Abstract
For many problems in Physics and Computational Fluid Dynamics (CFD), providing an accurate approximation of derivatives is a challenging task. This paper presents a class of high order numerical schemes for approximating the first derivative. These approximations are derived based on solving a special system of equations with some unknown coefficients. The construction method provides numerous types of schemes with different orders of accuracy. The accuracy of each scheme is analyzed by using Fourier analysis, which illustrates the dispersion and dissipation of the scheme. The polynomial technique is used to verify the order of accuracy of the proposed schemes by obtaining the error terms. Dispersion and dissipation errors are calculated and compared to show the features of high order schemes. Furthermore, there is a plan to study the stability and accuracy properties of the present schemes and apply them to standard systems of time dependent partial differential equations in CFD.
Highlights
Many phenomena in physics like turbulent fluid flows have a range of time scales and space scales
The length scales that are resolved by computation models are determined by the spectral resolution [1], and the accuracy with which these scales are represented depends upon the numerical scheme [2]
Explicit schemes express the nodal derivatives as an explicit weighted sum of the function values at the adjacent nodes
Summary
Many phenomena in physics like turbulent fluid flows have a range of time scales and space scales. Fourier analysis of finite difference schemes [3] shows that the errors in computing the derivatives can be quite large for smaller scales. Implicit or compact schemes equate a linear combination of the nodal derivatives to a weighted sum of the function values so that the derivatives must be calculated by solving for a matrix system implicitly. A generalized framework that can handle any stencil size on both sides will be derived, and a matrix form of the approximation will be built, so it is easy to obtain the coefficients in the scheme by solving a linear system.
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