1 Introduction
2 Solution Point Requirements
\(\wp \)
| 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|
\(N_p(\wp )\)
| 3 | 6 | 10 | 15 | 21 | 28 | 36 |
3 Rule Derivation
3.1 Background
3.2 Methodology
Orbit |
\(n_{\mathrm{pts}}\)
|
\(n_{\mathrm{dof}}\)
| Barycentric coordinates |
---|---|---|---|
\(S_3(\alpha =\tfrac{1}{3})\)
| 1 | 1 |
\({{\mathrm{Perm}}}(\alpha ,\alpha ,\alpha )\)
|
\(S_{21}(\alpha )\)
| 3 | 2 |
\({{\mathrm{Perm}}}(\alpha ,\alpha ,1-2\alpha )\)
|
\(S_{111}(\alpha ,\beta )\)
| 6 | 3 |
\({{\mathrm{Perm}}}(\alpha ,\beta ,1-\alpha -\beta )\)
|
4 Metrics
4.1 Background
4.2 Lebesgue constant
4.3 Truncation error
5 Candidate Point Sets
\(\wp \)
| 3 | 4 | 5 | 5 | 6 | 6 | 7 |
\(\varphi \)
| 5 | 7 | 8 | 9 | 10 | 11 | 12 |
\(\varphi ^{+}\)
| 6 | 9 | 10 | 10 | 12 | 12 | 14 |
\(N_r\)
| 95 | 66 | 722 | 473 | 412 | 12 | 136 |
\(N_r(\det \mathcal {V} \ne 0)\)
| 24 | 64 | 452 | 2 | 236 | 0 | 100 |
6 Numerical Experiments
6.1 Euler Equations
6.2 Error Estimation
7 Results and Discussion
\(\wp \)
|
\(\sigma (t = 100)\)
|
---|---|
3 | |
\(\alpha \)-opt | DNF \(t = 81.40\)
|
\(\Lambda \)-opt |
\(2.58 \times 10^{-2}\)
|
\(\xi \)-opt |
\(8.20 \times 10^{-3}\)
|
\(\sigma \)-opt |
\(8.20 \times 10^{-3}\)
|
\(\langle \sigma \rangle \)-opt |
\(8.61 \times 10^{-3}\)
|
WS |
\(8.27 \times 10^{-3}\)
|
4 | |
\(\alpha \)-opt | DNF \(t = 13.30\)
|
\(\Lambda \)-opt |
\(1.20 \times 10^{-3}\)
|
\(\xi \)-opt |
\(2.09 \times 10^{-3}\)
|
\(\sigma \)-opt |
\(6.59 \times 10^{-4}\)
|
\(\langle \sigma \rangle \)-opt |
\(6.76 \times 10^{-4}\)
|
WS |
\(1.15 \times 10^{-3}\)
|
5 | |
\(\alpha \)-opt | DNF \(t = 18.95\)
|
\(\Lambda \)-opt |
\(2.64 \times 10^{-3}\)
|
\(\xi \)-opt |
\(1.36 \times 10^{-4}\)
|
\(\sigma \)-opt |
\(9.69 \times 10^{-5}\)
|
\(\langle \sigma \rangle \)-opt |
\(9.69 \times 10^{-5}\)
|
WS |
\(6.92 \times 10^{-5}\)
|
6 | |
\(\alpha \)-opt | DNF \(t = 22.24\)
|
\(\Lambda \)-opt |
\(2.02 \times 10^{-4}\)
|
\(\xi \)-opt |
\(4.16 \times 10^{-5}\)
|
\(\sigma \)-opt |
\(2.38 \times 10^{-5}\)
|
\(\langle \sigma \rangle \)-opt |
\(2.42 \times 10^{-5}\)
|
WS |
\(3.16 \times 10^{-5}\)
|
7 | |
\(\alpha \)-opt | DNF \(t = 12.50\)
|
\(\Lambda \)-opt |
\(5.95 \times 10^{-6}\)
|
\(\xi \)-opt |
\(1.10 \times 10^{-5}\)
|
\(\sigma \)-opt |
\(5.95 \times 10^{-6}\)
|
\(\langle \sigma \rangle \)-opt |
\(5.95 \times 10^{-6}\)
|
WS |
\(8.00 \times 10^{-6}\)
|