TY - JOUR
T1 - A Novel Temporal Two-Grid Compact Finite Difference Scheme for the Viscous Burgers’ Equation
AU - Peng , Xiangyi
AU - Qiu , Wenlin
AU - Wang , Jiangxing
AU - Ma , Lina
JO - Advances in Applied Mathematics and Mechanics
VL - 6
SP - 1358
EP - 1380
PY - 2024
DA - 2024/10
SN - 16
DO - http://doi.org/10.4208/aamm.OA-2022-0302
UR - https://global-sci.org/intro/article_detail/aamm/23471.html
KW - Two-grid, compact finite difference, viscous Burgers, stability, error analysis.
AB - <p style="text-align: justify;">We present a novel two-grid compact finite difference scheme for the viscous Burgers’ equation in this paper, where the second-order Crank-Nicolson method
is used to deal with the time marching, the compact finite difference formula is used to
approximate the spatial second-order term, and the nonlinear convection term is discretized using the developed nonlinear fourth-order operator, providing the scheme
with both high fourth-order spatial convergence and a low computational cost. The
scheme is then established in three steps, with the first step being the construction of a
nonlinear coarse-grid compact finite difference scheme that is solved iteratively using
a fixed point iterative method, the second step being the application of the Lagrange
interpolation formula to obtain a rough solution on the fine grid, and the third step being the development of the linearized fine-grid compact finite difference scheme. We
also perform a convergence and stability analysis on the developed scheme, and the
results show that the scheme can achieve spatial fourth-order and temporal second-order convergence. Finally, a number of numerical examples are provided to validate
the theoretical predictions.</p>