1 Introduction
2 Solution Point Requirements
2.1 Polynomial Interpolation and Unisolvency
2.2 Quadrature Rules
3 Candidate Point Sets
\(\wp \)
| 3 | 4 | 5 |
\(\varphi \)
| 5 | 6 | 8 |
\(N_p\)
| 20 | 35 | 56 |
\(N_r\)
| 270 | 281 | 48 |
\(N_r(\det \mathcal {V} \ne 0)\)
| 193 | 275 | 25 |
4 Numerical Experiments
4.1 Euler Equations
4.2 Test Cases
\(\wp \)
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\(N = 4\)
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\(N = 5\)
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\(N = 6\)
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\(N = 8\)
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\(N = 12\)
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\(N = 16\)
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4.3 Error Estimation
4.4 Other Details
5 Results
5.1 Manufactured Sinusoidal Solution
5.2 Isentropic Euler vortex
5.3 Comparisons
Order | Set | Error | |
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M | V | ||
\(\wp = 3\)
| Top M | 0.73 | 1.22 |
Top V | 0.88 | 0.78 | |
\(\wp = 4\)
| Top M | 0.90 | 1.10 |
Top V | 0.97 | 1.01 | |
\(\wp = 5\)
| Top M | 0.64 | 0.68 |
Top V | 0.64 | 0.68 |
6 Stability
Case | Order | |||
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M | V |
\(\wp = 3\)
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\(\wp = 4\)
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\(\wp = 5\)
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| 180 | 50 | 5 |
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| 12 | 176 | 11 |
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| 0 | 1 | 0 |
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| 1 | 48 | 9 |