Please solve this problem using Matlab. Please show the modified code to get to the answers. Thanks in advance!!

CP$6[6$ marks$]$ Consider the complex Ginzburg Landau $($GL$)$ equation

$u_t=gra{d}^{2}u+u-u|u{|}^2,$

which arises in superconductor theory, quantum theory, and string theory. In this problem we aim to solve this GL

equation in the box $-1x,y1$ with periodic boundary conditions in space and white noise as an initial condition.

Using a finite difference approximations of $u_t$ and gradu, we arrive at the semi$-$implicit discretisation

$\frac{u_{n+1}-u_n}{t}=-\frac{}{h^2}A{u}_{n+1}+u_{n+1}-u_n|u_n{|}^2$

where $A$ is a suitably scaled version of the Poisson matrix. Notice we are treating the linear parts implicitly and the

nonlinear parts explicitly, which provides a good balance of stability and efficiency.

Starting from the initial condition $u_0,$ we must solve at each time step

$B{u}_{n+1}=N(u_n),$ where $B=(1-t)I+t\frac{}{h^2}$A and $N(u)=-u|u{|}^2.$

The code below

implements this method for a a certain choice of parameters $,,$ and $t,$ with

$N=100.$ Howevever, the code is inefficient and too slow.

$($a$)$ Modify the code to improve it as best you can, using a built$-$in direct method to solve any required linear systems of

equations. $($Hints: Consider sparsity, factorisations, and block operators.$)$ Show your computing at $t=10$ and report on

both the original and improved timings to compute it$.$

$($b$)$ Repeat $($a$),$ but now using a suitable iterative method to solve the linear systems. Compare with the timings from $($a$).$

Hint: It's very easy to make the code fast if you are willing to get a nonsense answer! This solutions using your modified

code should look at least qualitatively similar to those in the original version. If they don't, that's a good indication that

you've done something wrong!