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Fourierization of the Legendre-Galerkin method and a new space-time spectral method. (English) Zbl 1118.65111
An efficient and precise numerical solution for problems of the type $u_t={\mathcal L} u + {\mathcal N}(u,t)$ is investigated. Here $$\mathcal{L}$$ and $$\mathcal{N}$$ are high-order linear and lower order nonlinear operators respectively. This model can represent very important equations like Allen-Cahn, Burgers, Navier-Stokes, nonlinear Schrödinger and others. For approximating such problems, high order stable numerical schemes in space and time are desirable. In recent years several effective methods for solving problem formulated above were presented. Many of them are based on high-order spectral methods in space and a lower-order finite difference scheme in time which causes a mismatch in accuracy. So, it seems to be convenient for certain type of time dependent partial differential equations to use a spectral method for both, space and time.
A new space-time spectral method based on a Legendre-Galerkin method in space and a dual Petrov-Galerkin formulation in time is derived. Moreover the use of Fourier-like basis function in space may simplify the implemetnation of the new space-time spectral method. An optimal error analysis for model linear problems is proved. Implementations of the new algorithm also for nonlinear problems confirm numerical examples. They demonstrate that the proposed algorithm is unconditionally stable and effective.

##### MSC:
 65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs 65M15 Error bounds for initial value and initial-boundary value problems involving PDEs 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 65M20 Method of lines for initial value and initial-boundary value problems involving PDEs 35K55 Nonlinear parabolic equations 35Q53 KdV equations (Korteweg-de Vries equations) 35Q55 NLS equations (nonlinear Schrödinger equations)
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