In this paper, we discuss the stability and error estimates of the fully discrete schemes for linear conservation laws, which consists of an arbitrary Lagrangian–Eulerian discontinuous Galerkin method in space and explicit total variation diminishing Runge–Kutta (TVD-RK) methods up to third order accuracy in time. The scaling arguments and the standard energy analysis are the key techniques used in our work. We present a rigorous proof to obtain stability for the three fully discrete schemes under suitable CFL conditions. With the help of the reference cell, the error equations are easy to establish and we derive the quasi-optimal error estimates in space and optimal convergence rates in time. For the Euler-forward scheme with piecewise constant elements, the second order TVD-RK method with piecewise linear elements and the third order TVD-RK scheme with polynomials of any order, the usual CFL condition is required, while for other cases, stronger time step restrictions are needed for the results to hold true. More precisely, the Euler-forward scheme needs τ ≤ ρh2 and the second order TVD-RK scheme needs for higher order polynomials in space, where τ and h are the time and maximum space step, respectively, and ρ is a positive constant independent of τ and h.
Mots-clés : Arbitrary Lagrangian–Eulerian discontinuous Galerkin method, Runge–Kutta methods, stability, error estimates, conservation laws
@article{M2AN_2019__53_1_105_0,
author = {Zhou, Lingling and Xia, Yinhua and Shu, Chi-Wang},
title = {Stability analysis and error estimates of arbitrary {Lagrangian{\textendash}Eulerian} discontinuous {Galerkin} method coupled with {Runge{\textendash}Kutta} time-marching for linear conservation laws},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {105--144},
publisher = {EDP-Sciences},
volume = {53},
number = {1},
year = {2019},
doi = {10.1051/m2an/2018069},
mrnumber = {3933917},
zbl = {1418.65141},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2018069/}
}
TY - JOUR AU - Zhou, Lingling AU - Xia, Yinhua AU - Shu, Chi-Wang TI - Stability analysis and error estimates of arbitrary Lagrangian–Eulerian discontinuous Galerkin method coupled with Runge–Kutta time-marching for linear conservation laws JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2019 SP - 105 EP - 144 VL - 53 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2018069/ DO - 10.1051/m2an/2018069 LA - en ID - M2AN_2019__53_1_105_0 ER -
%0 Journal Article %A Zhou, Lingling %A Xia, Yinhua %A Shu, Chi-Wang %T Stability analysis and error estimates of arbitrary Lagrangian–Eulerian discontinuous Galerkin method coupled with Runge–Kutta time-marching for linear conservation laws %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2019 %P 105-144 %V 53 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2018069/ %R 10.1051/m2an/2018069 %G en %F M2AN_2019__53_1_105_0
Zhou, Lingling; Xia, Yinhua; Shu, Chi-Wang. Stability analysis and error estimates of arbitrary Lagrangian–Eulerian discontinuous Galerkin method coupled with Runge–Kutta time-marching for linear conservation laws. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 53 (2019) no. 1, pp. 105-144. doi : 10.1051/m2an/2018069. http://www.numdam.org/articles/10.1051/m2an/2018069/
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