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5. Выберите из представленных ниже разностных соотношений те,
которые составляют схему предиктор-корректор, аппроксимирующую дифференциальное уравнение:
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\(\displaystyle \frac{\partial u}{\partial t} = 7\frac{\partial^2 u}{\partial x^2} + 8\frac{\partial^2 u}{\partial y^2} + 9\frac{\partial^2 u}{\partial z^2} - 13ut.\) |
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\(\displaystyle \frac{u_{j,k,m}^{n+1/4} - u_{j,k,m}^n}{\Delta t/2} = 7\frac{u_{j+1,k,m}^{n+1/4} - 2u_{j,k,m}^{n+1/4} + u_{j-1,k,m}^{n+1/4}}{h_x^2}, \quad \frac{u_{j,k,m}^{n+1/2} - u_{j,k,m}^{n+1/4}}{\Delta t/2} = 8\frac{u_{j,k+1,m}^{n+1/2} - 2u_{j,k,m}^{n+1/2} + u_{j,k-1,m}^{n+1/2}}{h_y^2}, \quad \frac{u_{j,k,m}^{n+1} - u_{j,k,m}^{n+1/2}}{\Delta t} = 7\frac{u_{j+1,k,m}^{n+1/2} - 2u_{j,k,m}^{n+1/2} + u_{j-1,k,m}^{n+1/2}}{h_x^2} + 8\frac{u_{j,k+1,m}^{n+1/2} - 2u_{j,k,m}^{n+1/2} + u_{j,k-1,m}^{n+1/2}}{h_y^2} + 9\frac{u_{j,k,m+1}^{n+1/2} - 2u_{j,k,m}^{n+1/2} + u_{j,k,m-1}^{n+1/2}}{h_z^2} - 13u_{j,k,m}^{n+1/2}(n+1/2)\Delta t.\) |
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\(\displaystyle +13u_{j,k,m}^n \Delta t. \quad \frac{u_{j,k,m}^{n+1/2} - u_{j,k,m}^{n+1/4}}{\Delta t/2} = 8\frac{u_{j,k+1,m}^{n+1/2} - 2u_{j,k,m}^{n+1/2} + u_{j,k-1,m}^{n+1/2}}{h_y^2}, \quad \frac{u_{j,k,m}^{n+1} - u_{j,k,m}^{n+1/2}}{\Delta t} = 7\frac{u_{j+1,k,m}^{n+1/2} - 2u_{j,k,m}^{n+1/2} + u_{j-1,k,m}^{n+1/2}}{h_x^2} + 8\frac{u_{j,k+1,m}^{n+1/2} - 2u_{j,k,m}^{n+1/2} + u_{j,k-1,m}^{n+1/2}}{h_y^2} + 9\frac{u_{j,k,m+1}^{n+1/2} - 2u_{j,k,m}^{n+1/2} + u_{j,k,m-1}^{n+1/2}}{h_z^2} - 13u_{j,k,m}^{n+1/2}(n+1/2)\Delta t.\) |
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\(\displaystyle \frac{u_{j,k,m}^{n+1/4} - u_{j,k,m}^n}{\Delta t/2} = 7\frac{u_{j+1,k,m}^{n+1/4} - 2u_{j,k,m}^{n+1/4} + u_{j-1,k,m}^{n+1/4}}{h_x^2}, \quad \frac{u_{j,k,m}^{n+1/2} - u_{j,k,m}^{n+1/4}}{\Delta t/2} = 8\frac{u_{j,k+1,m}^{n+1/2} - 2u_{j,k,m}^{n+1/2} + u_{j,k-1,m}^{n+1/2}}{h_y^2}, \quad \frac{u_{j,k,m}^{n+1} - u_{j,k,m}^{n+1/2}}{\Delta t} = 9\frac{u_{j,k,m+1}^{n+1/2} - 2u_{j,k,m}^{n+1/2} + u_{j,k,m-1}^{n+1/2}}{h_z^2} - 13u_{j,k,m}^{n+1/2}(n+1/2)\Delta t.\) |
