Преобразование дифференциальной задачи в разностную

Тренажёры · Глава 2

Преобразование дифференциальной задачи в разностную

Задания для самоконтроля

6. Математическая модель трубчатого реактора с продольным перемешиванием в нестационарном режиме имеет вид:

\(\displaystyle \frac{\partial c}{\partial t} + v\frac{\partial c}{\partial x} = D\frac{\partial^2 c}{\partial x^2} - kc.\)

Выберите правильное решение задачи определения порядка аппроксимации явной разностной схемы, записанной для этого дифференциального уравнения:

\(\displaystyle \frac{c_j^{n+1} - c_j^n}{\Delta t} + v\frac{c_j^n - c_{j-1}^n}{h} = D\frac{c_{j+1}^n - 2c_j^n + c_{j-1}^n}{h^2} - kc_j^n.\)

	\(\displaystyle \left.\frac{\partial c}{\partial t}\right\|_j^n + \frac{1}{2}\left.\frac{\partial^2 c}{\partial t^2}\right\|_j^n\Delta t + \frac{1}{6}\left.\frac{\partial^3 c}{\partial t^3}\right\|_j^n(\Delta t)^2 + v\left(\left.\frac{\partial c}{\partial x}\right\|_j^n - \frac{1}{2}\left.\frac{\partial^2 c}{\partial x^2}\right\|_j^n h + \frac{1}{6}\left.\frac{\partial^3 c}{\partial x^3}\right\|_j^n h^2\right) = D\left(\left.\frac{\partial^2 c}{\partial x^2}\right\|_j^n + \frac{1}{12}\left.\frac{\partial^4 c}{\partial x^4}\right\|_j^n h^2\right) - kc_j^n; \quad \left.\frac{\partial c}{\partial t}\right\|_j^n + O(\Delta t) + v\left.\frac{\partial c}{\partial x}\right\|_j^n + O(h) = \sigma\left.\frac{\partial^2 c}{\partial x^2}\right\|_j^n + O(h^2) - kc_j^n \quad\Rightarrow\quad O(\Delta t, h).\)

	\(\displaystyle \left.\frac{\partial c}{\partial t}\right\|_j^n + \frac{1}{2}\left.\frac{\partial^2 c}{\partial t^2}\right\|_j^n\Delta t + \frac{1}{6}\left.\frac{\partial^3 c}{\partial t^3}\right\|_j^n(\Delta t)^2 + v\left(\left.\frac{\partial c}{\partial x}\right\|_j^n - \frac{1}{2}\left.\frac{\partial^2 c}{\partial x^2}\right\|_j^n h + \frac{1}{6}\left.\frac{\partial^3 c}{\partial x^3}\right\|_j^n h^2\right) = D\left(\left.\frac{\partial^2 c}{\partial x^2}\right\|_j^n + \frac{1}{12}\left.\frac{\partial^4 c}{\partial x^4}\right\|_j^n h^2\right) - kc_j^n; \quad \left.\frac{\partial c}{\partial t}\right\|_j^n + O(\Delta t) + v\left.\frac{\partial c}{\partial x}\right\|_j^n + O(h) = \sigma\left.\frac{\partial^2 c}{\partial x^2}\right\|_j^n + O(h^2) - kc_j^n \quad\Rightarrow\quad O(\Delta t, h^2).\)

	\(\displaystyle \left.\frac{\partial c}{\partial t}\right\|_j^n + \frac{1}{2}\left.\frac{\partial^2 c}{\partial t^2}\right\|_j^n\Delta t + \frac{1}{6}\left.\frac{\partial^3 c}{\partial t^3}\right\|_j^n(\Delta t)^2 + v\left(\left.\frac{\partial c}{\partial x}\right\|_j^n - \frac{1}{2}\left.\frac{\partial^2 c}{\partial x^2}\right\|_j^n h + \frac{1}{6}\left.\frac{\partial^3 c}{\partial x^3}\right\|_j^n h^2\right) = D\left(\left.\frac{\partial^2 c}{\partial x^2}\right\|_j^n + \frac{1}{12}\left.\frac{\partial^4 c}{\partial x^4}\right\|_j^n h^2\right) - kc_j^n; \quad \left.\frac{\partial c}{\partial t}\right\|_j^n + O(\Delta t) + v\left.\frac{\partial c}{\partial x}\right\|_j^n + O(h) = \sigma\left.\frac{\partial^2 c}{\partial x^2}\right\|_j^n + O(h) - kc_j^n \quad\Rightarrow\quad O(\Delta t, h).\)

	\(\displaystyle \left.\frac{\partial c}{\partial t}\right\|_j^n + \frac{1}{2}\left.\frac{\partial^2 c}{\partial t^2}\right\|_j^n\Delta t + \frac{1}{6}\left.\frac{\partial^3 c}{\partial t^3}\right\|_j^n(\Delta t)^2 + v\left(\left.\frac{\partial c}{\partial x}\right\|_j^n - \frac{1}{2}\left.\frac{\partial^2 c}{\partial x^2}\right\|_j^n h + \frac{1}{6}\left.\frac{\partial^3 c}{\partial x^3}\right\|_j^n h^2\right) = D\left(\left.\frac{\partial^2 c}{\partial x^2}\right\|_j^n + \frac{1}{12}\left.\frac{\partial^4 c}{\partial x^4}\right\|_j^n h^2\right) - kc_j^n; \quad \left.\frac{\partial c}{\partial t}\right\|_j^n + O(\Delta t^2) + v\left.\frac{\partial c}{\partial x}\right\|_j^n + O(h^2) = \sigma\left.\frac{\partial^2 c}{\partial x^2}\right\|_j^n + O(h^2) - kc_j^n \quad\Rightarrow\quad O(\Delta t^2, h^2).\)

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