# stability of numerical method

For the one dimensional transport equation :$\frac{\partial{u}}{\partial{t}}+c\frac{\partial{u}}{\partial{x}}=0$, where $c$ is a positive number, consider the following two numerical schemes:

(A) $\frac{u_j^{n+1}-u_j^n}{\delta t}+c\frac{u_j^n-u_{j-1}^n}{\delta x}=0$

(B) $\frac{u_j^{n+1}-u_j^n}{\delta t}+c\frac{u_{j+1}^n-u_{j-1}^n}{2\delta x}=0$

In the above, $j$ denotes space and $n$ is the time index.

Let $u_j^n=\xi^n \exp(ikj\delta x)$ and determine the amplification factor $(\xi=u_j^{n+1}/u_j^n)$ for both (A) and (B). Here $k$ is the wave number, and $i$ denotes the imaginary unit.

Every side over $u_j^n$, (A) will turn to be $\xi=1-\frac{c\delta t}{\delta x}(1-e^{-ik\delta x})$, and (B) will turn to be $\xi=1-\frac{c\delta t}{2\delta x}(e^{ik\delta x}-e^{-ik\delta x})$

Using the result above, determine the stability of both schemes?

Here is my solution:

$\xi=1-\frac{c\delta t}{\delta x}(1-e^{-ik\delta x})$, consider two points A $(1,0*i)$ and B $(cos(k\delta x),sin(k\delta x))$, then $1-e^{-ik\delta x}$ is vector $BA$,

then multiply some small number will be one point $C$ in line segment $AB$. If you draw some graph of unit, it is easy that $|\xi|\le 1$, then this scheme is stable.

As same method using above, we find scheme (B) is unstable.

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