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Local spectral time splitting method for first- and second-order partial differential equations. (English) Zbl 1075.65130
The paper deals with a solution of a class of partial differential equations (PDEs) with the aid of local spectral evolution kernels (LSEK) which allow to solve the mentioned PDEs with $$x$$-independent coefficients in a single step. They are utilized in an operator splitting scheme to arrive at a local spectral time-splitting (LSTS) method. The authors derive the LSEKs and review the time splitting schemes. Accuracy, stability, efficiency and parameter dependence of LSTS is investigated. Then LSTS is applied to Fisher’s equation, the Schrödinger equation and the Gross-Pitaevskii equations for Bose-Einstein condensation, and the numerical results are compared with reference ones.

##### MSC:
 65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs 35K15 Initial value problems for second-order parabolic equations 35Q40 PDEs in connection with quantum mechanics 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics 65T60 Numerical methods for wavelets 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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