TY - JOUR
T1 - High Order Finite Difference WENO Methods with Unequal-Sized Sub-Stencils for the Degasperis-Procesi Type Equations
AU - Lin , Jianfang
AU - Xu , Yan
AU - Xue , Huiwen
AU - Zhong , Xinghui
JO - Communications in Computational Physics
VL - 3
SP - 913
EP - 946
PY - 2022
DA - 2022/03
SN - 31
DO - http://doi.org/10.4208/cicp.OA-2021-0158
UR - https://global-sci.org/intro/article_detail/cicp/20303.html
KW - High order accuracy, weighted essentially non-oscillatory schemes, Degasperis-Procesi equation, $\mu$-Degasperis-Procesi equation, finite difference method, multi-resolution.
AB - <p style="text-align: justify;">In this paper, we develop two finite difference weighted essentially
non-oscillatory (WENO) schemes with unequal-sized sub-stencils for solving the
Degasperis-Procesi (DP) and $\mu$-Degasperis-Procesi ($\mu$DP) equations, which contain
nonlinear high order derivatives, and possibly peakon solutions or shock waves. By
introducing auxiliary variable(s), we rewrite the DP equation as a hyperbolic-elliptic
system, and the $\mu$DP equation as a first order system. Then we choose a linear finite
difference scheme with suitable order of accuracy for the auxiliary variable(s), and
two finite difference WENO schemes with unequal-sized sub-stencils for the primal
variable. One WENO scheme uses one large stencil and several smaller stencils, and
the other WENO scheme is based on the multi-resolution framework which uses a series of unequal-sized hierarchical central stencils. Comparing with the classical WENO
scheme which uses several small stencils of the same size to make up a big stencil, both
WENO schemes with unequal-sized sub-stencils are simple in the choice of the stencil
and enjoy the freedom of arbitrary positive linear weights. Another advantage is that
the final reconstructed polynomial on the target cell is a polynomial of the same degree as the polynomial over the big stencil, while the classical finite difference WENO
reconstruction can only be obtained for specific points inside the target interval. Numerical tests are provided to demonstrate the high order accuracy and non-oscillatory
properties of the proposed schemes.</p>