Crank-Nicolson's scheme with different time steps in subdomains for solving parabolic problems (Q2729356)

The article is devoted to constructing certain finite-difference schemes for parabolic problems with strongly gradient behavior of solutions.NEWLINENEWLINENEWLINEThe authors study the following initial-boundary value problem for the heat equation: NEWLINE\[NEWLINE \begin{alignedat}{2} &\frac{\partial u}{\partial t} - \Delta u = f(\vec x,t), &\quad & \vec x\in\Omega, \;t\in [0,T], \\ & u(\vec x,t) = u_0(\vec x), &\quad &\vec x\in\Omega, \\ & u(\vec x,t) = g(\vec x,t), &\quad &\vec x\in\Gamma, \;t\in [0,T]. \end{alignedat} NEWLINE\]NEWLINE where \(\Omega\) is a domain in \(\mathbb R^d\) and \(\Gamma =\partial\Omega\). It is assumed that there exist constants \(M_1\) and \(M_2\), \(M_2\gg M_1\), such that NEWLINE\[NEWLINE \left|\frac{\partial^3 u}{\partial t^3}\right|< 3M_1,\quad \vec x\in (\Omega_1)_t,\qquad \left|\frac{\partial^3 u}{\partial t^3}\right|< 3M_2,\quad \vec x\in (\Omega_2)_t\cup\widetilde{\Gamma}_t NEWLINE\]NEWLINE and \(\Omega = (\Omega_1)_t\cup\widetilde{\Gamma}_t\cup (\Omega_2)_t\). Moreover, it is assumed that the fourth spatial derivatives of a solution are bounded functions.NEWLINENEWLINENEWLINETo obtain an approximate solution to the problem, the authors expose a method of constructing difference schemes with different time steps in the subdomains \((\Omega_1)_t\) and \((\Omega_2)_t\) based on Crank-Nicolson's scheme. The authors prove that the approximate solution obtained converges in the uniform norm to an exact solution of the original differential problem with the order \(O(\tau^2 + h^2)\). Numerical test calculations are also discussed which illustrate the advantage of the suggested method.