I am trying to solve a fourth order partial differential equation $$ {\partial u\over \partial t} = i{\partial^4 u\over \partial x^4} $$ where $i=\sqrt{-1}$ and periodic boundary conditions.
I tried using first order in time and 2nd order in space finite difference scheme, even second order in time and space also. But I am unable to formulate a FD scheme with is stable. I am trying to check stability using von-Neumann stability criteria. Could somebody explain how to proceed in this case?
Let's try the second order accurate central finite difference for the spatial variable and Backward Euler for the temporal discretization. Then, your discretized PDE reads: $$\frac{u_j^{k+1} - u_j^k}{\Delta t} = i \frac{u_{j-2}^{k+1} - 4u_{j-1}^{k+1} + 6 u_{j}^{k+1} -4 u_{j+1}^{k+1} + u_{j+2}^{k+1} }{\Delta x^4}$$ Now make the usual Ansatz for a von-Neumann stability analysis $$u_j^k = e^{ikx_j}$$ and get your hands dirty: \begin{align} u_j^{k+1} &= e^{ikx_j} + \underbrace{r}_{i \Delta t / \Delta x^4} \Big( e^{i(k+1) (x_j - 2 \Delta x)} - 4 e^{i(k+1) (x_j - \Delta x)} + 6e^{i(k+1)x_j} \\ & \quad \quad \quad \quad \quad \quad \quad - 4 e^{i(k+1) (x_j + \Delta x)} + e^{i(k+1) (x_j + 2\Delta x)} \Big) \\ &= e^{ikx_j} + r \Big( e^{ik x_j } e^{i (x_j - 2 \Delta xk - 2 \Delta x)} - 4 e^{ikx_j} e^{i(x_j -k\Delta x - \Delta x) } + 6e^{ik x_j } e^{ix_j} \\ &\quad \quad \quad \quad \quad - 4 e^{ik x_j } e^{i (x_j + k\Delta x + \Delta x)} + e^{ik x_j } e^{i(x_j + 2\Delta x + 2 \Delta x k)} \Big) \\ &= e^{ik x_j } \Big[ 1 + r \Big( e^{i (x_j - 2 \Delta xk - 2 \Delta x)} - 4 e^{i(x_j -k\Delta x - \Delta x) } + 6 e^{ix_j} \\ & \quad \quad \quad \quad \quad \quad \quad - 4 e^{i (x_j + k\Delta x + \Delta x)} + e^{i(x_j + 2\Delta x + 2 \Delta x k)} \Big) \Big] \end{align}
By Wolfram Alpha, \begin{align}& e^{i (x_j - 2 \Delta xk - 2 \Delta x)} - 4 e^{i(x_j -k\Delta x - \Delta x) } + 6 e^{ix_j} - 4 e^{i (x_j + k\Delta x + \Delta x)} + e^{i(x_j + 2\Delta x + 2 \Delta x k)} \\ =& 16 e^{ix_j} \sin^4\Big( 0.5\Delta x [k+1] \Big)\end{align}
And using Euler's formula $e^{ix_j} = \cos(x_j) + i \sin(x_j)$
\begin{align} &r \Big( e^{i (x_j - 2 \Delta xk - 2 \Delta x)} - 4 e^{i(x_j -k\Delta x - \Delta x) } + 6 e^{ix_j} - 4 e^{i (x_j + k\Delta x + \Delta x)} + e^{i(x_j + 2\Delta x + 2 \Delta x k)} \Big) \\ =& \underbrace{16 \frac{\Delta t}{\Delta x^4} \sin^4\Big( 0.5\Delta x [k+1] \Big)}_{=: \alpha(k)} \big[ i \cos(x_j) - \sin(x_j) \big] \end{align}
Thus, the absolute value of the amplification $$ |G| = \Big \vert 1 + r \Big( e^{i (x_j - 2 \Delta xk - 2 \Delta x)} - 4 e^{i(x_j -k\Delta x - \Delta x) } + 6 e^{ix_j} - 4 e^{i (x_j + k\Delta x + \Delta x)} + e^{i(x_j + 2\Delta x + 2 \Delta x k)} \Big) \Big \vert$$ is given by \begin{align} |G|^2 &= \alpha(k)^2 \cos(x_j)^2 + (1 - \alpha(k) \sin(x_j)^2) \\ &= \alpha(k)^2 \cos(x_j)^2 + \alpha(k)^2 \sin(x_j)^2+ 1 - 2 \alpha(k) \sin(x_j) \\ &= \alpha(k)^2 + 1 - 2 \alpha(k) = (\alpha(k) - 1)^2 \end{align} So to have stability $|G| < 1$, we require \begin{align} \vert \alpha(k) - 1 \vert < 1 \end{align} Note that it holds $\forall k \in \mathbb{N}_0, \Delta t > 0, \Delta x >0$: $$0 < \alpha (k) \leq 16 \frac{\Delta t}{\Delta x^4} $$ Thus if you guarantee that $$ 16 \frac{\Delta t}{\Delta x^4} -1 < 1$$ you have stability.