Extrapolation techniques for first order hyperbolic partial differential equations

Abstract

A uniform grid of step size h is superimposed on the space variable x in the first order hyperbolic partial differential equation ∂u/∂t + a ∂u/∂x = 0 (a > 0, x > 0, t > 0). The space derivative is approximated by its backward difference and central difference replacements and the resulting linear systems of first order ordinary differential equations are solved employing Padé approximants to the exponential function. A number of difference schemes for solving the hyperbolic equation are thus developed and each is extrapolated to give higher order accuracy. The schemes, and their extrapolated forms, are applied to two problems, one of which has a discontinuity in the solution across a characteristic.