Computational Physics Lectures: Partial differential equations

Processing math: 100%

Error in CN scheme

For the Crank-Nicolson scheme we also need to Taylor expand

$u(x+\Delta x, t+\Delta t)$ and

$u(x-\Delta x, t+\Delta t)$ around

$t'=t+\Delta t/2$ .

$\begin{align} u(x+\Delta x, t+\Delta t)&=u(x,t')+\frac{\partial u(x,t')}{\partial x}\Delta x+\frac{\partial u(x,t')}{\partial t} \frac{\Delta t}{2} +\frac{\partial^2 u(x,t')}{2\partial x^2}\Delta x^2+\frac{\partial^2 u(x,t')}{2\partial t^2}\frac{\Delta t^2}{4} +\notag \tag{13}\\ \nonumber &\frac{\partial^2 u(x,t')}{\partial x\partial t}\frac{\Delta t}{2} \Delta x+ \mathcal{O}(\Delta t^3)\\ \nonumber u(x-\Delta x, t+\Delta t)&=u(x,t')-\frac{\partial u(x,t')}{\partial x}\Delta x+\frac{\partial u(x,t')}{\partial t} \frac{\Delta t}{2} +\frac{\partial^2 u(x,t')}{2\partial x^2}\Delta x^2+\frac{\partial^2 u(x,t')}{2\partial t^2}\frac{\Delta t^2}{4} - \notag\\ \nonumber &\frac{\partial^2 u(x,t')}{\partial x\partial t}\frac{\Delta t}{2} \Delta x+ \mathcal{O}(\Delta t^3)\\ u(x+\Delta x,t)&=u(x,t')+\frac{\partial u(x,t')}{\partial x}\Delta x-\frac{\partial u(x,t')}{\partial t} \frac{\Delta t}{2} +\frac{\partial^2 u(x,t')}{2\partial x^2}\Delta x^2+\frac{\partial^2 u(x,t')}{2\partial t^2}\frac{\Delta t^2}{4} -\notag \tag{14}\\ \nonumber &\frac{\partial^2 u(x,t')}{\partial x\partial t}\frac{\Delta t}{2} \Delta x+ \mathcal{O}(\Delta t^3)\\ \nonumber u(x-\Delta x,t)&=u(x,t')-\frac{\partial u(x,t')}{\partial x}\Delta x-\frac{\partial u(x,t')}{\partial t} \frac{\Delta t}{2} +\frac{\partial^2 u(x,t')}{2\partial x^2}\Delta x^2+\frac{\partial^2 u(x,t')}{2\partial t^2}\frac{\Delta t^2}{4} +\notag \\ \nonumber &\frac{\partial^2 u(x,t')}{\partial x\partial t}\frac{\Delta t}{2} \Delta x+ \mathcal{O}(\Delta t^3)\\ \nonumber u(x,t+\Delta t)&=u(x,t')+\frac{\partial u(x,t')}{\partial t}\frac{\Delta_t}{2} +\frac{\partial ^2 u(x,t')}{2\partial t^2}\Delta t^2 + \mathcal{O}(\Delta t^3)\\ \nonumber u(x,t)&=u(x,t')-\frac{\partial u(x,t')}{\partial t}\frac{\Delta t}{2}+\frac{\partial ^2 u(x,t')}{2\partial t^2}\Delta t^2 + \mathcal{O}(\Delta t^3) \tag{15} \end{align}$ We now insert these expansions in the approximations for the derivatives to find

$\begin{align} &\left[\frac{\partial u(x,t')}{\partial t}\right]_{\text{approx}} =\frac{\partial u(x,t')}{\partial t}+\mathcal{O}(\Delta t^2) , \tag{16}\\ \nonumber &\left[\frac{\partial^2 u(x,t')}{\partial x^2}\right]_{\text{approx}}=\frac{\partial^2 u(x,t')}{\partial x^2}+\mathcal{O}(\Delta x^2). \end{align}$