Consider the diffusion equation
$$
\frac{\partial v}{\partial t}=\mu \frac{\partial^2 v}{\partial x^2}, \quad v(x, 0)=\phi(x), \quad \int_a^b v(x, t) \mathrm{d} x=0
$$
with $x \in[a, b]$ and periodic boundary conditions. The solutio $\mathrm{n}$ is to be approximated using the central difference operator $L$ for the 1D Laplacian.
$$
L v_m=\frac{v_{m+1}-2 v_m+v_{m-1}}{h^2}
$$
and the following two finite different approximations, (i) Forw ard-Euler
$$
v_{n+1}=v_n+\mu k L v_n,(1)
$$
and (ii) Crank-Nicolson
$$
v_{n+1}=v_n+\mu k\left(L v_n+L v_{n+1}\right)
$$

Throughout, consider $[a, b]=[0,2 \pi]$ and the finite differenc e stencil to have periodic boundary conditions on the spatial lattice $[0, h, 2 h, \cdots,(N-1) h]$ where $h=\frac{2 \pi}{N}$ and $N$ is ev en.
(a) Determine the order of accuracy of the central difference operator $L v$ is approximating the second derivative $v_{x x}$.
(b) Using $v_m^n=\sum_{l=0}^{N-1} \hat{v}_l^n \exp \left(-i \frac{2 \pi l m}{N}\right)$ give the updates $\hat{v}_l^{n+1}$ in terms of $\hat{v}_l^n$ for each of the methods, including the ca se $l=0$.
(c) Give the solution for $v_m^n$ for each method when the initial condition is $\phi(m \Delta x)=(-1)^m$.
(d) What are the stability constraints on the time step $k$ for ea ch of the methods, if any, in equation (1) and (2)? Show there are either no constraints or express them in the form $k \leq F(h, \mu)$