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\(\displaystyle \begin{aligned}&\frac{u_{j,k,m}^{n+1/6}-u_{j,k,m}^n}{\Delta t/2}=7\frac{u_{j+1,k,m}^{n+1/6}-2u_{j,k,m}^{n+1/6}+u_{j-1,k,m}^{n+1/6}}{h_x^2},\\&\frac{u_{j,k,m}^{n+1/3}-u_{j,k,m}^{n+1/6}}{\Delta t/2}=8\frac{u_{j,k+1,m}^{n+1/3}-2u_{j,k,m}^{n+1/3}+u_{j,k-1,m}^{n+1/3}}{h_y^2},\\&\frac{u_{j,k,m}^{n+1/2}-u_{j,k,m}^{n+1/3}}{\Delta t/2}=9\frac{u_{j,k,m+1}^{n+1/2}-2u_{j,k,m}^{n+1/2}+u_{j,k,m-1}^{n+1/2}}{h_z^2},\\&\frac{u_{j,k,m}^{n+1}-u_{j,k,m}^n}{\Delta t}=7\frac{u_{j+1,k,m}^{n+1/6}-2u_{j,k,m}^{n+1/6}+u_{j-1,k,m}^{n+1/6}}{h_x^2}+8\frac{u_{j,k+1,m}^{n+1/3}-2u_{j,k,m}^{n+1/3}+u_{j,k-1,m}^{n+1/3}}{h_y^2}+9\frac{u_{j,k,m+1}^{n+1/2}-2u_{j,k,m}^{n+1/2}+u_{j,k,m-1}^{n+1/2}}{h_z^2}-13u_{j,k,m}^{n+1/2}\,n\,\Delta t.\end{aligned}\) |
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\(\displaystyle \begin{aligned}&\frac{u_{j,k,m}^{n+1/6}-u_{j,k,m}^n}{\Delta t/2}=7\frac{u_{j+1,k,m}^{n+1/6}-2u_{j,k,m}^{n+1/6}+u_{j-1,k,m}^{n+1/6}}{h_x^2},\\&\frac{u_{j,k,m}^{n+1/3}-u_{j,k,m}^{n+1/6}}{\Delta t/2}=8\frac{u_{j,k+1,m}^{n+1/3}-2u_{j,k,m}^{n+1/3}+u_{j,k-1,m}^{n+1/3}}{h_y^2},\\&\frac{u_{j,k,m}^{n+1/2}-u_{j,k,m}^{n+1/3}}{\Delta t/2}=9\frac{u_{j,k,m+1}^{n+1/2}-2u_{j,k,m}^{n+1/2}+u_{j,k,m-1}^{n+1/2}}{h_z^2},\\&\frac{u_{j,k,m}^{n+1}-u_{j,k,m}^n}{\Delta t}=7\frac{u_{j+1,k,m}^{n+1/6}-2u_{j,k,m}^{n+1/6}+u_{j-1,k,m}^{n+1/6}}{h_x^2}+8\frac{u_{j,k+1,m}^{n+1/3}-2u_{j,k,m}^{n+1/3}+u_{j,k-1,m}^{n+1/3}}{h_y^2}+9\frac{u_{j,k,m+1}^{n+1/2}-2u_{j,k,m}^{n+1/2}+u_{j,k,m-1}^{n+1/2}}{h_z^2}-13u_{j,k,m}^{n+1/2}(n+1/2)\,\Delta t.\end{aligned}\) |
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\(\displaystyle \begin{aligned}&\frac{u_{j,k,m}^{n+1/6}-u_{j,k,m}^n}{\Delta t/2}=7\frac{u_{j+1,k,m}^{n+1/6}-2u_{j,k,m}^{n+1/6}+u_{j-1,k,m}^{n+1/6}}{h_x^2},\\&\frac{u_{j,k,m}^{n+1/3}-u_{j,k,m}^{n+1/6}}{\Delta t/2}=8\frac{u_{j,k+1,m}^{n+1/3}-2u_{j,k,m}^{n+1/3}+u_{j,k-1,m}^{n+1/3}}{h_y^2},\\&\frac{u_{j,k,m}^{n+1/2}-u_{j,k,m}^{n+1/3}}{\Delta t/2}=9\frac{u_{j,k,m+1}^{n+1/2}-2u_{j,k,m}^{n+1/2}+u_{j,k,m-1}^{n+1/2}}{h_z^2},\\&\frac{u_{j,k,m}^{n+1}-u_{j,k,m}^n}{\Delta t}=7\frac{u_{j+1,k,m}^{n+1/6}-2u_{j,k,m}^{n+1/6}+u_{j-1,k,m}^{n+1/6}}{h_x^2}+8\frac{u_{j,k+1,m}^{n+1/3}-2u_{j,k,m}^{n+1/3}+u_{j,k-1,m}^{n+1/3}}{h_y^2}+9\frac{u_{j,k,m+1}^{n+1/2}-2u_{j,k,m}^{n+1/2}+u_{j,k,m-1}^{n+1/2}}{h_z^2}-13u_{j,k,m}^{n+1/2}\,n\,\Delta t.\end{aligned}\) |
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