×

An approximate inertial manifold for computing Burgers’ equation. (English) Zbl 0789.65069

The reviewer feels that he should neither change nor attempt to improve the authors’ own description that they give in their abstract:
We present a numerical scheme for the approximation of nonlinear evolution equations over large time intervals. Our algorithm is motivated from the dynamical systems point of view. In particular, we adapt the methodology of approximate inertial manifolds to a finite difference scheme. This leads to a differential treatment in which the higher (i.e. unresolved) modes are expressed in terms of the lower modes. As a particular example we derive an approximate inertial manifold for Burgers’ equation and develop a numerical algorithm suitable for computing. We perform a parameter study in which we compare the accuracy of a standard scheme with our modified scheme. For all values of the parameters (which are the coefficient of viscosity and the cell size), we obtain a decrease in the numerical error by at least a factor of 2.0 with the modified scheme. The decrease in error is substantially greater over large regions of the parameter space.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
35G10 Initial value problems for linear higher-order PDEs
35K25 Higher-order parabolic equations
35K55 Nonlinear parabolic equations
54H20 Topological dynamics (MSC2010)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] M. Chen and R. Temam, Incremental unknowns for solving partial differential equations, Num. Math., to appear.; M. Chen and R. Temam, Incremental unknowns for solving partial differential equations, Num. Math., to appear. · Zbl 0712.65103
[2] Ch. Devulder, M. Marion and E.S. Titi, On the rate of convergence of nonlinear Galerkin methods, submitted.; Ch. Devulder, M. Marion and E.S. Titi, On the rate of convergence of nonlinear Galerkin methods, submitted. · Zbl 0783.65053
[3] Fabes, E.; Luskin, M.; Sell, G., J. Diff. Eq., 89, 355-387 (1991) · Zbl 0728.34047
[4] Foias, C.; Manley, O. P.; Temam, R., Math. Mod. and Num. Anal. \(M^2 AN , 22, 93-114 (1988)\)
[5] Foias, C.; Sell, G.; Titi, E. S., J. Dynam. Diff. Eq., 1, No. 2, 199-243 (1989)
[6] Foias, C.; Temam, R., J. Math. Pures Appl., 58, 339-368 (1979) · Zbl 0454.35073
[7] Foias, C.; Titi, E. S., Nonlinearity, 4, 135-153 (1991) · Zbl 0714.34078
[8] Jolly, M. S.; Kevrekidis, I. G.; Titi, E. S., Physica D, 44, 38-60 (1990) · Zbl 0704.58030
[9] Marion, M., J. Dynam. Diff. eqs., 1, 245-267 (1989) · Zbl 0702.35127
[10] M. Marion and R. Temam, 57 (1990) 205-226.; M. Marion and R. Temam, 57 (1990) 205-226. · Zbl 0702.65081
[11] Temam, R., (J. Fac. of Sci., 36 (1989), Univ. of Tokyo: Univ. of Tokyo IA), 629-647, 1989 · Zbl 0698.58040
[12] Temam, R., SIAM J. Math. Anal., 21, 154-178 (1990) · Zbl 0715.35039
[13] R. Temam, Math. Comput., to appear.; R. Temam, Math. Comput., to appear.
[14] Titi, E. S., J. Math. Anal. Appl., 149, 540-557 (1990) · Zbl 0723.35063
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.