We consider an incompressible viscous flow without surface tension in a finite-depth domain of two dimensions, with free top boundary and fixed bottom boundary. This system is governed by the Navier–Stokes equations in this moving domain and the transport equation on the moving boundary. In this paper, we construct a stable numerical scheme to simulate the evolution of this system by discontinuous Galerkin method, and discuss the error analysis of the fluid under certain assumptions. Our formulation is mainly based on the geometric structure introduced in [Y. Guo and Ian Tice, Anal. PDE 6 (2013) 287–369; Y. Guo and Ian Tice, Arch. Ration. Mech. Anal. 207 (2013) 459–531; L. Wu, SIAM J. Math. Anal. 46 (2014) 2084–2135], and the natural energy estimate, which is rarely used in the numerical study of this system before.
DOI : 10.1051/m2an/2014065
Keywords: Stability, free boundary, Navier–Stokes equation
Wu, Lei 1 ; Shu, Chi-Wang 1
@article{M2AN_2015__49_4_1019_0,
author = {Wu, Lei and Shu, Chi-Wang},
title = {Numerical {Solution} of the {Viscous} {Surface} {Wave} with {Discontinuous} {Galerkin} {Method}},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1019--1046},
year = {2015},
publisher = {EDP Sciences},
volume = {49},
number = {4},
doi = {10.1051/m2an/2014065},
mrnumber = {3371902},
zbl = {1321.35143},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2014065/}
}
TY - JOUR AU - Wu, Lei AU - Shu, Chi-Wang TI - Numerical Solution of the Viscous Surface Wave with Discontinuous Galerkin Method JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2015 SP - 1019 EP - 1046 VL - 49 IS - 4 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2014065/ DO - 10.1051/m2an/2014065 LA - en ID - M2AN_2015__49_4_1019_0 ER -
%0 Journal Article %A Wu, Lei %A Shu, Chi-Wang %T Numerical Solution of the Viscous Surface Wave with Discontinuous Galerkin Method %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2015 %P 1019-1046 %V 49 %N 4 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/m2an/2014065/ %R 10.1051/m2an/2014065 %G en %F M2AN_2015__49_4_1019_0
Wu, Lei; Shu, Chi-Wang. Numerical Solution of the Viscous Surface Wave with Discontinuous Galerkin Method. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 4, pp. 1019-1046. doi: 10.1051/m2an/2014065
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