Sufficient conditions for the preservation of the boundedness in a numerical method for a physical model with transport memory and nonlinear damping

Departing from a finite-difference scheme to approximate solutions of a nonlinear, hyperbolic partial differential equation which generalizes the Burgers–Huxley equation from fluid dynamics, we investigate conditions on the model coefficients and the computational parameters under which positive and bounded initial data evolve into positive and bounded new approximations. The model under investigation includes nonlinear coefficients of damping and advection, and the reaction term extends the reaction law of the classical Fisher–Kolmogorov–Petrovsky–Piscounov equation. The method can be expressed in vector form in terms of a multiplicative matrix which, under certain parametric conditions, becomes an M-matrix. Using the fact that every M-matrix is non-singular and that the entries of its inverse are positive, real numbers, we establish sufficient conditions under which the method provides new, positive and bounded approximations from previous, positive and bounded data and boundary conditions. The numerical results confirm the fact that the conditions derived here are sufficient for the positivity and the boundedness of the approximations; moreover, computational experiments evidence the fact that the method still preserves these properties for values of the model and the numerical parameters outside of the analytic regions of positivity and boundedness. We point out that our simulations show a good agreement between the numerical approximations computed through our method and the corresponding, analytical solutions.

► Introduction of a method for the solutions of a generalized Burgers–Huxley equation.
► Derivation of conditions for the positivity and the boundedness of the method.
► Use of the method to approximate traveling-wave solutions.