Monotone iterative finite volume algorithms for coupled systems of first‐order nonlinear PDEs

Abstract

AbstractThis paper studies coupled systems of first‐order nonlinear partial differential equations, where the first equation is an advection equation with a source term. The system is known to model physical phenomena such as general blood–tissue exchange (BTEX) and the gas–solid interphase heat transfer for the fast igniting catalytic converter. We propose a finite volume implicit–explicit approximation for the system and establish the existence and uniqueness of the classical solution of the system using the method of upper and lower solutions. The error estimates for the numerical scheme are also derived for each iteration of the monotone iterative method. Numerical tests show that the proposed scheme can accurately describe the behavior of various physical phenomena. The performance of the scheme is compared with the existing results in the literature and the numerical solutions are shown to preserve physical properties of the solutions such as positivity, blow up behavior in finite time, and concentration of the impulse.