A Comparative study of finite difference and multiquadric schemes for the Euler equations

Abstract

We present a true scattered-data, grid free spatial approximation scheme called multiquadrics (MQ) first developed by Hardy. We have modified the original MQ scheme and demonstrated that it is exceptionally accurate for not only surface approximations, but also partial derivative estimates. Unlike familiar polynomial ap proximations, MQ is excellent for moderate to steep gradient regions, but rather poor in relatively flat regions. For that reason, we advocate a hybrid scheme which uses MQ in moderate to steep regions, but switches to the familiar polynomial methods in relatively flat regions. In this paper, we compare the accuracy and operation counts of finite differences and MQ schemes for the solution of time depen dent nonlinear set of hyperbolic partial differential equations. We have demonstrated both from an operations count and accuracy that MQ is superior to the familiar finite difference scheme.