Solution of non-linear partial differential equations with the Chebyshev Spectral method

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Virginia Tech


The Spectral method is a powerful numerical technique for solving engineering differential equations. The method is a specialization of the method of weighted residuals. Trial functions that are easily and exactly differentiable are used. Often the functions used also satisfy an orthogonality equation, which can improve the efficiency of the approximation. Generally, the entire domain is modeled, but multiple sub-domains may be used.

A Chebyshev-Collocation Spectral method is used to solve a variety of ordinary and partial differential equations. The Chebyshev Polynomial series follows a well established recursion relation for calculation of the polynomials and their derivatives. Two different schemes are studied for formulation of the problems, a Fast Fourier Transform approach, and a matrix multiplication approach. First, the one-dimensional ordinary differential equation representing the deflection of a tapered bar under its own weight is studied. Next, the two dimensional Poisson's equation is examined. Lastly, a two dimensional, highly non-linear, two parameter Bratu's equation is solved.

Each problem's results are compared to results from other methods or published data. Accuracy is very good, with a significant improvement in computer time.