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 - <p style="text-align: justify;">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.</p>