An accelerated technique for the coupled system of reaction-diffusion-transport equations arising from catalytic converters (Q2157669)

The authors consider the coupled system \(\frac{\partial u}{\partial x} +\alpha ^{\prime }u=\frac{\alpha ^{\prime }u}{1+K^{\prime }e^{\gamma w}}\), \( \frac{\partial v}{\partial x}+\beta ^{\prime }v=\beta ^{\prime }w\), \(\frac{ \partial w}{\partial t}-D\frac{\partial ^{2}w}{\partial x^{2}}+\beta w=\beta v+\frac{\alpha u}{1+Ke^{-\gamma w}}\), posed in \((0,L)\times (0,T)\), for different constants, and which simulates the behavior of a catalytic converter. The initial conditions \(u(0,t)=h_{1}(t)\), \(v(0,t)=h_{2}(t)\) are added with non-negative initial data and the boundary conditions \( w(x,0)=w_{0}(x)\), \(w_{x}(0,t)=w_{x}(L,t)=0\) are imposed. Assuming that the surface converter is known and solving the second equation, the authors end with the simplified model \(v(x,t)=h_{2}(t)e^{-\beta ^{\prime }x}+\beta ^{\prime }\int_{0}^{x}e^{-\beta ^{\prime }(x-y)}w(y,t)dy\), \(\frac{\partial w }{\partial t}-D\frac{\partial ^{2}w}{\partial x^{2}}+\beta w=\beta v+B(x,t)e^{\gamma w}\), with the above initial and boundary conditions for \(w\). The purpose of the authors is to develop and analyze five monotone iterative methods for these models. They define the notion of lower (resp. upper) solution to the second problem as a pair \((v,w)\in \mathcal{X}_{2}=C( \overline{Q_{T}})\times (C^{2,1}(\overline{Q_{T}})\cap C(\overline{Q_{T}}))\) , which satisfies inequalities \(\leq \) (resp. \(\geq \)) instead of equalities in both the equations and initial and boundary conditions. They also define the notion of ordered lower and upper solution. The first main results of the paper prove the existence of a solution to the first or second problem if ordered lower and upper solutions exist, and the convergence of algorithms. Then the authors compare the rate of convergence for the different algorithms. For the proof, the authors first prove regularity and positivity results for parabolic equations generalizing the above problem. They finally prove error estimates for these algorithms.