Semi-discrete and fully discrete HDG methods for Burgers' equation

Zimo Zhu; Gang Chen; Xiaoping Xie

doi:10.3934/cpaa.2021132

Article Contents

2023, Volume 22, Issue 1: 58-81. Doi: 10.3934/cpaa.2021132

This issue Previous Article A linear, decoupled and positivity-preserving numerical scheme for an epidemic model with advection and diffusion Next Article Asymptotic stability of intermittently damped semi-linear hyperbolic-type equations

Semi-discrete and fully discrete HDG methods for Burgers' equation

School of Mathematics, Sichuan University, Chengdu 610064, China

^* Corresponding author
^* Corresponding author

Received: January 2021

Revised: June 2021

Early access: July 2021

Published: January 2023

This work was supported by National Natural Science Foundation of China (11771312)

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Abstract

This paper proposes semi-discrete and fully discrete hybridizable discontinuous Galerkin (HDG) methods for the Burgers' equation in two and three dimensions. In the spatial discretization, we use piecewise polynomials of degrees $ k \ (k \geq 1), k-1 $ and $ l \ (l = k-1; k) $ to approximate the scalar function, flux variable and the interface trace of scalar function, respectively. In the full discretization method, we apply a backward Euler scheme for the temporal discretization. Optimal a priori error estimates are derived. Numerical experiments are presented to support the theoretical results.

Keywords:

Mathematics Subject Classification: Primary: 65M12, 65M22, 65M60.

Citation:

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Figure 1. The domain : $ 4\times 4 $ (left) and $ 8\times 8 $ (right) meshes

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Figure 2. The domain: $ 2\times 2\times 2 $(left) and $ 4\times 4\times 4 $(right) meshes

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Table 1. History of convergence for Example 5.1 with $ \nu = 1, k = 1 $

(a) Method: HDG-I(l = 1)
mesh	$ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $		$ \frac{\lVert \mathit{\boldsymbol{q}}(T)- \mathit{\boldsymbol{q}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{q}}(T)\rVert_0} $
mesh	error	order	error	order
$ 4\times 4 $	2.1597e-01	–	2.9311e-01	–
$ 8\times 8 $	5.4132e-02	2.00	1.4864e-01	0.98
$ 16\times 16 $	1.3543e-02	2.00	7.4578e-02	0.99
$ 32\times 32 $	3.3865e-03	2.00	3.7322e-02	1.00
$ 64\times 64 $	8.4665e-04	2.00	1.8665e-02	1.00
(b) Method: HDG-II(l = 0)
mesh	$ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $		$ \frac{\lVert \mathit{\boldsymbol{q}}(T)- \mathit{\boldsymbol{q}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{q}}(T)\rVert_0} $
mesh	error	order	error	order
$ 4\times 4 $	2.4145e-01	-	3.1279e-01	-
$ 8\times 8 $	6.0180e-02	2.00	1.5806e-01	0.98
$ 16\times 16 $	1.5038e-02	2.00	7.9255e-02	1.00
$32\times 32$	3.7593e-03	2.00	3.9656e-02	1.00
$64\times 64$	9.3980e-04	2.00	1.9832e-02	1.00

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Table 2. History of convergence for Example 5.1 with $ \nu = 0.01, k = 1 $

(a) Method: HDG-I(l = 1)
mesh	$ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $		$ \frac{\lVert \boldsymbol{q}(T)- \boldsymbol{q}_{h}(T)\rVert_0}{\lVert \boldsymbol{q}(T)\rVert_0} $
mesh	error	order	error	order
$ 4\times 4 $	1.1207e-01	-	2.9063e-01	-
$ 8\times 8 $	3.3444e-02	1.74	1.4978e-01	0.96
$ 16\times 16 $	8.5650e-03	1.97	7.4851e-02	1.00
$ 32\times 32 $	2.1460e-03	2.00	3.7359e-02	1.00
$ 64\times 64 $	5.3674e-04	2.00	1.8669e-02	1.00
(b) Method: HDG-II(l = 0)
mesh	$ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $		$ \frac{\lVert \boldsymbol{q}(T)- \boldsymbol{q}_{h}(T)\rVert_0}{\lVert \boldsymbol{q}(T)\rVert_0} $
mesh	error	order	error	order
$ 4\times 4 $	1.3157e-01	-	3.0275e-01	-
$ 8\times 8 $	4.1890e-02	1.65	1.6350e-01	0.89
$ 16\times 16 $	1.0394e-02	2.01	8.0699e-02	1.02
$32\times 32$	2.5616e-03	2.02	3.9968e-02	1.01
$64\times 64$	6.3806e-04	2.01	1.9936e-02	1.00

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Table 3. History of convergence for Example 5.1 with $ \nu = 1, k = 2 $

