KM2016 / Hyperbolic / testhyp
.pdf1
Тестирование программы hyperbol
При решении начальной задачи методом Рунге-Кутта число обраще
ний к функции для вычисления правой части равно 290.
Поточечные оценки точности аппроксимации в узловых точках по про
странству дается выражениями
|
|
|
= |
1 − |
~ |
|
, ′ = |
1 − |
∂~ /∂ |
|
, |
′′ = |
1 |
− |
∂2 ~ /∂ 2 |
|
, |
|||||
|
|
|
|
∂ /∂ |
∂2 /∂ 2 |
|||||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
где |
|
|
|
|
|
, |
~ |
– |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
= ( |
, ) |
|
|
|
результаты численной аппроксимации решения исход |
|||||||||||||||
ной задачи в точке ( , ).
Результаты представлены в таблице.
2
Момент времени 0 = 0
|
1 |
1.2 |
1.4 |
1.6 |
1.8 |
2 |
2.2 |
2.4 |
2.6 |
2.8 |
3 |
|
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
|
|
|
|
|
|
|
|
|
|
|
|
˜ |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
|
|
|
|
|
|
|
|
|
|
|
|
(%) |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
|
|
|
|
|
|
|
|
|
|
|
|
∂ /∂ |
1 |
1.2 |
1.4 |
1.6 |
1.8 |
2 |
2.2 |
2.4 |
2.6 |
2.8 |
3 |
|
|
|
|
|
|
|
|
|
|
|
|
∂˜ /∂ |
1 |
1.2 |
1.4 |
1.6 |
1.8 |
2 |
2.2 |
2.4 |
2.6 |
2.8 |
3 |
|
|
|
|
|
|
|
|
|
|
|
|
′ (%) |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
|
|
|
|
Момент времени 1 = 0.2 |
|
|
|
|
|||
|
|
|
|
|
|
|
|
|
|
|
|
|
1 |
1.2 |
1.4 |
1.6 |
1.8 |
2 |
2.2 |
2.4 |
2.6 |
2.8 |
3 |
|
|
|
|
|
|
|
|
|
|
|
|
|
0.208 |
0.2538 |
0.302 |
0.3528 |
0.4067 |
0.464 |
0.5252 |
0.5906 |
0.6606 |
0.7356 |
0.816 |
|
|
|
|
|
|
|
|
|
|
|
|
˜ |
0.2078 |
0.2536 |
0.3017 |
0.3525 |
0.4064 |
0.4637 |
0.5248 |
0.5902 |
0.6602 |
0.7352 |
0.8155 |
|
|
|
|
|
|
|
|
|
|
|
|
(%) |
0.0799 |
0.075 |
0.0744 |
0.0726 |
0.0709 |
0.069 |
0.0671 |
0.0651 |
0.063 |
0.061 |
0.0586 |
|
|
|
|
|
|
|
|
|
|
|
|
∂ /∂ |
1.12 |
1.407 |
1.729 |
2.092 |
2.5 |
2.96 |
3.478 |
4.059 |
4.709 |
5.434 |
6.24 |
|
|
|
|
|
|
|
|
|
|
|
|
∂˜ /∂ |
1.118 |
1.404 |
1.726 |
2.088 |
2.496 |
2.955 |
3.472 |
4.053 |
4.703 |
5.428 |
6.233 |
|
|
|
|
|
|
|
|
|
|
|
|
′ (%) |
0.221 |
0.203 |
0.195 |
0.184 |
0.173 |
0.162 |
0.152 |
0.142 |
0.133 |
0.123 |
0.116 |
∂2 /∂ 2 |
1.2 |
2.074 |
3.293 |
4.915 |
6.998 |
9.6 |
12.78 |
16.59 |
21.09 |
26.34 |
32.4 |
∂2 ˜ /∂ 2 |
1.176 |
2.045 |
3.259 |
4.877 |
6.955 |
9.552 |
12.72 |
16.53 |
21.03 |
26.28 |
32.33 |
′′ (%) |
