Numerical analysis of a method for a partial integro-differential equation model in regulatory gene networks

In this paper, we propose a semi-Lagrangian Runge-Kutta method to approximate the solution of a multidimensional partial integro-differential equation (PIDE) model for regulatory networks involving multiple genes with self- and cross-regulations. For the first time in the literature, we address the numerical analysis of a semi-Lagrangian method for a PIDE model without second-order derivative terms. Prom this analysis, we obtain second-order convergence in time and space. Moreover, some examples with analytical solution in one spatial dimension illustrate the theoretical results, while others in higher dimensions show the expected behavior of the solution. Finally, the scalability of the method and the comparison with a previously proposed first-order semi-Lagrangian method are discussed.