Numerical Approximation of the In Situ Combustion Model Using the Nonlinear Mixed Complementarity Method

In this work, we study a numerical method to approximate the exact solution of a simple in situ combustion model. To achieve this, we use the mixed nonlinear complementarity method (MNCP), a variation of the Newton method for solving nonlinear systems, incorporating a single Hadamard product in its formulation. The method is based on an implicit finite difference scheme and a mixed nonlinear complementarity algorithm (FDA-MNCP). One of its main advantages is that it ensures global convergence, unlike the finite difference method and the Newton method, which only guarantee local convergence. We apply this theory to an in situ combustion model, reformulating it in terms of mixed complementarity. Additionally, we compare it with the FDA-NCP method, demonstrating that the FDA-MNCP is computationally more efficient when the spatial discretization is refined.