On some dissipative fully discrete nonlinear Galerkin schemes for the Kuramoto-Sivashinsky equation. (English) Zbl 0941.65526
Summary: We show that two fully discrete nonlinear Galerkin schemes based on explicit approximate inertial manifolds preserve the dissipativity of the Kuramoto-Sivashinsky equation (KSE). The radius of the absorbing ball is shown to be uniform in both the time step and number of modes, so that the result holds in the PDE limit. While the schemes are specifically designed to deal with the difficulty of the linear instability in the KSE, simpler schemes can be derived following this approach for other dissipative nonlinear evolutionary equations, such as the 2D Navier-Stokes equations.
MSC:
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 35Q35 | PDEs in connection with fluid mechanics |
| 47H20 | Semigroups of nonlinear operators |
| 34G20 | Nonlinear differential equations in abstract spaces |
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