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Parallelization, initialization, and boundary treatments for the diamond scheme

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Abstract

We study a class of general purpose linear multisymplectic integrators for Hamiltonian wave equations based on a diamond-shaped mesh. On each diamond, the PDE is discretized by a symplectic Runge–Kutta method. The scheme advances in time by filling in each diamond locally. We demonstrate that this leads to greater efficiency and parallelization and easier treatment of boundary conditions compared with methods based on rectangular meshes. We develop a variety of initial and boundary value treatments and present numerical evidence of their performance. In all cases, the observed order of convergence is equal to or greater than the number of stages of the underlying Runge–Kutta method.

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Notes

  1. For example, for wide-stencil finite difference methods, a significant development effort over many years has yielded stable methods using the summation by parts and simultaneous approximation term approaches [3, 6, 12, 19]. These finite difference operators approximate ux (resp. uxx) at all points, using different finite differences near the boundary. Stability is achieved by requiring that the finite difference is skew- (resp. self-) adjoint with respect to an inner product, designed along with the method.

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Funding

This research was supported by the Marsden Fund of the Royal Society Te Ap\(\bar {\mathrm {a}}\)rangi and by a Massey University PhD Scholarship.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Victoria University of Wellington, Wellington, New Zealand

    Stephen Marsland

  2. School of Fundamental Sciences, Massey University, Palmerston North, New Zealand

    Robert I. McLachlan & Matthew C. Wilkins

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  1. Stephen Marsland
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  2. Robert I. McLachlan
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  3. Matthew C. Wilkins
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Correspondence to Robert I. McLachlan.

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Marsland, S., McLachlan, R.I. & Wilkins, M.C. Parallelization, initialization, and boundary treatments for the diamond scheme. Numer Algor 84, 761–779 (2020). https://doi.org/10.1007/s11075-019-00778-8

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  • DOI: https://doi.org/10.1007/s11075-019-00778-8

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