2.01 |
1.39 |
1.02 |
0.783 |
0.618 |
0.501 |
0.414 |
0.347 |
0.297 |
0.253 |
0.23 |
|
|
|
|
Момент времени 2 = 0.3 |
|
|
|
|
|||
|
|
|
|
|
|
|
|
|
|
|
|
|
1 |
1.2 |
1.4 |
1.6 |
1.8 |
2 |
2.2 |
2.4 |
2.6 |
2.8 |
3 |
|
|
|
|
|
|
|
|
|
|
|
|
|
0.327 |
0.4067 |
0.4941 |
0.5906 |
0.6975 |
0.816 |
0.9475 |
1.093 |
1.255 |
1.433 |
1.629 |
|
|
|
|
|
|
|
|
|
|
|
|
˜ |
0.3264 |
0.406 |
0.4933 |
0.5897 |
0.6965 |
0.8149 |
0.9463 |
1.092 |
1.253 |
1.431 |
1.627 |
|
|
|
|
|
|
|
|
|
|
|
|
(%) |
0.168 |
0.159 |
0.153 |
0.146 |
0.14 |
0.133 |
0.126 |
0.119 |
0.112 |
0.105 |
0.101 |
|
|
|
|
|
|
|
|
|
|
|
|
∂ /∂ |
1.27 |
1.667 |
2.141 |
2.706 |
3.375 |
4.16 |
5.075 |
6.132 |
7.346 |
8.727 |
10.29 |
|
|
|
|
|
|
|
|
|
|
|
|
∂˜ /∂ |
1.265 |
1.66 |
2.133 |
2.697 |
3.365 |
4.149 |
5.063 |
6.119 |
7.331 |
8.712 |
10.27 |
|
|
|
|
|
|
|
|
|
|
|
|
′ (%) |
0.422 |
0.391 |
0.354 |
0.32 |
0.289 |
0.26 |
0.235 |
0.212 |
0.192 |
0.172 |
0.163 |
∂2 /∂ 2 |
1.8 |
3.11 |
4.939 |
7.373 |
10.5 |
14.4 |
19.17 |
24.88 |
31.64 |
39.51 |
48.6 |
∂2 ˜ /∂ 2 |
1.767 |
3.066 |
4.889 |
7.315 |
10.43 |
14.33 |
19.09 |
24.8 |
31.54 |
39.41 |
48.49 |
′′ (%) |
1.82 |
1.43 |
1.02 |
0.788 |
0.622 |
0.504 |
0.416 |
0.35 |
0.297 |
0.255 |
0.236 |
|
|
|
|
Момент времени 3 = 0.4 |
|
|
|
|
|||
|
|
|
|
|
|
|
|
|
|
|
|
|
1 |
1.2 |
1.4 |
1.6 |
1.8 |
2 |
2.2 |
2.4 |
2.6 |
2.8 |
3 |
|
0.464 |
0.5906 |
0.7356 |
0.9021 |
1.093 |
1.312 |
1.561 |
1.845 |
2.165 |
2.525 |
2.928 |
|
|
|
|
|
|
|
|
|
|
|
|
˜ |
0.4627 |
0.589 |
0.7338 |
0.9001 |
1.091 |
1.309 |
1.559 |
1.842 |
2.162 |
2.521 |
2.924 |
|
|
|
|
|
|
|
|
|
|
|
|
(%) |
0.271 |
0.262 |
0.244 |
0.228 |
0.211 |
0.196 |
0.181 |
0.167 |
0.154 |
0.142 |
0.135 |
|
|
|
|
|
|
|
|
|
|
|
|
∂ /∂ |
1.48 |
2.029 |
2.717 |
3.566 |
4.599 |
5.84 |
7.311 |
9.036 |
11.04 |
13.34 |
15.96 |
|
|
|
|
|
|
|
|
|
|
|
|
∂˜ /∂ |
1.471 |
2.018 |
2.704 |
3.551 |
4.582 |
5.821 |
7.29 |
9.012 |
11.01 |
13.31 |
15.93 |
|
|
|
|
|
|
|
|
|
|
|
|
′ (%) |
0.6 |
0.583 |
0.497 |
0.434 |
0.378 |
0.331 |
0.291 |
0.257 |
0.227 |
0.204 |
0.186 |
∂2 /∂ 2 |
2.4 |
4.147 |
6.586 |
9.83 |
14 |
19.2 |
25.56 |
33.18 |
42.18 |
52.68 |
64.8 |
∂2 ˜ /∂ 2 |
2.363 |
4.085 |
6.518 |
9.752 |
13.91 |
19.1 |
25.45 |
33.06 |
42.06 |
52.54 |
64.66 |
′′ (%) |
1.55 |
1.49 |
1.03 |
0.8 |
0.63 |
0.51 |
0.421 |
0.355 |
0.296 |
0.271 |
0.219 |