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Volume 25, Issue 2
Hermite WENO Schemes with Lax-Wendroff Type Time Discretizations for Hamilton-Jacobi Equations

Jianxian Qiu

J. Comp. Math., 25 (2007), pp. 131-144.

Published online: 2007-04

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  • Abstract

In this paper, we use Hermite weighted essentially non-oscillatory (HWENO) schemes with a Lax-Wendroff time discretization procedure, termed HWENO-LW schemes, to solve Hamilton-Jacobi equations. The idea of the reconstruction in the HWENO schemes comes from the original WENO schemes, however both the function and its first derivative values are evolved in time and are used in the reconstruction. One major advantage of HWENO schemes is its compactness in the reconstruction. We explore the possibility in avoiding the nonlinear weights for part of the procedure, hence reducing the cost but still maintaining non-oscillatory properties for problems with strong discontinuous derivative. As a result, comparing with HWENO with Runge-Kutta time discretizations schemes (HWENO-RK) of Qiu and Shu [19] for Hamilton-Jacobi equations, the major advantages of HWENO-LW schemes are their saving of computational cost and their compactness in the reconstruction. Extensive numerical experiments are performed to illustrate the capability of the method.

  • Keywords

WENO scheme, Hermite interpolation, Hamilton-Jacobi equation, Lax-Wendroff type time discretization, High order accuracy.

  • AMS Subject Headings

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COPYRIGHT: © Global Science Press

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@Article{JCM-25-131, author = {Jianxian Qiu}, title = {Hermite WENO Schemes with Lax-Wendroff Type Time Discretizations for Hamilton-Jacobi Equations}, journal = {Journal of Computational Mathematics}, year = {2007}, volume = {25}, number = {2}, pages = {131--144}, abstract = {

In this paper, we use Hermite weighted essentially non-oscillatory (HWENO) schemes with a Lax-Wendroff time discretization procedure, termed HWENO-LW schemes, to solve Hamilton-Jacobi equations. The idea of the reconstruction in the HWENO schemes comes from the original WENO schemes, however both the function and its first derivative values are evolved in time and are used in the reconstruction. One major advantage of HWENO schemes is its compactness in the reconstruction. We explore the possibility in avoiding the nonlinear weights for part of the procedure, hence reducing the cost but still maintaining non-oscillatory properties for problems with strong discontinuous derivative. As a result, comparing with HWENO with Runge-Kutta time discretizations schemes (HWENO-RK) of Qiu and Shu [19] for Hamilton-Jacobi equations, the major advantages of HWENO-LW schemes are their saving of computational cost and their compactness in the reconstruction. Extensive numerical experiments are performed to illustrate the capability of the method.

}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/8680.html} }
TY - JOUR T1 - Hermite WENO Schemes with Lax-Wendroff Type Time Discretizations for Hamilton-Jacobi Equations AU - Jianxian Qiu JO - Journal of Computational Mathematics VL - 2 SP - 131 EP - 144 PY - 2007 DA - 2007/04 SN - 25 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/8680.html KW - WENO scheme, Hermite interpolation, Hamilton-Jacobi equation, Lax-Wendroff type time discretization, High order accuracy. AB -

In this paper, we use Hermite weighted essentially non-oscillatory (HWENO) schemes with a Lax-Wendroff time discretization procedure, termed HWENO-LW schemes, to solve Hamilton-Jacobi equations. The idea of the reconstruction in the HWENO schemes comes from the original WENO schemes, however both the function and its first derivative values are evolved in time and are used in the reconstruction. One major advantage of HWENO schemes is its compactness in the reconstruction. We explore the possibility in avoiding the nonlinear weights for part of the procedure, hence reducing the cost but still maintaining non-oscillatory properties for problems with strong discontinuous derivative. As a result, comparing with HWENO with Runge-Kutta time discretizations schemes (HWENO-RK) of Qiu and Shu [19] for Hamilton-Jacobi equations, the major advantages of HWENO-LW schemes are their saving of computational cost and their compactness in the reconstruction. Extensive numerical experiments are performed to illustrate the capability of the method.

Jianxian Qiu. (2007). Hermite WENO Schemes with Lax-Wendroff Type Time Discretizations for Hamilton-Jacobi Equations. Journal of Computational Mathematics. 25 (2). 131-144. doi:
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