## Une réhabilitation des méthodes spectrales de type Laguerre. (Reappraisal of Laguerre type spectral method).(French)Zbl 0604.42026

The authors consider the numerical solution of Partial Differential equations involving space and time. The methods used involve step by step integration in time, with special approximations by Laguerre Polynomials. The stability conditions are given for various types of step by step integration, and a number of bounds are given for the norms of solutions. As an illustration, results are given for the problem of solving the Euler equations in one dimension for a homentropic gas obeying the equation of state $$p=1/3\rho^ 3$$ for which an exact solution is known. This includes the application of a shock wave.
Reviewer: Ll.G.Chambers

### MSC:

 42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.) 65Z05 Applications to the sciences 35K35 Initial-boundary value problems for higher-order parabolic equations 35L40 First-order hyperbolic systems 76N99 Compressible fluids and gas dynamics 65N35 Spectral, collocation and related methods for boundary value problems involving PDEs