# Solving a nonlinear problem with CDF

I'm trying to solve this problem:

$\begin{cases} \partial_t E=-k\left([f(\rho)-i.\left[\delta+\frac{1}{2}a\left[\dfrac{\nabla^2_{\bot}}{4}+1-\rho^2\right]\right]]E - 2CP\right)\\ \partial_t P=-\gamma_{\bot}[(1+i\delta)P + ED]\\ \partial_t D=-\gamma_{\parallel}(D-\chi(\rho)-\frac{1}{2}(E^*P+EP^*)) \end{cases}$

Here $f(\rho)$ and $\chi(\rho)$ are radial functions. and the Laplacian is only en x and y (transversal Laplacian).

$\rho$ is the radial coordinate in a cylindrical coordinate system.

with

$E=E_x,E_y$ ;and $E_x , E_y \in \mathbb{C}$

$P=P_x,P_y$ ; and $P_x , P_y \in \mathbb{C}$

so $E_x,\ E_y,\ P_x,\ P_y$ are complex functions. which amounts to a total of 9 equations.

I don't know much about CDF, I tried doing some explicit forward method, since I saw a Navier-Stokes problem being solved that way, but its unstable:

from the equation for the evolution of $P$: $$P^{j+1}_{n,m}=-\gamma_{\bot}[(1+i\delta)P^j_{n,m}+E^j_{n,m}D^j_{n,m}]\Delta t + P^j_{n,m}$$

from the equation for the evolution of $D$: $$D^{j+1}_{n,m}=-\gamma_{\parallel}\Delta t (D^j_{n,m}-\chi(\rho)_{n,m}-\frac{1}{2}((E^j_{n,m})^*P^j_{n,m}+E^j_{n,m}(P^j_{n,m})^*))+D^j_{n,m}$$

from the equation fo the evolution of $E$:

$\partial_t E=i\dfrac{ka}{8}\nabla^2_{\bot}E-k([f(\rho)E-i[\delta+\frac{1}{2}a[1-\rho^2]]E-P)\\$

then: $$\Re(E)^{j+1}_{n,m}= -\frac{ka}{8}\Delta t \left(\frac{\Im(E)_{n+1,m}^{j} - 2\Im(E)_{n,m}^{j} + \Im(E)_{n-1,m}^{j}}{\Delta x^2}+\frac{\Im(E)_{n,m+1}^{j} - 2\Im(E)_{n,l}^{m} + \Im(E)_{n,m-1}^{j}}{\Delta y^2}\right) -\left(f(\rho)_{n,m}\Re(E)^j_{n,m} + (\delta+\frac{a}{2}[1-\rho^2])\Im(E)^j_{n,m} - \Re(P)^j_{n,m}\right)k\Delta t + \Re(E)^{j}_{n,m}\\$$

witch holds for $\Re(E_x)$ and $\Re(E_y)$

$$\Im(E)^{j+1}_{n,m}= \frac{ka}{8}\Delta t \left(\frac{\Re(E)_{n+1,m}^{j} - 2\Re(E)_{n,m}^{j}+\Re(E)_{n-1,m}^{j}}{\Delta x^2} + \frac{\Re(E)_{n,m+1}^{j} - 2\Re(E)_{n,m}^{j} + \Re(E)_{n,m-1}^{j}}{\Delta y^2}\right) - \left(f(\rho)_{n,m}\Im(E)^j_{n,m} - (\delta+\frac{a}{2}[1-\rho^2])\Re(E)^j_{n,m} - \Im(P)^j_{n,m}\right)k\Delta t + \Im(E)^{j}_{n,m}\\$$ witch holds for $\Im(E_x)$ and $\Im(E_y)$

I have been reading up on CDF for several weeks, but still feel kind of lost.

Any suggestions about winch method should I use, and why, or maybe some book.

• Which CDF do you mean? Central difference? – Biswajit Banerjee Dec 26 '15 at 20:16
• I think that maybe the reference is to CFD and not CDF. – nicoguaro Dec 27 '15 at 18:54
• The one i've tried in the example is forward on the time derivate, And central for the space one(hence the expresión for the laplacian). But as it is, it's not consistent (at least with the parameters I am using) Since it blow up after some time, and it does so even faster as I improve the grid stepping. Now I'm looking into implicit methods and galerkin polinomial methods(simulating the equations assuming the answer is near a function that can be expressed with Laguerre polinomial) But I'm not sure what are the cons and pros of each method. Or if they are worth the trouble. – alxg Dec 27 '15 at 21:26