WENO-Z scheme with new nonlinear weights for Hamilton-Jacobi equations and adaptive approximation

Kwangil Kim; Kwanhung Ri; Wonho Han

Applications of Mathematics (2025)

  • Issue: 3, page 413-439
  • ISSN: 0862-7940

Abstract

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A new fifth-order weighted essentially nonoscillatory (WENO) scheme is designed to approximate Hamilton-Jacobi equations. As employing a fifth-order linear approximation and three third-order ones on the same six-point stencil as before, a newly considered WENO-Z methodology is adapted to define nonlinear weights and the final WENO reconstruction results in a simple and clear convex combination. The scheme has formal fifth-order accuracy in smooth regions of the solution and nonoscillating behavior nearby singularities. A full account is given of the key role of parameters in WENO reconstruction and their selection. The latter half describes the adaptive stage on WENO approximation in convergence framework, which enables us to get the numerical solution to converge still achieving high-order accuracy for the nonconvex problems where the pure WENO scheme fails to converge. Detailed numerical experiments are performed to demonstrate the ability of the proposed numerical methods.

How to cite

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Kim, Kwangil, Ri, Kwanhung, and Han, Wonho. "WENO-Z scheme with new nonlinear weights for Hamilton-Jacobi equations and adaptive approximation." Applications of Mathematics (2025): 413-439. <http://eudml.org/doc/299990>.

@article{Kim2025,
abstract = {A new fifth-order weighted essentially nonoscillatory (WENO) scheme is designed to approximate Hamilton-Jacobi equations. As employing a fifth-order linear approximation and three third-order ones on the same six-point stencil as before, a newly considered WENO-Z methodology is adapted to define nonlinear weights and the final WENO reconstruction results in a simple and clear convex combination. The scheme has formal fifth-order accuracy in smooth regions of the solution and nonoscillating behavior nearby singularities. A full account is given of the key role of parameters in WENO reconstruction and their selection. The latter half describes the adaptive stage on WENO approximation in convergence framework, which enables us to get the numerical solution to converge still achieving high-order accuracy for the nonconvex problems where the pure WENO scheme fails to converge. Detailed numerical experiments are performed to demonstrate the ability of the proposed numerical methods.},
author = {Kim, Kwangil, Ri, Kwanhung, Han, Wonho},
journal = {Applications of Mathematics},
keywords = {Hamilton-Jacobi equation; WENO-Z scheme; nonlinear weight; adaptive approximation; convergence},
language = {eng},
number = {3},
pages = {413-439},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {WENO-Z scheme with new nonlinear weights for Hamilton-Jacobi equations and adaptive approximation},
url = {http://eudml.org/doc/299990},
year = {2025},
}

TY - JOUR
AU - Kim, Kwangil
AU - Ri, Kwanhung
AU - Han, Wonho
TI - WENO-Z scheme with new nonlinear weights for Hamilton-Jacobi equations and adaptive approximation
JO - Applications of Mathematics
PY - 2025
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
IS - 3
SP - 413
EP - 439
AB - A new fifth-order weighted essentially nonoscillatory (WENO) scheme is designed to approximate Hamilton-Jacobi equations. As employing a fifth-order linear approximation and three third-order ones on the same six-point stencil as before, a newly considered WENO-Z methodology is adapted to define nonlinear weights and the final WENO reconstruction results in a simple and clear convex combination. The scheme has formal fifth-order accuracy in smooth regions of the solution and nonoscillating behavior nearby singularities. A full account is given of the key role of parameters in WENO reconstruction and their selection. The latter half describes the adaptive stage on WENO approximation in convergence framework, which enables us to get the numerical solution to converge still achieving high-order accuracy for the nonconvex problems where the pure WENO scheme fails to converge. Detailed numerical experiments are performed to demonstrate the ability of the proposed numerical methods.
LA - eng
KW - Hamilton-Jacobi equation; WENO-Z scheme; nonlinear weight; adaptive approximation; convergence
UR - http://eudml.org/doc/299990
ER -

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