Нуль мне, конечно, не нужен. Достаточно будет точности в третьем знаке после запятой.
Проблема с этим методом в том, что я не могу его реализовать правильно, либо чего-то недопонимаю.
Беру за начально приближение
![$% MathType!MTEF!2!1!+-
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\[
u(x,y,0) = u_0 (x,y) = \sin (2\pi xy) + 2\cos ((x - 1)\frac{\pi }{2}(y - 1))\cos (\frac{\pi }{2}x)\cos (\frac{\pi }{2}y)
\]
$ $% MathType!MTEF!2!1!+-
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\[
u(x,y,0) = u_0 (x,y) = \sin (2\pi xy) + 2\cos ((x - 1)\frac{\pi }{2}(y - 1))\cos (\frac{\pi }{2}x)\cos (\frac{\pi }{2}y)
\]
$](https://dxdy-02.korotkov.co.uk/f/9/a/a/9aa7d432b038e9eb16613ea0d10a7e1382.png)
, оно удовлетворяет граничным условиям, подставляю его в
![$% MathType!MTEF!2!1!+-
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% WcbaGaamyBaiaacYcacaWGUbaabaGaamyAaaaakiaacMcaaaa!77A9!
\[
u_{m,n}^{i + 1} = u_{m,n}^i + \frac{\alpha }{4}(u_{m + 1,n}^i + u_{m,n + 1}^i + u_{m - 1,n}^{i + 1} + u_{m,n - 1}^{i + 1} - 4u_{m,n}^i ) + \frac{{dx^2 }}{4}U(u_{m,n}^i )
\]
$ $% MathType!MTEF!2!1!+-
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% WcbaGaamyBaiaacYcacaWGUbaabaGaamyAaaaakiaacMcaaaa!77A9!
\[
u_{m,n}^{i + 1} = u_{m,n}^i + \frac{\alpha }{4}(u_{m + 1,n}^i + u_{m,n + 1}^i + u_{m - 1,n}^{i + 1} + u_{m,n - 1}^{i + 1} - 4u_{m,n}^i ) + \frac{{dx^2 }}{4}U(u_{m,n}^i )
\]
$](https://dxdy-04.korotkov.co.uk/f/f/2/2/f227f086036c7768ad93560323e32aa782.png)
, прогоняю это уравнение через все внутренние узлы сетки, затем для каждогу узла высчитываю разницу
![$% MathType!MTEF!2!1!+-
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% WGPbGaey4kaSIaaGymaaaakiabgkHiTiaadwhadaqhaaWcbaGaamyB
% aiaacYcacaWGUbaabaGaamyAaaaaaaa!49D9!
\[
\varepsilon _{m,n}^{i + 1} = u_{m,n}^{i + 1} - u_{m,n}^i
\]
$ $% MathType!MTEF!2!1!+-
% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
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% WGPbGaey4kaSIaaGymaaaakiabgkHiTiaadwhadaqhaaWcbaGaamyB
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\[
\varepsilon _{m,n}^{i + 1} = u_{m,n}^{i + 1} - u_{m,n}^i
\]
$](https://dxdy-01.korotkov.co.uk/f/4/5/f/45f9011e6fc6a0d398a6cbb9c6b2c09c82.png)
если эта разница меня удовлетворяет (я, к примеру, сравниваю среднее арифметическое ошибок в каждой точке с желаемой точностью в 0.001), значит достигнута желаемая точность, в противном случае вся процедура повторяется.