# American Institute of Mathematical Sciences

March  2021, 29(1): 1897-1923. doi: 10.3934/era.2020097

## A weak Galerkin finite element method for nonlinear conservation laws

 1 Department of Mathematics, University of Arkansas at Little Rock, Little Rock, AR 72204, USA 2 Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA 3 College of Mathematics, Physics and Information Engineering, Jiaxing University, Jiaxing, Zhejiang 314001, China

* Corresponding author: Xiu Ye

Received  April 2020 Revised  July 2020 Published  March 2021 Early access  September 2020

Fund Project: The first author is supported in part by NSF grant DMS-1620016. The third author is supported in part by Zhejiang provincial NSF of China grant LY19A010008

A weak Galerkin (WG) finite element method is presented for nonlinear conservation laws. There are two built-in parameters in this WG framework. Different choices of the parameters will lead to different approaches for solving hyperbolic conservation laws. The convergence analysis is obtained for the forward Euler time discrete and the third order explicit TVDRK time discrete WG schemes respectively. The theoretical results are verified by numerical experiments.

Citation: Xiu Ye, Shangyou Zhang, Peng Zhu. A weak Galerkin finite element method for nonlinear conservation laws. Electronic Research Archive, 2021, 29 (1) : 1897-1923. doi: 10.3934/era.2020097
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##### References:
WG solution for Example 2, $T = 1/\pi, \lambda_1 = 2,\lambda_2 = 1,N = 128$
The $P_1$ WG solution and DG solution for Example 3, $T = 2\pi, N = 512$
$L^2$ errors and corresponding convergence rates of Example 1. $T = 2\pi$, $\lambda_1 = \lambda_2 = 1$
 $P_1$ element $P_2$ element $P_3$ element N $\|u-u_0\|$ Rate $\|u-u_0\|$ Rate $\|u-u_0\|$ Rate 8 1.29E-01 3.36E-03 2.66E-04 16 3.02E-02 2.10 3.99E-04 3.08 1.94E-05 3.78 32 7.22E-03 2.06 4.93E-05 3.02 1.27E-06 3.93 64 1.78E-03 2.02 6.14E-06 3.00 8.06E-08 3.98 128 4.42E-04 2.01 7.67E-07 3.00 5.06E-09 4.00
 $P_1$ element $P_2$ element $P_3$ element N $\|u-u_0\|$ Rate $\|u-u_0\|$ Rate $\|u-u_0\|$ Rate 8 1.29E-01 3.36E-03 2.66E-04 16 3.02E-02 2.10 3.99E-04 3.08 1.94E-05 3.78 32 7.22E-03 2.06 4.93E-05 3.02 1.27E-06 3.93 64 1.78E-03 2.02 6.14E-06 3.00 8.06E-08 3.98 128 4.42E-04 2.01 7.67E-07 3.00 5.06E-09 4.00
$L^2$ errors and corresponding convergence rates of Example 2. $T = 0.2$, $\lambda_1 = \lambda_2 = 2.5$
 $P_1$ element $P_2$ element $P_3$ element N $\|u-u_0\|$ Rate $\|u-u_0\|$ Rate $\|u-u_0\|$ Rate 8 1.68E-02 6.60E-03 1.89E-03 16 6.11E-03 1.46 7.86E-04 3.07 2.22E-04 3.09 32 1.42E-03 2.10 1.63E-04 2.27 9.96E-06 4.48 64 3.49E-04 2.03 2.85E-05 2.51 8.19E-07 3.60 128 8.67E-05 2.01 4.98E-06 2.51 5.81E-08 3.82
 $P_1$ element $P_2$ element $P_3$ element N $\|u-u_0\|$ Rate $\|u-u_0\|$ Rate $\|u-u_0\|$ Rate 8 1.68E-02 6.60E-03 1.89E-03 16 6.11E-03 1.46 7.86E-04 3.07 2.22E-04 3.09 32 1.42E-03 2.10 1.63E-04 2.27 9.96E-06 4.48 64 3.49E-04 2.03 2.85E-05 2.51 8.19E-07 3.60 128 8.67E-05 2.01 4.98E-06 2.51 5.81E-08 3.82
$L^2$ errors and corresponding convergence rates of Example 3. $T = 2\pi$, $\lambda_1 = 2,\lambda_2 = 1$
 $P_1$ element $P_2$ element N $\|u-u_0\|$ Rate $\|u-u_0\|$ Rate 8 5.93E-01 4.23E-01 16 5.01E-01 0.24 3.25E-01 0.38 32 3.93E-01 0.35 2.52E-01 0.37 64 3.26E-01 0.27 1.98E-01 0.35 128 2.72E-01 0.26 1.58E-01 0.33 256 2.26E-01 0.27 1.27E-01 0.32 512 1.89E-01 0.26 1.03E-01 0.30
 $P_1$ element $P_2$ element N $\|u-u_0\|$ Rate $\|u-u_0\|$ Rate 8 5.93E-01 4.23E-01 16 5.01E-01 0.24 3.25E-01 0.38 32 3.93E-01 0.35 2.52E-01 0.37 64 3.26E-01 0.27 1.98E-01 0.35 128 2.72E-01 0.26 1.58E-01 0.33 256 2.26E-01 0.27 1.27E-01 0.32 512 1.89E-01 0.26 1.03E-01 0.30
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