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A second-order accurate linearized difference scheme for the two-dimensional Cahn-Hilliard equation. (English) Zbl 0847.65056
The Cahn-Hilliard equation is the nonlinear evolution equation of first order \[ u_t+ \Delta^2 u= \Delta \varphi (u), \qquad (x, y, t)\in \Omega \times (0, T\rangle \] for \(u(x, y, t)\), subject to the boundary conditions \[ {{\partial u} \over {\partial v}}= 0, \quad {\partial \over {\partial v}} (\varphi (u)- \Delta u)=0 \quad \text{on } \partial \Omega \times (0, T\rangle \] and the initial condition \[ u(x, y, 0)= u_0 (x, y), \qquad (x,y)\in \overline {\Omega}, \] where \(\varphi (\cdot)= \psi' (\cdot)\), \(\psi (u)= \gamma (u^2- \beta^2)^2 /4\), \(\gamma >0\), \(\Omega\) is the interior of the rectangle \(\langle 0,L_1 \rangle \times \langle 0,L_2 \rangle\), and \(v\) is the outward pointing normal to \(\partial \Omega\).
A linearized finite difference scheme is derived by the method of order reduction. It is proved that the derived scheme is uniquely solvable and convergent. The rate of convergence is estimated by the order two in the discrete \(L_2\) norm. The coefficient matrix of the difference system is symmetric and positive definite, so many well-known iterative methods (e.g. Gauss-Seidel or SOR method) can be used to solve the system.

MSC:
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35K55 Nonlinear parabolic equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
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