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5. Выберите из представленных ниже разностных соотношений те, которые
составляют схему предиктор-корректор, аппроксимирующую дифференциальное уравнение:
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\(\displaystyle \frac{\partial u}{\partial t} = 7\frac{\partial^2 u}{\partial x^2} + 8\frac{\partial^2 u}{\partial y^2} - 19ut.\) |
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\(\displaystyle \frac{u_{j,k}^{n+1/4} - u_{j,k}^n}{\Delta t} = 7\frac{u_{j+1,k}^{n+1/4} - 2u_{j,k}^{n+1/4} + u_{j-1,k}^{n+1/4}}{h_x^2}, \quad \frac{u_{j,k}^{n+1/2} - u_{j,k}^{n+1/4}}{\Delta t} = 8\frac{u_{j,k+1}^{n+1/2} - 2u_{j,k}^{n+1/2} + u_{j,k-1}^{n+1/2}}{h_y^2}\\ \frac{u_{j,k}^{n+1} - u_{j,k}^{n+1/2}}{\Delta t} = 7\frac{u_{j+1,k}^{n+1/2} - 2u_{j,k}^{n+1/2} + u_{j-1,k}^{n+1/2}}{h_x^2} + 8\frac{u_{j,k+1}^{n+1/2} - 2u_{j,k}^{n+1/2} + u_{j,k-1}^{n+1/2}}{h_y^2} - 19u_{j,k}^{n+1/2}(n+1/2)\,\Delta t.\) |
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\(\displaystyle \frac{u_{j,k}^{n+1/4} - u_{j,k}^n}{\Delta t/2} = 7\frac{u_{j+1,k}^{n+1/4} - 2u_{j,k}^{n+1/4} + u_{j-1,k}^{n+1/4}}{h_x^2}, \quad \frac{u_{j,k}^{n+1/2} - u_{j,k}^{n+1/4}}{\Delta t/2} = 8\frac{u_{j,k+1}^{n+1/2} - 2u_{j,k}^{n+1/2} + u_{j,k-1}^{n+1/2}}{h_y^2},\\ \frac{u_{j,k}^{n+1} - u_{j,k}^{n+1/2}}{\Delta t} = 7\frac{u_{j+1,k}^{n+1/2} - 2u_{j,k}^{n+1/2} + u_{j-1,k}^{n+1/2}}{h_x^2} + 8\frac{u_{j,k+1}^{n+1/2} - 2u_{j,k}^{n+1/2} + u_{j,k-1}^{n+1/2}}{h_y^2} - 19u_{j,k}^{n+1/2}(n+1/2)\,\Delta t.\) |
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\(\displaystyle \frac{u_{j,k}^{n+1/4} - u_{j,k}^n}{\Delta t/2} = 7\frac{u_{j+1,k}^{n+1/4} - 2u_{j,k}^{n+1/4} + u_{j-1,k}^{n+1/4}}{h_x^2}, \quad \frac{u_{j,k}^{n+1/2} - u_{j,k}^{n+1/4}}{\Delta t/2} = 8\frac{u_{j,k+1}^{n+1/2} - 2u_{j,k}^{n+1/2} + u_{j,k-1}^{n+1/2}}{h_y^2},\\ \frac{u_{j,k}^{n+1} - u_{j,k}^{n+1/2}}{\Delta t} = 7\frac{u_{j+1,k}^{n+1/2} - 2u_{j,k}^{n+1/2} + u_{j-1,k}^{n+1/2}}{h_x^2} + 8\frac{u_{j,k+1}^{n+1} - 2u_{j,k}^{n+1} + u_{j,k-1}^{n+1}}{h_y^2} - 19u_{j,k}^{n+1}\,n\,\Delta t.\) |
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\(\displaystyle \frac{u_{j,k}^{n+1/4} - u_{j,k}^n}{\Delta t} = 7\frac{u_{j+1,k}^{n+1/4} - 2u_{j,k}^{n+1/4} + u_{j-1,k}^{n+1/4}}{h_x^2}, \quad \frac{u_{j,k}^{n+1/2} - u_{j,k}^{n+1/4}}{\Delta t} = 8\frac{u_{j,k+1}^{n+1/2} - 2u_{j,k}^{n+1/2} + u_{j,k-1}^{n+1/2}}{h_y^2}\\ \frac{u_{j,k}^{n+1} - u_{j,k}^{n+1/2}}{\Delta t} = 7\frac{u_{j+1,k}^{n+1/2} - 2u_{j,k}^{n+1/2} + u_{j-1,k}^{n+1/2}}{h_x^2} + 8\frac{u_{j,k+1}^{n+1} - 2u_{j,k}^{n+1} + u_{j,k-1}^{n+1}}{h_y^2} - 19u_{j,k}^{n+1}\,n\,\Delta t.\) |
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\(\displaystyle \frac{u_{j,k}^{n+1/4} - u_{j,k}^n}{\Delta t/2} = 7\frac{u_{j+1,k}^{n+1/4} - 2u_{j,k}^{n+1/4} + u_{j-1,k}^{n+1/4}}{h_x^2}, \quad \frac{u_{j,k}^{n+1/2} - u_{j,k}^{n+1/4}}{\Delta t/2} = 8\frac{u_{j,k+1}^{n+1/4} - 2u_{j,k}^{n+1/4} + u_{j,k-1}^{n+1/4}}{h_y^2},\\ \frac{u_{j,k}^{n+1} - u_{j,k}^{n+1/2}}{\Delta t} = 7\frac{u_{j+1,k}^{n+1/2} - 2u_{j,k}^{n+1/2} + u_{j-1,k}^{n+1/2}}{h_x^2} + 8\frac{u_{j,k+1}^{n+1/2} - 2u_{j,k}^{n+1/2} + u_{j,k-1}^{n+1/2}}{h_y^2} - 19u_{j,k}^{n+1/2}(n+1/2)\,\Delta t.\) |
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\(\displaystyle \frac{u_{j,k}^{n+1} - u_{j,k}^n}{\Delta t} = 7\frac{u_{j+1,k}^{n+1} - 2u_{j,k}^{n+1} + u_{j-1,k}^{n+1}}{h_x^2}, \quad \frac{u_{j,k}^{n+1} - u_{j,k}^n}{\Delta t} = 8\frac{u_{j,k+1}^{n+1} - 2u_{j,k}^{n+1} + u_{j,k-1}^{n+1}}{h_y^2},\\ \frac{u_{j,k}^{n+1} - u_{j,k}^n}{\Delta t} = 7\frac{u_{j+1,k}^{n+1} - 2u_{j,k}^{n+1} + u_{j-1,k}^{n+1}}{h_x^2} + 8\frac{u_{j,k+1}^{n+1} - 2u_{j,k}^{n+1} + u_{j,k-1}^{n+1}}{h_y^2} - 19u_{j,k}^{n+1/2}\,n\,\Delta t.\) |
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