(a) Method: HDG-I(l = 2)
mesh	$ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $		$ \frac{\lVert \boldsymbol{q}(T)- \boldsymbol{q}_{h}(T)\rVert_0}{\lVert \boldsymbol{q}(T)\rVert_0} $
mesh	error	order	error	order
$ 4\times 4 $	1.3700e-02	-	4.1763e-02	-
$ 8\times 8 $	1.5937e-03	3.10	1.0597e-02	1.98
$ 16\times 16 $	1.9284e-04	3.05	2.6642e-03	1.99
$ 32\times 32 $	2.3736e-05	3.02	6.6756e-04	2.00
$ 64\times 64 $	2.9449e-06	3.01	1.6705e-04	2.00
(b) Method: HDG-II(l = 1)
mesh	$ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $		$ \frac{\lVert \boldsymbol{q}(T)- \boldsymbol{q}_{h}(T)\rVert_0}{\lVert \boldsymbol{q}(T)\rVert_0} $
mesh	error	order	error	order
$ 4\times 4 $	1.4714e-02	-	4.3007e-02	-
$ 8\times 8 $	1.7267e-03	3.09	1.0913e-02	1.98
$ 16\times 16 $	2.1016e-04	3.04	2.7449e-03	1.99
$ 32\times 32 $	2.5953e-05	3.02	6.8800e-04	2.00
$ 64\times 64 $	3.2256e-06	3.01	1.7220e-04	2.00

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Table 4. History of convergence for Example 5.1 with $ \nu = 0.01, k = 2 $

(a) Method: HDG-I(l = 2)
mesh	$ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $		$ \frac{\lVert \mathit{\boldsymbol{q}}(T)- \mathit{\boldsymbol{q}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{q}}(T)\rVert_0} $
mesh	error	order	error	order
$ 4\times 4 $	9.4294e-02	–	1.0218e-01	–
$ 8\times 8 $	1.2210e-02	2.95	1.6179e-02	2.66
$ 16\times 16 $	1.5272e-03	3.00	3.0797e-03	2.39
$ 32\times 32 $	1.9068e-04	3.00	6.9507e-04	2.15
$ 64\times 64 $	2.3817e-05	3.00	1.6880e-04	2.04
(b) Method: HDG-II(l = 1)
mesh	$ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $		$ \frac{\lVert \mathit{\boldsymbol{q}}(T)- \mathit{\boldsymbol{q}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{q}}(T)\rVert_0} $
mesh	error	order	error	order
$ 4\times 4 $	9.4922e-02	–	1.0298e-01	–
$ 8\times 8 $	1.2259e-02	2.95	1.6577e-02	2.64
$ 16\times 16 $	1.5310e-03	3.00	3.1770e-03	2.38
$ 32\times 32 $	1.9114e-04	3.00	7.2041e-04	2.14
$ 64\times 64 $	2.3877e-05	3.00	1.7525e-04	2.04

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Table 5. History of convergence with $ k = 3 $: Example 5.2

$ \Delta t $	HDG-I$ (l=3) $		HDG-II$ (l=2) $
	$ \frac{\lVert \mathit{\boldsymbol{u}}(T)-\mathit{\boldsymbol{u}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{u}}(T)\rVert_0} $		$ \frac{\lVert \mathit{\boldsymbol{u}}(T)-\mathit{\boldsymbol{u}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{u}}(T)\rVert_0} $
	error	order	error	order
0.2	2.2145e-03	–	2.2145e-03	–
0.1	3.7353e-04	2.57	3.7353e-04	2.57
0.05	5.7074e-05	2.71	5.7074e-05	2.71
0.025	8.1013e-06	2.82	8.1013e-06	2.82
0.0125	1.0947e-06	2.89	1.0947e-06	2.89

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Table 6. History of convergence with $ k = 1 $: Example 5.2

(a) Method: HDG-I(l = 1)
mesh	$ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $		$ \frac{\lVert \mathit{\boldsymbol{q}}(T)- \mathit{\boldsymbol{q}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{q}}(T)\rVert_0} $
mesh	error	order	error	order
$ 8\times 8 $	1.5445e-01	–	4.2222e-01	–
$ 16\times 16 $	4.2801e-02	1.85	1.9709e-01	1.10
$ 32\times 32 $	1.0908e-02	1.97	9.8335e-02	1.00
$ 64\times 64 $	2.7412e-03	1.99	4.9190e-02	1.00
$ 128\times 128 $	6.8606e-04	2.00	2.4598e-02	1.00
(b) Method: HDG-II(l = 0)
mesh	$ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $		$ \frac{\lVert \mathit{\boldsymbol{q}}(T)- \mathit{\boldsymbol{q}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{q}}(T)\rVert_0} $
mesh	error	order	error	order
$ 8\times 8 $	1.7900e-01	–	4.5177e-01	–
$ 16\times 16 $	4.7423e-02	1.92	2.0942e-01	1.11
$ 32\times 32 $	1.1971e-02	1.99	1.0380e-01	1.01
$ 64\times 64 $	3.0019e-03	2.00	5.1851e-02	1.00
$ 128\times 128 $	7.5102e-04	2.00	2.5920e-02	1.00

