Numerical approximation of the insitu combustion model using the nonlinear mixed complementarity method

TL;DR

This paper addresses numerical solution of a parabolic in-situ combustion model by formulating it as a nonlinear mixed complementarity problem and solving it with a feasible interior-point method (FDA-MNCP). It combines a Crank-Nicolson-based implicit finite-difference discretization with an MCP framework, enabling global convergence and robust handling of nonnegativity constraints on temperature-like variables. Numerical experiments show FDA-MNCP yields solutions very close to those obtained by the FDA-NCP method across various spatial discretizations, with scalability advantages at finer meshes. The study demonstrates the potential of FDA-MNCP for parabolic (and extendable to hyperbolic) problems that admit MCP reformulations, providing a robust and efficient tool for large-scale discretizations in combustion modeling and related PDE systems.

Abstract

In this work, we will study a numerical method that allows finding an approximation of the exact solution for a in-situ combustion model using the nonlinear mixed complementary method, which is a variation of the Newtons method for solving nonlinear systems based on an implicit finite difference scheme and a nonlinear algorithm mixed complementarity, FDA-MNCP. The method has the advantage of provide a global convergence in relation to the finite difference method and method of Newton that only has local convergence. The theory is applied to model in-situ combustion, which can be rewritten in the form of mixed complementarity also we do a comparison with the FDA-NCP method

Figures (7)

• Figure 1: Comparison of the FDA-MNCP and FDA-NCP methods for M = 50 at time instants $t = 0.000, 0.002, 0.004, 0.006, 0.008, 0.010$. The values of $\theta$ are represented by green dots and a solid red line, the values of $\eta$ are represented by pink dots and a solid blue line.
• Figure 2: Comparison of the FDA-MNCP and FDA-NCP methods for$M = 100$ at times $t = 0.000, 0.002, 0.004, 0.006, 0.008, 0.010$. The values of $\theta$ are represented by green dots and a solid red line, the values of $\eta$ are represented by pink dots and a solid blue line.
• Figure 3: Comparison of the FDA-MNCP and FDA-NCP methods for $M = 200$ at times $t = 0.000, 0.002, 0.004, 0.006, 0.008, 0.010$. The values of $\theta$ are represented by green dots and a solid red line, the values of $\eta$ are represented by pink dots and a solid blue line.
• Figure 4: Comparison of the FDA-MNCP and FDA-NCP methods for $M = 400$ at times $t = 0.000, 0.002, 0.004, 0.006, 0.008, 0.010$. The values of $\theta$ are represented by green dots and a solid red line, the values of $\eta$ are represented by pink dots and a solid blue line.
• Figure 5: Difference between DA-NCP and FDA-NCP methods.