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Numerical Simulation of Singularity Propagation Modeled by Linear Convection Equations with Spatially Heterogeneous Nonlocal Interactions

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Abstract

We study the propagation of singularities in solutions of linear convection equations with spatially heterogeneous nonlocal interactions. A spatially varying nonlocal horizon parameter is adopted in the model, which measures the range of nonlocal interactions. Via heterogeneous localization, this can lead to the seamless coupling of the local and nonlocal models. We are interested in understanding the impact on singularity propagation due to the heterogeneities of the nonlocal horizon and the local and nonlocal transition. We first analytically derive equations to characterize the propagation of different types of singularities for various forms of nonlocal horizon parameters in the nonlocal regime. We then use asymptotically compatible schemes to discretize the equations and carry out numerical simulations to illustrate the propagation patterns in different scenarios.

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Data Availability Statement

The datasets generated during the current study are available from the corresponding author on reasonable request. They support our published claims and comply with field standards.

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Acknowledgements

The authors would like to thanks Jiwei Zhang for helpful discussions on the subject.

Funding

Research of Yan Xu is supported by NSFC Grant No. 12071455. Research of Qiang Du is supported by NSF DMS-2012562.

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Authors and Affiliations

  1. School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui, 230026, People’s Republic of China

    Xiaoxuan Yu & Yan Xu

  2. Department of Applied Physics and Applied Mathematics, and Data Science Institute, Columbia University, New York, NY, 10027, USA

    Qiang Du

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  1. Xiaoxuan Yu
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  2. Yan Xu
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Contributions

All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by Xiaoxuan Yu. The first draft of the manuscript was written by Xiaoxuan Yu and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

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Correspondence to Yan Xu.

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The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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Yan Xu: Research supported by NSFC grants 12071455.

Qiang Du: Research supported in part by NSF DMS-2012562.

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Yu, X., Xu, Y. & Du, Q. Numerical Simulation of Singularity Propagation Modeled by Linear Convection Equations with Spatially Heterogeneous Nonlocal Interactions. J Sci Comput 92, 59 (2022). https://doi.org/10.1007/s10915-022-01915-7

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  • DOI: https://doi.org/10.1007/s10915-022-01915-7

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