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Volume 13, Issue 2
High-Order Multi-Resolution WENO Schemes with Lax-Wendroff Method for Fractional Differential Equations

Yan Zhang, Liang Li & Jun Zhu

East Asian J. Appl. Math., 13 (2023), pp. 320-339.

Published online: 2023-04

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  • Abstract

New high-order finite difference multi-resolution weighted essentially non-oscillatory (WENO) schemes with the Lax-Wendroff (LW) time discretization method for solving fractional differential equations with the Caputo fractional derivative are considered. The fractional derivative of order $α ∈ (1, 2)$ is split into an integral part and a second derivative term. The Gauss-Jacobi quadrature method is employed to solve the integral part, and a new multi-resolution WENO method for discretizing the second derivative term is developed. High-order spatial reconstruction procedures use any positive numbers (with a minor restriction) as linear weights. The new multi-resolution WENO-LW methods are one-step explicit high-order finite difference schemes. They are more compact than multi-resolution WENO-RK schemes of the same order. The LW time discretization is more cost efficient than the Runge-Kutta time discretization method. The construction of new multi-resolution WENO-LW schemes is simple and easy to generalize to arbitrary high-order accuracy in multi-dimensions. One- and two-dimensional examples with strong discontinuities verify the good performance of new fourth-, sixth-, and eighth-order multi-resolution WENO-LW schemes.

  • AMS Subject Headings

65M06, 35R11

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COPYRIGHT: © Global Science Press

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@Article{EAJAM-13-320, author = {Zhang , YanLi , Liang and Zhu , Jun}, title = {High-Order Multi-Resolution WENO Schemes with Lax-Wendroff Method for Fractional Differential Equations}, journal = {East Asian Journal on Applied Mathematics}, year = {2023}, volume = {13}, number = {2}, pages = {320--339}, abstract = {

New high-order finite difference multi-resolution weighted essentially non-oscillatory (WENO) schemes with the Lax-Wendroff (LW) time discretization method for solving fractional differential equations with the Caputo fractional derivative are considered. The fractional derivative of order $α ∈ (1, 2)$ is split into an integral part and a second derivative term. The Gauss-Jacobi quadrature method is employed to solve the integral part, and a new multi-resolution WENO method for discretizing the second derivative term is developed. High-order spatial reconstruction procedures use any positive numbers (with a minor restriction) as linear weights. The new multi-resolution WENO-LW methods are one-step explicit high-order finite difference schemes. They are more compact than multi-resolution WENO-RK schemes of the same order. The LW time discretization is more cost efficient than the Runge-Kutta time discretization method. The construction of new multi-resolution WENO-LW schemes is simple and easy to generalize to arbitrary high-order accuracy in multi-dimensions. One- and two-dimensional examples with strong discontinuities verify the good performance of new fourth-, sixth-, and eighth-order multi-resolution WENO-LW schemes.

}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.2022-202.260922}, url = {http://global-sci.org/intro/article_detail/eajam/21651.html} }
TY - JOUR T1 - High-Order Multi-Resolution WENO Schemes with Lax-Wendroff Method for Fractional Differential Equations AU - Zhang , Yan AU - Li , Liang AU - Zhu , Jun JO - East Asian Journal on Applied Mathematics VL - 2 SP - 320 EP - 339 PY - 2023 DA - 2023/04 SN - 13 DO - http://doi.org/10.4208/eajam.2022-202.260922 UR - https://global-sci.org/intro/article_detail/eajam/21651.html KW - Multi-resolution WENO-LW scheme, fractional differential equation, Lax-Wendroff time discretization, Gauss-Jacobi quadrature, high-order accuracy. AB -

New high-order finite difference multi-resolution weighted essentially non-oscillatory (WENO) schemes with the Lax-Wendroff (LW) time discretization method for solving fractional differential equations with the Caputo fractional derivative are considered. The fractional derivative of order $α ∈ (1, 2)$ is split into an integral part and a second derivative term. The Gauss-Jacobi quadrature method is employed to solve the integral part, and a new multi-resolution WENO method for discretizing the second derivative term is developed. High-order spatial reconstruction procedures use any positive numbers (with a minor restriction) as linear weights. The new multi-resolution WENO-LW methods are one-step explicit high-order finite difference schemes. They are more compact than multi-resolution WENO-RK schemes of the same order. The LW time discretization is more cost efficient than the Runge-Kutta time discretization method. The construction of new multi-resolution WENO-LW schemes is simple and easy to generalize to arbitrary high-order accuracy in multi-dimensions. One- and two-dimensional examples with strong discontinuities verify the good performance of new fourth-, sixth-, and eighth-order multi-resolution WENO-LW schemes.

Yan Zhang, Liang Li & Jun Zhu. (2023). High-Order Multi-Resolution WENO Schemes with Lax-Wendroff Method for Fractional Differential Equations. East Asian Journal on Applied Mathematics. 13 (2). 320-339. doi:10.4208/eajam.2022-202.260922
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