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Optimal error estimates of the Chebyshev-Legendre spectral method for solving the generalized Burgers equation. (English) Zbl 1050.65083
The authors consider a one-dimensional Burgers equation with nonlinear flux function and approximate it using Chebyshev-Legendre collocation (semidiscretized case) and Legendre-Galerkin-Chebyshev collocation (fully discretized case, with Crank-Nicolson stepping in time for the diffusion part and leapfrog for the convection part). Stability and convergence is shown for Dirichlet conditions, whereas for the other boundary conditions there are some comments. In the fully discretized case, $$\tau\sqrt{N}$$ is required to be sufficiently small.
Some notations are not explained in the paper (like $$\tau_0$$ etc.); it is probably better to be understood if one reads it together with the 1994 paper of W. S. Don and D. Gottlieb [SIAM J. Numer. Anal. 31, No. 6, 1519–1534 (1994; Zbl 0815.65106)].
The authors also show numerical results which demonstrate the high accuracy reachable and the fact that Legendre-Galerkin-Chebyshev collocation is of equal accuracy but considerably faster than Legendre collocation on fine grids.

##### MSC:
 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 35Q53 KdV equations (Korteweg-de Vries equations) 65M70 Spectral, collocation and related 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 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
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