Application of a collocation method to unsteady flow problems

Abstract

Abstract A Chebyshev collocation numerical model is developed for solving time-dependent linear and non-linear partial differential equations. This method approximates a solution to a differential equation at certain selected points with a series expansion of orthogonal polynomials. Chebyshev polynomials are used in the numerical model and the selected points at which the numerical approximation is done are known as the collocation points. These collocation points are the extrema of the Chebyshev polynomial. The usage of the Chebyshev polynomial as the interpolating function enables one to evaluate the spectral representation (in terms of the Chebyshev coefficients) of the actual function using the Fast Fourier Transform (FFT) technique. The numerical approximations of the spatial derivatives of the function in question are derived from the global consideration of the Chebyshev coefficients of the function. Numerical solutions to three problems are obtained and are discussed in this paper. Agreement with analytical and other numerical solutions is excellent.