1 Introduction
2 Local Discontinuous Galerkin Method
3 New Immersed Boundary Method
3.1 Identify the Information of the Ghost Cell
3.2 Reconstruction
4 Numerical Example
4.1 Boundary Condition is Imposed Through Numerical Fluxes
i
|
N
s
|
N
B
|
N
G
|
N
T
|
---|---|---|---|---|
1 | 20 × 20 | 316 | 0 | 2740 |
2 | 30 × 30 | 476 | 0 | 4560 |
3 | 40 × 40 | 636 | 0 | 6580 |
4 | 50 × 50 | 796 | 0 | 8800 |
5 | 60 × 60 | 950 | 0 | 11220 |
6 | 80 × 80 | 1276 | 0 | 16660 |
i
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q
1
|
u
| ||||||
---|---|---|---|---|---|---|---|---|
L∞ | Order | L2 | Order | L∞ | Order | L2 | Order | |
1 | 2.55 × 10−5 | 8.79 × 10−5 | 4.63 × 10−3 | 7.17 × 10−4 | ||||
2 | 1.05 × 10−5 | 3.48 | 4.72 × 10−5 | 2.44 | 3.10 × 10−3 | 1.58 | 4.81 × 10−4 | 1.57 |
3 | 5.51 × 10−6 | 3.51 | 3.05 × 10−5 | 2.39 | 2.32 × 10−3 | 1.56 | 3.63 × 10−4 | 1.54 |
4 | 3.30 × 10−6 | 3.53 | 2.17 × 10−5 | 2.33 | 1.87 × 10−3 | 1.53 | 2.91 × 10−4 | 1.51 |
5 | 2.14 × 10−6 | 3.57 | 1.65 × 10−5 | 2.28 | 1.56 × 10−3 | 1.49 | 2.43 × 10−4 | 1.48 |
6 | 1.04 × 10−6 | 3.66 | 1.07 × 10−5 | 2.20 | 1.17 × 10−3 | 1.45 | 1.83 × 10−4 | 1.44 |
4.2 Boundary Condition is Imposed Through Reconstruction
i
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N
s
|
N
B
|
N
G
|
N
T
|
---|---|---|---|---|
1 | 20 × 20 | 472 | 316 | 1804 |
2 | 30 × 30 | 712 | 476 | 3020 |
3 | 40 × 40 | 952 | 636 | 4300 |
4 | 50 × 50 | 1192 | 796 | 5680 |
5 | 60 × 60 | 1432 | 956 | 7160 |
6 | 80 × 80 | 1912 | 1276 | 10420 |
i
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q
1
|
u
| ||||||
---|---|---|---|---|---|---|---|---|
L∞ | Order | L2 | Order | L∞ | Order | L2 | Order | |
1 | 1.59 × 10−4 | 2.21 × 10−4 | 3.85 × 10−2 | 5.10 × 10−3 | ||||
2 | 9.94 × 10−5 | 1.84 | 1.21 × 10−4 | 2.35 | 2.63 × 10−2 | 1.49 | 3.48 × 10−3 | 1.50 |
3 | 7.23 × 10−5 | 1.75 | 7.89 × 10−5 | 2.35 | 2.00 × 10−2 | 1.50 | 2.64 × 10−3 | 1.51 |
4 | 5.68 × 10−5 | 1.68 | 5.66 × 10−5 | 2.31 | 1.61 × 10−2 | 1.49 | 2.13 × 10−3 | 1.50 |
5 | 4.68 × 10−5 | 1.62 | 4.31 × 10−5 | 2.28 | 1.35 × 10−2 | 1.47 | 1.78 × 10−3 | 1.47 |
6 | 3.46 × 10−5 | 1.56 | 2.80 × 10−5 | 2.22 | 1.02 × 10−2 | 1.44 | 1.35 × 10−3 | 1.44 |
4.3 Boundary Condition is Imposed Hybrid
i
|
N
s
|
N
B
|
N
G
|
N
T
|
---|---|---|---|---|
1 | 20 × 20 | 324 | 156 | 2072 |
2 | 30 × 30 | 484 | 220 | 3352 |
3 | 40 × 40 | 644 | 288 | 4808 |
4 | 50 × 50 | 811 | 383 | 6587 |
5 | 60 × 60 | 964 | 456 | 8760 |
6 | 80 × 80 | 1284 | 572 | 12124 |
i
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q
1
|
u
| ||||||
---|---|---|---|---|---|---|---|---|
L∞ | Order | L2 | Order | L∞ | Order | L2 | Order | |
1 | 1.05 × 10−4 | 1.27 × 10−4 | 8.39 × 10−3 | 6.75 × 10−4 | ||||
2 | 3.63 × 10−4 | − 5.05 | 6.83 × 10−5 | 2.52 | 5.31 × 10−3 | 1.86 | 3.34 × 10−4 | 2.87 |
3 | 8.76 × 10−5 | 7.48 | 4.59 × 10−5 | 2.09 | 4.93 × 10−3 | 0.39 | 2.61 × 10−4 | 1.29 |
4 | 4.82 × 10−5 | 3.94 | 3.27 × 10−5 | 2.22 | 5.21 × 10−3 | − 0.36 | 2.18 × 10−4 | 1.20 |
5 | 1.22 × 10−5 | 9.44 | 2.51 × 10−5 | 1.83 | 4.85 × 10−3 | 0.48 | 1.71 × 10−4 | 1.67 |
1.22 × 10−5 | 0.00 | 1.62 × 10−5 | 2.65 | 3.15 × 10−3 | 2.62 | 1.02 × 10−4 | 3.15 |
i
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q
1
|
u
| ||||||
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L∞ | Order | L2 | Order | L∞ | Order | L2 | Order | |
1 | 1.25 × 10−4 | 1.25 × 10−4 | 8.65 × 10−3 | 7.06 × 10−4 | ||||
2 | 3.59 × 10−4 | − 4.33 | 6.72 × 10−5 | 2.48 | 5.32 × 10−3 | 1.99 | 3.56 × 10−4 | 3.05 |
3 | 5.80 × 10−5 | 9.60 | 4.51 × 10−5 | 2.10 | 5.22 × 10−3 | 0.10 | 2.83 × 10−4 | 1.20 |
4 | 4.51 × 10−5 | 1.66 | 3.22 × 10−5 | 2.23 | 5.70 × 10−3 | − 0.57 | 2.33 × 10−4 | 1.26 |
5 | 2.92 × 10−5 | 2.99 | 2.48 × 10−5 | 1.79 | 5.30 × 10−3 | 0.50 | 1.86 × 10−4 | 1.58 |
6 | 2.77 × 10−7 | 28.21 | 1.59 × 10−5 | 2.69 | 3.23 × 10−3 | 3.00 | 1.11 × 10−4 | 3.10 |