TY - JOUR
T1 - Convergence of a second-order energy-decaying method for the viscous rotating shallow water equation
AU - Akrivis, Georgios
AU - Li, Buyang
AU - Wang, Jilu
N1 - Funding Information:
\ast Received by the editors March 30, 2020; accepted for publication (in revised form) October 19, 2020; published electronically January 26, 2021. https://doi.org/10.1137/20M1328051 \bfF \bfu \bfn \bfd \bfi \bfn \bfg : The work of the authors was partially supported by the Hong Kong Polytechnic University project P0031035 ZZKQ and the National Natural Science Foundation of China grants U1930402 and 12071020. \dagger Department of Computer Science and Engineering, University of Ioannina, 451 10 Ioannina, Greece, and Institute of Applied and Computational Mathematics, FORTH, 700 13 Heraklion, Crete, Greece ([email protected]). \ddagger Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong ([email protected]). \S Corresponding author. Beijing Computational Science Research Center, Beijing 100193, China ([email protected]).
Publisher Copyright:
© 2021 Society for Industrial and Applied Mathematics.
PY - 2021/1/26
Y1 - 2021/1/26
N2 - An implicit energy-decaying modified Crank-Nicolson time-stepping method is constructed for the viscous rotating shallow water equation on the plane. Existence, uniqueness, and convergence of semidiscrete solutions are proved by using Schaefer's fixed point theorem and H2 estimates of the discretized hyperbolic-parabolic system. For practical computation, the semidiscrete method is further discretized in space, resulting in a fully discrete energy-decaying finite element scheme. A fixed-point iterative method is proposed for solving the nonlinear algebraic system. The numerical results show that the proposed method requires only a few iterations to achieve the desired accuracy, with second-order convergence in time, and preserves energy decay well.
AB - An implicit energy-decaying modified Crank-Nicolson time-stepping method is constructed for the viscous rotating shallow water equation on the plane. Existence, uniqueness, and convergence of semidiscrete solutions are proved by using Schaefer's fixed point theorem and H2 estimates of the discretized hyperbolic-parabolic system. For practical computation, the semidiscrete method is further discretized in space, resulting in a fully discrete energy-decaying finite element scheme. A fixed-point iterative method is proposed for solving the nonlinear algebraic system. The numerical results show that the proposed method requires only a few iterations to achieve the desired accuracy, with second-order convergence in time, and preserves energy decay well.
KW - Energy decay
KW - Error estimate
KW - Modified Crank-Nicolson
KW - Viscous shallow water equation
UR - http://www.scopus.com/inward/record.url?scp=85103785929&partnerID=8YFLogxK
U2 - 10.1137/20M1328051
DO - 10.1137/20M1328051
M3 - Journal article
AN - SCOPUS:85103785929
SN - 0036-1429
VL - 59
SP - 265
EP - 288
JO - SIAM Journal on Numerical Analysis
JF - SIAM Journal on Numerical Analysis
IS - 1
ER -