A collocation-Galerkin method for a first order hyperbolic equation with space and time dependent coefficient

A collocation-Galerkin scheme is proposed for an initial-boundary value problem for a first order hyperbolic equation in one space dimension. The Galerkin equations satisfied by the approximating solution are obtained from a weak-weak formulation of the initial-boundary value problem. The collocation points are taken to be affine images of the roots of the Jacobian polynomials of degree $r - 1$ on [0, 1] with respect to the weight function $x(1 - x)$. Optimal ${L^\infty }({L^2})$-norm estimates of the error are derived.