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Corner Cases, Singularities, and Dynamic Factoring

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Abstract

In Eikonal equations, rarefaction is a common phenomenon known to degrade the rate of convergence of numerical methods. The “factoring” approach alleviates this difficulty by deriving a PDE for a new (locally smooth) variable while capturing the rarefaction-related singularity in a known (non-smooth) “factor”. Previously this technique was successfully used to address rarefaction fans arising at point sources. In this paper we show how similar ideas can be used to factor the 2D rarefactions arising due to nonsmoothness of domain boundaries or discontinuities in PDE coefficients. Locations and orientations of such rarefaction fans are not known in advance and we construct a “just-in-time factoring” method that identifies them dynamically. The resulting algorithm is a generalization of the Fast Marching Method originally introduced for the regular (unfactored) Eikonal equations. We show that our approach restores the first-order convergence and illustrate it using a range of maze navigation examples with non-permeable and “slowly permeable” obstacles.

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Notes

  1. This formula for \(u(\mathbf{x })\) is derived on the unbounded domain \(\varOmega _{\infty } = \{ \mathbf{x }\in R^2 \mid F(\mathbf{x }) > 0 \}\) but remains valid on \(\varOmega = [0,1]\times [0,1]\) as long as \(\varOmega _{\infty }\)-optimal trajectories from every \(\mathbf{x }\in \varOmega \) to \(\mathbf{x }_0\) stay entirely inside \(\varOmega ,\) which is the case for all examples considered in this section. The linearity of \(F(\mathbf{x })\) can be used to show that all optimal paths are circular arcs, whose radii are monotone decreasing in \(|\mathbf{v }|\). (When \(\mathbf{v }=0,\) these radii are infinite; i.e., all optimal paths are straight lines and \(u(\mathbf{x }) = s_0 |\mathbf{x }-\mathbf{x }_0|.\))

  2. Throughout the paper we refer to this approach as “additive factoring” to stay consistent with the terminology used in prior literature.

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Acknowledgements

The authors are grateful to anonymous reviewers for their suggestions on improving this paper.

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  1. Center for Applied Mathematics, Cornell University, Ithaca, NY, 14853, USA

    Dongping Qi

  2. Department of Mathematics and Center for Applied Mathematics, Cornell University, Ithaca, NY, 14853, USA

    Alexander Vladimirsky

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  1. Dongping Qi
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  2. Alexander Vladimirsky
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Correspondence to Alexander Vladimirsky.

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The second author’s work is supported in part by the National Science Foundation Grant DMS-1738010 and the Simons Foundation Fellowship.

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Qi, D., Vladimirsky, A. Corner Cases, Singularities, and Dynamic Factoring. J Sci Comput 79, 1456–1476 (2019). https://doi.org/10.1007/s10915-019-00905-6

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  • DOI: https://doi.org/10.1007/s10915-019-00905-6

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