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