|
4. Численное решение дифференциального уравнения
| |
\(\displaystyle 5\left( \dfrac{\partial^2 u}{\partial x^2} + \dfrac{\partial^2 u}{\partial y^2} \right) = 7u - 6xy\) |
|
проводится методом установления с использованием схемы переменных направлений. Из представленных
ниже разностных схем выберите ту, которая записана без ошибок.
|
\(\displaystyle \begin{cases} \dfrac{u_{j,k}^{n+1/2} - u_{j,k}^n}{\Delta t} = 5\lambda_{xx} u_{j,k}^{n+1/2} - 7u_{j,k}^{n+1/2} + 6(j-1)(k-1)h^2 \\[2ex] \dfrac{u_{j,k}^{n+1} - u_{j,k}^{n+1/2}}{\Delta t} = 5\lambda_{yy} u_{j,k}^{n+1} \end{cases}\) |
|
|
\(\displaystyle \begin{cases} \dfrac{u_{j,k}^{n+1/2} - u_{j,k}^n}{\Delta t} = 5\lambda_{xx} u_{j,k}^{n+1/2} + 5\lambda_{yy} u_{j,k}^n - 7u_{j,k}^{n+1/2} + 6(j-1)(k-1)h^2 \\[2ex] \dfrac{u_{j,k}^{n+1} - u_{j,k}^{n+1/2}}{\Delta t} = 5\lambda_{xx} u_{j,k}^{n+1/2} + 5\lambda_{yy} u_{j,k}^{n+1} \end{cases}\) |
|
|
\(\displaystyle \begin{cases} \dfrac{u_{j,k}^{n+1/2} - u_{j,k}^n}{\Delta t} + \dfrac{5}{2}\lambda_{xx} u_{j,k}^{n+1/2} + \dfrac{5}{2}\lambda_{yy} u_{j,k}^n = 7u_{j,k}^{n+1/2} - 6(j-1)(k-1)h^2 \\[2ex] \dfrac{u_{j,k}^{n+1} - u_{j,k}^{n+1/2}}{\Delta t} + \dfrac{5}{2}\lambda_{xx} u_{j,k}^{n+1/2} + \dfrac{5}{2}\lambda_{yy} u_{j,k}^{n+1} = 0 \end{cases}\) |
|
|
\(\displaystyle \begin{cases} \dfrac{u_{j,k}^{n+1/2} - u_{j,k}^n}{\Delta t} = \dfrac{5}{2}\lambda_{xx} u_{j,k}^{n+1/2} + \dfrac{5}{2}\lambda_{yy} u_{j,k}^n + 7u_{j,k}^{n+1/2} - 6(j-1)(k-1)h^2 \\[2ex] \dfrac{u_{j,k}^{n+1} - u_{j,k}^{n+1/2}}{\Delta t} = \dfrac{5}{2}\lambda_{xx} u_{j,k}^{n+1/2} + \dfrac{5}{2}\lambda_{yy} u_{j,k}^{n+1} \end{cases}\) |
|
|
\(\displaystyle \begin{cases} \dfrac{u_{j,k}^{n+1/2} - u_{j,k}^n}{\Delta t} = \dfrac{5}{2}\lambda_{xx} u_{j,k}^{n+1/2} + \dfrac{5}{2}\lambda_{yy} u_{j,k}^n - 7u_{j,k}^{n+1/2} + 6(j-1)(k-1)h^2 \\[2ex] \dfrac{u_{j,k}^{n+1} - u_{j,k}^{n+1/2}}{\Delta t} = \dfrac{5}{2}\lambda_{xx} u_{j,k}^{n+1/2} + \dfrac{5}{2}\lambda_{yy} u_{j,k}^{n+1} \end{cases}\) |
|
|
\(\displaystyle \begin{cases} \dfrac{u_{j,k}^{n+1/2} - u_{j,k}^n}{\Delta t} = \dfrac{5}{2}\lambda_{xx} u_{j,k}^{n+1/2} + \dfrac{5}{2}\lambda_{yy} u_{j,k}^n - 7u_{j,k}^{n+1/2} + 6(j-1)(k-1)h^2 \\[2ex] \dfrac{u_{j,k}^{n+1} - u_{j,k}^{n+1/2}}{\Delta t} = \dfrac{5}{2}\lambda_{xx} u_{j,k}^{n+1/2} + \dfrac{5}{2}\lambda_{yy} u_{j,k}^{n+1} \end{cases}\) |
|