TY - JOUR
T1 - Boundary treatment of linear multistep methods for hyperbolic conservation laws
AU - Zuo, Hujian
AU - Zhao, Weifeng
AU - Lin, Ping
N1 - Funding Information:
This work was supported by the National Natural Science Foundation of China (Grant Nos. 11771040 , 11801030 and 11861131004 ) and the Fundamental Research Funds for the Central Universities (No. 06500073 ).
Publisher Copyright:
© 2022 Elsevier Inc.
PY - 2022/7/15
Y1 - 2022/7/15
N2 - When using high-order schemes to solve hyperbolic conservation laws in bounded domains, it is necessary to properly treat boundary conditions so that the overall accuracy and stability are maintained. In [1, 2] a finite difference boundary treatment method is proposed for Runge-Kutta methods of hyperbolic conservation laws. The method combines an inverse Lax-Wendroff procedure and a WENO type extrapolation to achieve desired accuracy and stability. In this paper, we further develop the boundary treatment method for high-order linear multistep methods (LMMs) of hyperbolic conservation laws. We test the method through both 1D and 2D benchmark numerical examples for two third-order LMMs, one with a constant time step and the other with a variable time step. Numerical examples show expected high order accuracy and excellent stability. In addition, the approach in [3] may be adopted to deal with an exceptional case where eigenvalues of the flux Jacobian matrix change signs at the boundary. These results demonstrate that the combined boundary treatment method works very well for LMMs of hyperbolic conservation laws.
AB - When using high-order schemes to solve hyperbolic conservation laws in bounded domains, it is necessary to properly treat boundary conditions so that the overall accuracy and stability are maintained. In [1, 2] a finite difference boundary treatment method is proposed for Runge-Kutta methods of hyperbolic conservation laws. The method combines an inverse Lax-Wendroff procedure and a WENO type extrapolation to achieve desired accuracy and stability. In this paper, we further develop the boundary treatment method for high-order linear multistep methods (LMMs) of hyperbolic conservation laws. We test the method through both 1D and 2D benchmark numerical examples for two third-order LMMs, one with a constant time step and the other with a variable time step. Numerical examples show expected high order accuracy and excellent stability. In addition, the approach in [3] may be adopted to deal with an exceptional case where eigenvalues of the flux Jacobian matrix change signs at the boundary. These results demonstrate that the combined boundary treatment method works very well for LMMs of hyperbolic conservation laws.
KW - Boundary treatment
KW - Hyperbolic conservation laws
KW - Inverse lax-Wendroff
KW - Linear multistep methods
UR - http://www.scopus.com/inward/record.url?scp=85126556433&partnerID=8YFLogxK
U2 - 10.1016/j.amc.2022.127079
DO - 10.1016/j.amc.2022.127079
M3 - Article
AN - SCOPUS:85126556433
SN - 0096-3003
VL - 425
JO - Applied Mathematics and Computation
JF - Applied Mathematics and Computation
M1 - 127079
ER -