The Galerkin method for first order hyperbolic equations

The first question which is addressed is the problem of stability. Just as in finite difference theory, many Galerkin methods, stable and convergent for the Cauchy problem, are unstable for hyperbolic boundary value problems because of improper treatment of the boundary conditions. The usual Galerkin method is seen to be unstable for a linear, constant coefficient, well-posed hyperbolic system on [0, 1]. A method of treating boundary conditions is then proposed which will yield a stable and convergent method for any well-posed, linear hyperbolic system in one dimension. This idea generalizes in a natural way to problems in more than one space dimension. Examples in one and two dimensions are considered.

The approximate solution, by two dissipative Galerkin methods, of the initial-value problem

∂u / ∂t = ∂u / ∂x , u(x,0) = v(x)

is the second problem considered. Convergence estimates in L₂ and ℓ₂ are derived for smooth solutions for semidiscrete schemes. When v is taken to have a jump discontinuity at zero that discontinuity will propogate along x + t = 0 . Estimates in L₂ and L_∞ of the pollution effects of the discontinuity are found. These estimates show these effects to decay exponentially in h^-1 in regions a fixed distance d from the discontinuity and exponentially in d for fixed h. Finally, analogous estimates are shown to hold under the Crank-Nicholson time discretization.

Finally, asymptotic error estimates are derived for an H^-1 method for an initial-boundary value problem for a hyperbolic equation. This method is optimal in L² and computes the approximation in piecewise polynomial spaces.