- #1

pomekrank

- 13

- 0

Let's consider the heat equation as [itex]\frac{\partial T}{\partial t}=\alpha \frac{{{\partial }^{2}}T}{\partial {{x}^{2}}}[/itex]

In order to have a second accuracy system, one can use the Crank-Nicolson method as [itex]\frac{{{\partial }^{2}}T}{\partial {{x}^{2}}}\approx \frac{1}{2}\left( \frac{T_{i+1}^{k+1}-2T_{i}^{k+1}+T_{i-1}^{k+1}}{\Delta {{x}^{2}}}+\frac{T_{i+1}^{k}-2T_{i}^{k}+T_{i-1}^{k}}{\Delta {{x}^{2}}} \right)+O\left( \Delta {{t}^{2}}+\Delta {{x}^{2}} \right)[/itex]

However, when the finite difference method is use with respect to time, usually a forward Euler method like [itex] \frac{\partial T}{\partial t}\approx \frac{T_{i}^{k+1}-T_{i}^{k}}{\Delta t}+O\left( \Delta t,\,\,\Delta x \right)[/itex]

Does it make the entire system an accuracy [itex]O\left( \Delta t,\Delta {{x}} \right)[/itex] ? If so, why don't use a much simpler method like center difference [itex]\frac{\partial T}{\partial x}\approx \frac{T_{i+1}^{k}-T_{i-1}^{k}}{2\Delta x}+O\left( \Delta t,\,\Delta {{x}^{2}} \right)[/itex] instead of Crank-Nicolson ? Furthermore, if one wants to get a full 2nd order system, how could it be possible ? Thank you,

Steven