Numerical methods for coupled systems of nonlinear parabolic boundary value problems (Q750098)

The coupled system of nonlinear equations \(u_ t-D^{(1)}\nabla^ 2u=f^{(1)}(x,t,u,v),\) \(v_ t-D^{(2)}\nabla^ 2v=f^{(2)}(x,t,u,v),\) (t\(\in [0,T]\), \(x\in \Omega)\) with the initial- boundary conditions \(\alpha^{(1)}\partial u/\partial \mu +\beta^{(1)}u=g^{(1)}(x,t),\alpha^{(2)}\partial v/\partial \nu +\beta^{(2)}v=g^{(2)}(x,t),\) \(u(x,0)=\psi^{(1)}_{(x)},\) \(v(x,0)=\psi^{(2)}(x)\) is considered. Here \(\Omega\) is a bounded domain in \(R^ p\), \(D^{(\ell)}\equiv D^{(\ell)}(x,t)>0\), \(\alpha^{(\ell)}=\alpha^{(\ell)}(x)\geq 0\), \(\beta^{(\ell)}=\beta^{(\ell)}(x)\geq 0\) with \(\alpha^{(\ell)}+\beta^{(\ell)}>0\) on \(\partial \Omega\) and \(f^{(\ell)}\), \(g^{(\ell)}\), \(\psi^{(\ell)}\) are Hölder continuous functions of their respective arguments. Using the method of upper-lower solutions monotone sequences for the finite difference equations are constructed. The convergence of the finite difference solution to the differential solution as the mesh size decreases to zero is proved.