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Table 7. History of convergence with $ k = 2 $: Example 5.2

(a) Method: HDG-I(l = 2)
mesh	$ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $		$ \frac{\lVert \mathit{\boldsymbol{q}}(T)- \mathit{\boldsymbol{q}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{q}}(T)\rVert_0} $
mesh	error	order	error	order
$ 8\times 8 $	2.8115e-02	–	4.9545e-02	–
$ 16\times 16 $	4.7515e-03	2.56	2.1126e-02	1.23
$ 32\times 32 $	6.1910e-04	2.94	5.6085e-03	1.91
$ 64\times 64 $	7.7864e-05	2.99	1.4177e-03	1.98
$ 128\times 128 $	9.7479e-06	3.00	3.5571e-04	1.99
(b) Method: HDG-II(l = 1)
mesh	$ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $		$ \frac{\lVert \mathit{\boldsymbol{q}}(T)- \mathit{\boldsymbol{q}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{q}}(T)\rVert_0} $
mesh	error	order	error	order
$ 8\times 8 $	2.9126e-02	–	5.0053e-02	–
$ 16\times 16 $	4.8563e-03	2.58	2.1191e-02	1.24
$ 32\times 32 $	6.3425e-04	2.94	5.6117e-03	1.92
$ 64\times 64 $	7.9961e-05	2.99	1.4199e-03	1.98
$ 128\times 128 $	1.0021e-05	3.00	3.5663e-04	1.99

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Table 8. History of convergence with $ k = 1 $: Example 5.3

(a) Method: HDG-I(l = 1)
mesh	$ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $		$ \frac{\lVert \mathit{\boldsymbol{q}}(T)- \mathit{\boldsymbol{q}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{q}}(T)\rVert_0} $
mesh	error	order	error	order
$ 2\times 2\times 2 $	6.9815e-01	–	5.6066e-01	–
$ 4\times 4\times 4 $	1.7672e-01	1.98	3.0285e-01	0.89
$ 8\times 8\times 8 $	4.4207e-02	2.00	1.5438e-01	0.97
$ 16\times 16\times 16 $	1.1050e-02	2.00	7.7566e-02	0.99
$ 32\times 32\times 32 $	2.7621e-03	2.00	3.8830e-02	1.00
(b) Method: HDG-II(l = 0)
mesh	$ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $		$ \frac{\lVert \mathit{\boldsymbol{q}}(T)- \mathit{\boldsymbol{q}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{q}}(T)\rVert_0} $
mesh	error	order	error	order
$ 2\times 2\times 2 $	8.8064e-01	–	6.1105e-01	–
$ 4\times 4\times 4 $	2.0917e-01	2.07	3.1971e-01	0.93
$ 8\times 8\times 8 $	5.1528e-02	2.02	1.6186e-01	0.98
$ 16\times 16\times 16 $	1.2835e-02	2.00	8.1193e-02	0.99
$ 32\times 32\times 32 $	3.2055e-03	2.00	4.0630e-02	1.00

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Table 9. History of convergence with $ k = 2 $: Example 5.3

(a) Method: HDG-I(l = 2)
mesh	$ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $		$ \frac{\lVert \mathit{\boldsymbol{q}}(T)- \mathit{\boldsymbol{q}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{q}}(T)\rVert_0} $
mesh	error	order	error	order
$ 2\times 2\times 2 $	1.3095e-01	–	1.9090e-01	–
$ 4\times 4\times 4 $	1.4825e-02	3.14	5.3052e-02	1.85
$ 8\times 8\times 8 $	1.7531e-03	3.08	1.3659e-02	1.96
$ 16\times 16\times 16 $	2.1463e-04	3.03	3.4459e-03	1.99
(b) Method: HDG-II(l = 1)
mesh	$ \frac{\lVert u(T)-u_{h}(T) \rVert_0}{\lVert u(T)\rVert_0} $		$ \frac{\lVert \mathit{\boldsymbol{q}}(T)- \mathit{\boldsymbol{q}}_{h}(T)\rVert_0}{\lVert \mathit{\boldsymbol{q}}(T)\rVert_0} $
mesh	error	order	error	order
$ 2\times 2\times 2 $	1.4705e-01	–	1.9582e-01	–
$ 4\times 4\times 4 $	1.6034e-02	3.20	5.3962e-02	1.86
$ 8\times 8\times 8 $	1.8868e-03	3.09	1.3871e-02	1.96
$ 16\times 16\times 16 $	2.3104e-04	3.03	3.4979e-03	1.99
$ 32\times 32\times 32 $	2.8681e-05	3.01	8.7713e-04	2.00

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