An orthogonal spline collocation alternating direction implicit method for second-order hyperbolic problems

Abstract

On a rectangular region, we consider a linear second-order hyperbolic initial-boundary value problem involving a mixed derivative term, continuous variable coefficients and nonhomogeneous Dirichlet boundary conditions. In comparison to the alternating direction implicit Laplace-modified method of Fernandes (1997), we formulate and analyse a new parameter-free alternating direction implicit scheme in which the standard central difference formula is used for the time approximation and orthogonal spline collocation is used for the spatial discretization. We establish unconditional stability of the scheme, and its optimal order in the discrete maximum norm in time and the H 1 norm in space. Numerical experiments indicate that the new scheme, which has the same order as the method of Fernandes (1997, Numer. Math., 77, 223-241), is more accurate. We also show that the new scheme is easily generalized to the second-order hyperbolic problems on rectangular polygons. Extensions of the scheme to problems with discontinuous coefficients, nonlinear problems, and problems with other boundary conditions are also discussed.