k^th-Order Closed Form Difference Equations for 1-Dimensional, First and Second order, Linear Hyperbolic Equations, with Applications to Quasilinear Systems

Abstract

In Huang and Cushman (1981) a high order accurate nonstandard finite element technique was presented for deriving finite difference schemes of k^{^tk}-order accuracy for the linear in- viscid Burger’s equation and a second order linear hyperbolic equation. The schemes however were not presented in closed form and as we will see for linear equations the schemes are similar to characteristic schemes. The technique when used on nonlinear equations (e.g. the shallow water equations) requires the inversion of a matrix at each node (the dimension of the matrix increases with increasing order of interpolation). This inversion operation can be a very time-consuming process. We thus propose to present a closed form equation to eliminate the necessity of inverting a matrix at each node.