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A Prolate-Element Method for Nonlinear PDEs on the Sphere

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Abstract

A p-type spectral-element method using prolate spheroidal wave functions (PSWFs) as basis functions, termed as the prolate-element method, is developed for solving partial differential equations (PDEs) on the sphere. The gridding on the sphere is based on a projection of the prolate-Gauss-Lobatto points by using the cube-sphere transform, which is free of singularity and leads to quasi-uniform grids. Various numerical results demonstrate that the proposed prolate-element method enjoys some remarkable advantages over the polynomial-based element method: (i) it can significantly relax the time step size constraint of an explicit time-marching scheme, and (ii) it can increase the accuracy and enhance the resolution.

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Authors and Affiliations

  1. Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, 637371, Singapore

    Jing Zhang & Li-Lian Wang

  2. School of Mathematical Sciences, Xiamen University, Xiamen, 361005, China

    Zhijian Rong

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  1. Jing Zhang
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  2. Li-Lian Wang
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  3. Zhijian Rong
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Correspondence to Li-Lian Wang.

Additional information

This work is partially supported by Singapore AcRF Tier 1 Grant RG58/08, Singapore MOE Grant T207B2202, and Singapore NRF2007IDM-IDM002-010.

The work of the second author is also partially supported by a Leading Academic Discipline Project of Shanghai Municipal Education Commission, China (No. J50101).

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Zhang, J., Wang, LL. & Rong, Z. A Prolate-Element Method for Nonlinear PDEs on the Sphere. J Sci Comput 47, 73–92 (2011). https://doi.org/10.1007/s10915-010-9421-y

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  • DOI: https://doi.org/10.1007/s10915-010-9421-y

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