(In Preparation) High order time stepping methods for precise Fourier pseudo-spectral discretizations of the sine Gordon equation


This paper is in preparation.

Abstract: We examine different time stepping methods for solving the sine Gordon equation using Fourier pseudo-spectral spatial discretizations. We also compare the accuracy of the schemes in single, double and quadruple precision arithmetic. We are particularly interested in accurate results over long times. We find that in double precision, sixth and eighth order implicit Gauss Legendre Runge-Kutta schemes give the most accurate results at the lowest computational cost. We also find that integrating factor methods, often result in larger errors over long integration times. Finally, we show that high order methods can accurately compute solutions near unstable homoclinic orbits.

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