An enhanced pseudospectral Chebyshev method for dissipative partial differential equations
Inertial manifolds were introduced by R. Temam and co-workers in 1985 [cf. \textit{R. Temam}, Infinite-dimensional dynamical systems in mechanics and physics (1988; Zbl 0662.35001)] for the study of nonlinear dissipative partial differential equations from a dynamical systems points of view. For the numerical solution of such problems, approximate inertial manifolds (AIM) were also considered. As it was analyzed by \textit{B. Garcia-Archilla}, \textit{J. Novo} and \textit{E. Titi} [SIAM J. Numer. Anal. 35, No. 3, 941-972 (1998; Zbl 0914.65105)] for a spectral Galerkin method using eigenfunctions, it is sufficient to evaluate the AIM only once after the time integration. With the paper in hand, the authors extend this idea of postprocessing to more realistic methods using orthogonal Chebyshev polynomials instead of unknown eigenfunctions. In addition, this new method is shown to be convergent and efficient even for limited regularity of the exact solution. Numerical examples for the viscous Burgers equation illustrate the method. For the time integration, a variable stepsize variable order backward differentiation formula is used.
dissipative partial differential equations
scientific article; zbMATH DE number 1423314