Long time behavior of finite difference approximation to the 3D nonlinear Sobolev-Galpern equation. (English) Zbl 1240.65264
Summary: A fully discrete difference scheme to the 3D nonlinear Sobolev-Galpern equation with Dirichlet initial-value boundary conditions is presented. The unique solvability of the numerical solution is shown by using discrete space functional analysis and the energy estimate method. The long-time stability and convergence of the scheme are proved. Furthermore, the existence of a global attractor for the discrete dynamical system and the upper semicontinuity \(d(\mathcal{A}_{h,\tau},\mathcal{A})\to 0\) are proved. Results show that the difference scheme can effectively simulate infinite dimensional dynamical systems.
MSC:
65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
35B41 | Attractors |
35Q35 | PDEs in connection with fluid mechanics |
37C70 | Attractors and repellers of smooth dynamical systems and their topological structure |