# Are linear, CTCS codes always stable?

I would like to solve some equations which basically look like this $$\frac{\partial u}{\partial x}=F\left(v,\frac{\partial v}{\partial y},\frac{\partial^2 v}{\partial y^2}\right),$$ $$\frac{\partial v}{\partial x}=G\left(u,\frac{\partial u}{\partial y},\frac{\partial^2 u}{\partial y^2}\right).$$ My current plan is to solve this by replacing every derivative with a centred-difference equation and so the numerical scheme looks like this $$u_{i+1,j}=u_{i,j} + \Delta x\,F\left(v_{i+1/2,j},v_{i+1/2,j+1},v_{i+1/2,j-1}\right)$$ $$v_{i+3/2,j}=v_{i+1/2,j} + \Delta x\,G\left(u_{i+1/2,j},u_{i+1/2,j+1},u_{i+1/2,j-1}\right),$$ where I define each $$v_{i+1/2,j}$$ to be located halfway between $$u_{i,j}$$ and $$u_{i+1,j}$$ in the $$x$$-direction. I initialisise $$u$$ and $$v$$ at $$x=x_{min}$$ and then iterate along in $$x$$. Is this scheme stable? Do you know if a scheme for solving such a problem is documented anywhere. I have coded it up and it seems to be stable for certain values of $$\Delta x$$ and $$\Delta y$$. However, I do not know if there is a fundamental problem with my scheme or if I have simply made a silly coding error somewhere.

Edit:

I have linked a pdf showing the expressions for F and G. They are highlighted in red text on page 3.

https://www.dropbox.com/s/43q5kkkqrrkbvrq/Normal_mode_code.pdf?dl=0

• Without knowing the details of your implementation it’s impossible for us to know if something is wrong or not. Also, if it is possible, specify $F$ and $G$ operators explicitly. Feb 22 '20 at 3:30
• I suppose that CTCS refers to centred in time and centred in space. If that's the case, then the answer is no. One counterexample is the advection equation. Feb 22 '20 at 13:06
• @nicoguaro I would say that's the answer. Feb 22 '20 at 19:41
• @AntonMenshov, I'm not that sure. I just calculated the stability condition using von Neumann analysis and it seemed to be stable (if CFL≤1). Feb 22 '20 at 23:57
• Thank you for your comments. The expressions for F and G are quite complicated. They involve complex numbers. I have attached a pdf showing the equations. Feb 24 '20 at 10:29