The cubic spline solution of practical problems modelled by hyperbolic partial differential equations

An approximation is developed for a general one-dimensional hyperbolic partial differential equation with constant coefficients and function value boundary conditions. The time derivative is replaced by a finite difference representation and the space derivative by a cubic spline. As expected, a three-level finite difference formula in time is obtained giving the solution at each succeeding time level. The spline approximation produces a spline function which can be used on each time level to obtain the solution at any points intermediate to the mesh points. The numerical scheme is extended to the more general variable-coefficient case and for derivative boundary conditions. Truncation errors and stability criteria are produced, and the scheme is rigorously tested on practical problems, a comparison being made with a more well-known fully implicit finite difference scheme.