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A Hermite spectral method for solitons. (English) Zbl 1173.65069
We use the analytical soliton solutions of the Korteweg-de Vries (KdV) equation to test a new spectral numerical method for partial differential evolution equations with unbounded spatial domain. The proposed spatial discretization uses Hermite functions in the spectral space while the temporal discretization is performed by a symmetric exponential integrator coupled with fixed point iterations. The algorithm could be used to numerically describe the soliton behaviour, such as small-amplitude long waves on the free surface of water.
65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
47H10 Fixed-point theorems