Von Neumann's stability analysis on non linear and coupled equations

I'd like to know if is possible to make a Von Neumann's stability analysis on an system of coupled equations, featuring quadratics:

\begin{aligned} \frac{\partial u_1}{\partial t}&=D_1\Delta u_1 - s_1\left(\left(\frac{\nabla u}{(k+a)^2}-2u\frac{\nabla a}{(k+a)^3}\right)\nabla(a) + \frac{u}{(k+a)^2}\left(\Delta(a) + 2\frac{\partial^2a}{\partial x \partial y}\right)\right) \\&\quad+ r_1u_1 \left(\frac{n^2}{\beta + n^2}-u_1\right)\\ \frac{\partial a}{\partial t} &= D_a\Delta a + c_1n\frac{u_1^2}{\alpha_1 + u_1^2} - \gamma a\\ \frac{\partial n}{\partial t}&= D_s\Delta n - l_1u_1\frac{n^2}{\beta + n^2} \end{aligned}

My idea was to make different analysis for each of terms of the equations. There are no problems for most of the terms, but I don't know how to deal with the coupled ones and the nonlinear one. Can someone give me hints or link me a paper dealing with these?

Here is the numerical scheme :

\begin{align*} &\frac{U_{i,j}^{k+1}-U_{i,j}^k}{\Delta t}= D_1 \frac{U_{i+1,j}^k+U_{i-1,j}^k+U_{i,j+1}^k+U_{i,j-1}^k-4U_{i,j}^k}{{\Delta z}^2} + r_1 U_{i,j}^k\bigg(\frac{{N_{i,j}^k}^2}{\beta + {N_{i,j}^k}^2} - U_{i,j}^k\bigg)\\ &\qquad\qquad\qquad-s_1\bigg( \frac{U_{i+1,j}^k-U_{i-1,j}^k+U_{i,j+1}^k-U_{i,j-1}^k}{2\big(k_1+A_{i,j}^k\big)^2\Delta z} -2U_{i,j}^k\frac{A_{i+1,j}-A_{i-1,j}^k+A_{i,j+1}^k-A_{i,j-1}^k}{2\Delta z(k_1+A_{i,j}^k)^3} \bigg)\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times\frac{A_{i+1,j}^k-A_{i-1,j}^k+A_{i,j+1}^k-A_{i,j-1}^k}{2\Delta z}\\ &+ \frac{s_1U_{i,j}^k}{(k_1+{A_{i,j}^k}^2)}\times \bigg( \frac{A_{i+1,j}^k+A_{i-1,j}^k+A_{i,j+1}^k+A_{i,j-1}^k-4A_{i,j}^k}{{\Delta z}^2} + \frac{A_{i+1,j+1}^k-A_{i+1,j-1}^k-A_{i-1,j+1}^k+A_{i+1,j+1}^k}{2{\Delta z}^2}\bigg) \\ \end{align*}

\begin{align*} &\frac{A_{i,j}^{k+1}-A_{i,j}^k}{\Delta t}= D_a \frac{A_{i+1,j}^k+A_{i-1,j}^k+A_{i,j+1}^k+A_{i,j-1}^k-4A_{i,j}^k}{{\Delta z}^2}-\gamma A_{i,j}^k+c_1N_{i,j}^k\frac{{U_{i,j}^k}^2}{\alpha_1+{U_{i,j}^k}^2}+c_2N_{i,j}^k\frac{{V_{i,j}^k}^2}{\alpha_2+{V_{i,j}^k}^2} \end{align*}

\begin{align*} &\qquad\frac{N_{i,j}^{k+1}-N_{i,j}^k}{\Delta t}= D_a \frac{N_{i+1,j}^k+N_{i-1,j}^k+N_{i,j+1}^k+N_{i,j-1}^k-4N_{i,j}^k}{{\Delta z}^2}-\frac{{N_{i,j}^k}^2}{\beta + {N_{i,j}^k}^2}\big(l_1{U_{i,j}^k}^2+l_2{V_{i,j}^k}^2\big)\\ \end{align*}

• Von Neumann stability analysis is performed for finite-difference schemes, so discretized equations are kind of the focus of the question - or at least a thorough explanation of what FD-scheme is used for each term. This will highly increase the probability of a good answer from the community; however, maybe someone has particular references for coupled and nonlinear terms handling ready that will suffice you. – Anton Menshov Jun 3 '18 at 15:21
• I just added it. The discretization could certainly be done in a smarter way – nougatine Jun 3 '18 at 16:11

1 Answer

Von Neumann stability analysis does not only apply to FD schemes... it can always be applied to finite volume, finite element... or whatever method you use. The only requirement that must be fulfilled is the linearity of the equations and the orthogonality/uniformity of the mesh to apply Fourier methods for the following linear discretised system:

$$d_t\vec{u}+A\vec{u}=\vec{f}$$

where in your case $\vec{u}=(\vec{u}_1,\vec{a},\vec{n})^{T}$ and $\vec{f}=\vec{0}$.

If you have a nonlinear system, i.e. $A=A(u)$, you can always linearise it $A\approx A({\vec{u}_0})$ and see what occurs in the vicinity of a selected solution $\vec{u}_0$, usually the one that makes the system stationary: $$A(\vec{u}_0)\vec{u}_0=\vec{f}$$ and perform the stability analysis.