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Derivation of CN scheme

To derive the Crank-Nicolson equation, we start with the forward Euler scheme and Taylor expand u(x,t+\Delta t) , u(x+\Delta x, t) and u(x-\Delta x,t) \begin{align} u(x+\Delta x,t)&=u(x,t)+\frac{\partial u(x,t)}{\partial x} \Delta x+\frac{\partial^2 u(x,t)}{2\partial x^2}\Delta x^2+\mathcal{O}(\Delta x^3), \tag{10}\\ \nonumber u(x-\Delta x,t)&=u(x,t)-\frac{\partial u(x,t)}{\partial x}\Delta x+\frac{\partial^2 u(x,t)}{2\partial x^2} \Delta x^2+\mathcal{O}(\Delta x^3),\\ \nonumber u(x,t+\Delta t)&=u(x,t)+\frac{\partial u(x,t)}{\partial t}\Delta t+ \mathcal{O}(\Delta t^2). \tag{11} \end{align}