General hyperbolic difference formulas for linear and quasilinear hyperbolic equations

The method of non‐standard finite elements was used to develop multilevel difference schemes for linear and quasilinear hyperbolic equations with Dirichlet boundary conditions. A closed form equation of k th‐order accuracy in space and time ( O (Δ t k , Δ x k )) was developed for one‐dimensional systems of linear hyperbolic equations with Dirichlet boundary conditions. This same equation is also applied to quasilinear systems. For the quasilinear systems a simple iteration technique was used to maintain the k th‐order accuracy. Numerical results are presented for the linear and non‐linear inviscid Burger's equation and a system of shallow water equations with Dirichlet boundary conditions.