The Construction and Analysis of Compact and Noncompact Schemes

Abstract

In this paper, several families of schemes are obtained based on the idea of nth order polynomial. Each family has different schemes with different types and orders. Also, the error terms, which determine the order of the scheme, of each scheme in all families are computed using the same methodology. Traditional finite different schemes beside backward, central, and forward compact schemes at each family are introduced. This work proposes a clear and simple method of constructing finite difference schemes, and it presents the flexibility and the properties of compact schemes. Beside the feature of the high order accuracy, which are gained by compact schemes without increasing the width of points set, compact schemes achieve better spectrum resolution compared to the traditional noncompact ones. Additionally, comparing the numerical dissipation of many schemes illustrates the favor of compact schemes when using problems with high frequency. Finally, there is an effort to solve and compare solutions of some standard problems from Computational Fluid Dynamics (CFD) using compact and noncompact schemes.