Volume 27, Issue 3
A New Type of High-Order WENO Schemes for Hamilton-Jacobi Equations on Triangular Meshes

Jun Zhu & Jianxian Qiu

Commun. Comput. Phys., 27 (2020), pp. 897-920.

Published online: 2020-02

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  • Abstract

In this paper, a new type of third-order and fourth-order weighted essentially non-oscillatory (WENO) schemes is designed for simulating the Hamilton-Jacobi equations on triangular meshes. We design such schemes with the use of the nodal information defined on five unequal-sized spatial stencils, the application of monotone Hamiltonians as a building block, the artificial set of positive linear weights to make up high-order approximations in smooth regions simultaneously avoiding spurious oscillations nearby discontinuities of the derivatives of the solutions. The spatial reconstructions are convex combinations of the derivatives of a modified cubic/quartic polynomial defined on a big spatial stencil and four quadratic polynomials defined on small spatial stencils, and a third-order TVD Runge-Kutta method is used for the time discretization. The main advantages of these WENO schemes are their efficiency, simplicity, and can be easily implemented to higher dimensional unstructured meshes.Extensive numerical tests are performed to illustrate the good performance of such new WENO schemes.

  • Keywords

Unequal-sized stencil, weighted essentially non-oscillatory scheme, high-order ap- proximation, Hamilton-Jacobi equation, triangular mesh.

  • AMS Subject Headings

65M60, 35L65

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

zhujun@nuaa.edu.cn (Jun Zhu)

jxqiu@xmu.edu.cn (Jianxian Qiu)

  • BibTex
  • RIS
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@Article{CiCP-27-897, author = {Zhu , Jun and Qiu , Jianxian }, title = {A New Type of High-Order WENO Schemes for Hamilton-Jacobi Equations on Triangular Meshes}, journal = {Communications in Computational Physics}, year = {2020}, volume = {27}, number = {3}, pages = {897--920}, abstract = {

In this paper, a new type of third-order and fourth-order weighted essentially non-oscillatory (WENO) schemes is designed for simulating the Hamilton-Jacobi equations on triangular meshes. We design such schemes with the use of the nodal information defined on five unequal-sized spatial stencils, the application of monotone Hamiltonians as a building block, the artificial set of positive linear weights to make up high-order approximations in smooth regions simultaneously avoiding spurious oscillations nearby discontinuities of the derivatives of the solutions. The spatial reconstructions are convex combinations of the derivatives of a modified cubic/quartic polynomial defined on a big spatial stencil and four quadratic polynomials defined on small spatial stencils, and a third-order TVD Runge-Kutta method is used for the time discretization. The main advantages of these WENO schemes are their efficiency, simplicity, and can be easily implemented to higher dimensional unstructured meshes.Extensive numerical tests are performed to illustrate the good performance of such new WENO schemes.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2018-0156}, url = {http://global-sci.org/intro/article_detail/cicp/13921.html} }
TY - JOUR T1 - A New Type of High-Order WENO Schemes for Hamilton-Jacobi Equations on Triangular Meshes AU - Zhu , Jun AU - Qiu , Jianxian JO - Communications in Computational Physics VL - 3 SP - 897 EP - 920 PY - 2020 DA - 2020/02 SN - 27 DO - http://dor.org/10.4208/cicp.OA-2018-0156 UR - https://global-sci.org/intro/cicp/13921.html KW - Unequal-sized stencil, weighted essentially non-oscillatory scheme, high-order ap- proximation, Hamilton-Jacobi equation, triangular mesh. AB -

In this paper, a new type of third-order and fourth-order weighted essentially non-oscillatory (WENO) schemes is designed for simulating the Hamilton-Jacobi equations on triangular meshes. We design such schemes with the use of the nodal information defined on five unequal-sized spatial stencils, the application of monotone Hamiltonians as a building block, the artificial set of positive linear weights to make up high-order approximations in smooth regions simultaneously avoiding spurious oscillations nearby discontinuities of the derivatives of the solutions. The spatial reconstructions are convex combinations of the derivatives of a modified cubic/quartic polynomial defined on a big spatial stencil and four quadratic polynomials defined on small spatial stencils, and a third-order TVD Runge-Kutta method is used for the time discretization. The main advantages of these WENO schemes are their efficiency, simplicity, and can be easily implemented to higher dimensional unstructured meshes.Extensive numerical tests are performed to illustrate the good performance of such new WENO schemes.

Jun Zhu & Jianxian Qiu. (2020). A New Type of High-Order WENO Schemes for Hamilton-Jacobi Equations on Triangular Meshes. Communications in Computational Physics. 27 (3). 897-920. doi:10.4208/cicp.OA-2018-0156
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