High-order transverse schemes for the numerical solution of PDEs

Many existing numerical schemes for the solution of initial-boundary value problems for partial differential equations can be derived by the method of lines. The PDEs are converted into a system of ordinary differential equations either with initial conditions (longitudinal scheme) or with boundary conditions (transverse scheme). In particular, this paper studies the performance of the transverse scheme in combination with boundary value methods. Moreover, we do not restrict the semi-discretization by the usual first- or second-order finite-difference approximations to replace the derivative with respect to time, but we use high-order formulae.