Finite difference Hermite WENO schemes for the Hamilton-Jacobi equations

Feng Zheng Chi-Wang Shu Brown University Jianxian Qiu

Numerical Analysis and Scientific Computing mathscidoc:1804.25007

Journal of Computational Physics, 337, 27-41, 2017
In this paper, a new type of finite difference Hermite weighted essentially non-oscillatory (HWENO) schemes are constructed for solving Hamilton-Jacobi (HJ) equations. Point values of both the solution and its first derivatives are used in the HWENO reconstruction and evolved via time advancing. While the evolution of the solution is still through the classical numerical fluxes to ensure convergence to weak solutions, the evolution of the first derivatives of the solution is through a simple dimension-by-dimension non-conservative procedure to gain efficiency. The main advantages of this new scheme include its compactness in the spatial field and its simplicity in the reconstructions. Extensive numerical experiments in one and two dimensional cases are performed to verify the accuracy, high resolution and efficiency of this new scheme.
HWENO method; Hamilton-Jacobi equation; finite difference method
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  title={Finite difference Hermite WENO schemes for the Hamilton-Jacobi equations},
  author={Feng Zheng, Chi-Wang Shu, and Jianxian Qiu},
  booktitle={Journal of Computational Physics},
Feng Zheng, Chi-Wang Shu, and Jianxian Qiu. Finite difference Hermite WENO schemes for the Hamilton-Jacobi equations. 2017. Vol. 337. In Journal of Computational Physics. pp.27-41.
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