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arXiv · 1701.06030

Fourth-order time-stepping for stiff PDEs on the sphere

Hadrien Montanelli·Yuji Nakatsukasa
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Abstract

We present in this paper algorithms for solving stiff PDEs on the unit sphere with spectral accuracy in space and fourth-order accuracy in time. These are based on a variant of the double Fourier sphere method in coefficient space with multiplication matrices that differ from the usual ones, and implicit-explicit time-stepping schemes. Operating in coefficient space with these new matrices allows one to use a sparse direct solver, avoids the coordinate singularity and maintains smoothness at the poles, while implicit-explicit schemes circumvent severe restrictions on the time-steps due to stiffness. A comparison is made against exponential integrators and it is found that implicit-explicit schemes perform best. Implementations in MATLAB and Chebfun make it possible to compute the solution of many PDEs to high accuracy in a very convenient fashion.

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Hadrien Montanelli, Yuji Nakatsukasa. 2017-01-21. Fourth-order time-stepping for stiff PDEs on the sphere. https://arxiv.org/abs/1701.06030

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