensional Burgers-Huxley equations where two analytical solutions are known. The determination of exact solitary wave solutions remains nevertheless an open problem. The author proposes an exact numerical approximation which provides an adequate model for analytical solutions and therefore also for proposed solitary wave solutions. The approach is shortly described as following: the author discretizes the equation by a symmetric finite difference scheme in spatial and temporal directions. He computes then the approximate solution for the next temporal node by exactly solving a cubic polynomial equation arising from the spatial discretization. The author proves that the solutions, derived by this method under suitable constraints on the arameters, has the same mathematical properties as the analytical solutions: they are positive, bounded and monotone in spatial and temporal direction. By simulation results he confirms further that his approach provides a good approximations of the analytical solutions. His method, by using a symmetric finite difference scheme, is limited to problems in which advection is not dominating. For advection-dominated problems one should use instead upwind or downwind discretization schemes to avoid instabilities.
scientific article; zbMATH DE number 6363087
scientific article