A Legendre spectral method in time for first-order hyperbolic equations

In this paper, we take first-order hyperbolic equations with periodic boundary conditions as a model to present a Legendre spectral method in time with Fourier approximation in spatial. Convergence analysis of the spectral scheme is given and the L2-optimal error estimate in spatial is achieved. Also, the method is valid for variable coefficient case. Numerical results show the efficiency of the method.