TY - JOUR
T1 - Optimal Error Estimates of the Local Discontinuous Galerkin Method with Generalized Numerical Fluxes for One-Dimensional KdV Type Equations
AU - Zhang , Hongjuan
AU - Meng , Xiong
AU - Zhang , Dazhi
AU - Wu , Boying
JO - Journal of Computational Mathematics
VL - 1
SP - 63
EP - 88
PY - 2024
DA - 2024/11
SN - 43
DO - http://doi.org/10.4208/jcm.2307-m2022-0278
UR - https://global-sci.org/intro/article_detail/jcm/23530.html
KW - Korteweg-de Vries type equations, Local discontinuous Galerkin method, Generalized fluxes, Error estimates.
AB - <p style="text-align: justify;">In this paper, we investigate the local discontinuous Galerkin method with generalized
numerical fluxes for one-dimensional nonlinear Korteweg-de Vries type equations. The
numerical flux for the nonlinear convection term is chosen as the generalized Lax-Friedrichs
flux, and the generalized alternating flux and upwind-biased flux are used for the dispersion
term. The generalized Lax-Friedrichs flux with anti-dissipation property will compensate
the numerical dissipation of the dispersion term, resulting in a nearly energy conservative
scheme that is useful in resolving waves and is beneficial for long time simulations. To
deal with the nonlinearity and different numerical flux weights, a suitable numerical initial
condition is constructed, for which a modified global projection is designed. By establishing
relationships between the prime variable and auxiliary variables in combination with sharp
bounds for jump terms, optimal error estimates are obtained. Numerical experiments are
shown to confirm the validity of theoretical results.</p>