Numerical Approximations Using Chebyshev Polynomial Expansions

Abstract

We present numerical solutions for differential equations by expanding the unknown function in terms of Chebyshev polynomials and solving a system of linear equations directly for the values of the function at the extrema (or zeros) of the Chebyshev polynomial of order N (El-gendi's method). The solutions are exact at these points, apart from round-off computer errors and the convergence of other numerical methods used in connection to solving the linear system of equations. Applications to initial value problems in time-dependent quantum field theory, and second order boundary value problems in fluid dynamics are presented.

minor wording changes, some typos have been eliminated

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• Real-time dynamics of the O(N) model in 1+1 dimensions
• Fermion pair production in QED and the backreaction problem in (1+1)-dimensional boost-invariant coordinates revisited
• Backreaction and Particle Production in (3+1)-dimensional QED
• Integral representation for scattering phase shifts via the phase-amplitude approach
• Parallel algorithm with spectral convergence for nonlinear integro-differential equations
• Continuum versus periodic lattice Monte Carlo approach to classical field theory
• A well-conditioned collocation method using pseudospectral integration matrix
• Production of Anticentauro Events in Ultra-Relativistic Heavy Ion Collisions
• The class of Lucas-Lehmer polynomials
• Comparing Pure Yang-Mills SU(2) and SU(3) Propagators
• An accurate spectral method for solving the Schroedinger equation