## Finite Differences: Partial Differential Equations

### Week 5

##### Problem 8.1

(a) Looking at Equation (8.20) in the textbook for inspiration in terms of second order derivatives, we can easily derive the straightforward approximation. $$\frac{\delta^2 u}{\delta t^2} = v^2 \frac{\delta^2 u}{\delta x^2}$$ $$\frac{\delta^2 u}{\delta t^2} = \frac{u_j^{n+1}-2u_j^n+u_j^{n-1}}{(\Delta t)^2}\, ,\quad \frac{\delta^2 u}{\delta x^2} = \frac{u_{j+1}^n-2u_j^n+u_{j-1}^n}{(\Delta x)^2}$$ $$\frac{u_j^{n+1}-2u_j^n+u_j^{n-1}}{(\Delta t)^2} = v^2 \left(\frac{u_{j+1}^n-2u_j^n+u_{j-1}^n}{(\Delta x)^2}\right)$$

(b) We can clearly see that this is a second order approximation in both space and time.

(c) We can substitute Equation (8.9) in for all of our u terms and solve in terms of A to find the mode amplitudes. $$u_j^n=A(k)^n e^{ijk\Delta x} \quad (8.9)$$ $$u_j^{n+1}=2u_j^n-u_j^{n-1}+\frac{v^2(\Delta t^2)}{(\Delta x^2)}\bigl(u_{j+1}^n-2u_j^n+u_{j-1}^n\bigr)$$ $$A^{n+1}e^{ijk\Delta x}=2A^n e^{ijk\Delta x}-A^{n-1}e^{ijk\Delta x}+ \frac{v^2 (\Delta t)^2}{(\Delta x)^2}\bigl(A^n e^{i(j+1)k\Delta x}-2A^n e^{ijk\Delta x}+A^n e^{i(j-1)k\Delta x}\bigr)$$ $$A=2-A^{-1}+\frac{v^2 (\Delta t)^2}{(\Delta x)^2}\bigl(e^{ik\Delta x}-2+e^{-ik\Delta x}\bigr)$$ $$A+A^{-1}=2+\frac{2v^2 (\Delta t)^2}{(\Delta x)^2}\bigl(\cos(k\Delta x)-1\bigr)$$ $$\Bigl(A+A^{-1}=2+\frac{2v^2 (\Delta t)^2}{(\Delta x)^2}\bigl(\cos(k\Delta x)-1\bigr)\Bigr)\cdot A$$ $$A^2-\alpha A +1\, ,\quad \alpha =2+\frac{2v^2 (\Delta t)^2}{(\Delta x)^2}\bigl(\cos(k\Delta x)-1\bigr)$$ $$\frac{-b\pm \sqrt{b^2-4ac}}{2a}\quad \Rightarrow \quad A=\frac{-\alpha \pm \sqrt{\alpha ^2-4}}{2}$$ I am leaving the solution for A in this simplified form.

(d) We know that |A|2=1. Because there are two solutions, A1 and A2, we know that A1*A2=1. This is only possible if the two solutions are complex conjugates of each other. Therefore, the square root term in the quadratic formula needs to be negative. $$\alpha ^2-4\lt 0$$ $$\alpha ^2 \lt 4$$ $$-2 \lt \alpha \lt 2$$ $$-2 \lt 2+\frac{2v^2 (\Delta t)^2}{(\Delta x)^2}\bigl(\cos(k\Delta x)-1\bigr) \lt 2$$ $$-4 \lt \frac{2v^2 (\Delta t)^2}{(\Delta x)^2}\bigl(\cos(k\Delta x)-1\bigr) \lt 0$$ $$-2 \lt \frac{v^2 (\Delta t)^2}{(\Delta x)^2}\bigl(\cos(k\Delta x)-1\bigr) \lt 0$$ Because the cosine term fluctuates between 0 and 1, the (cos - 1) term will fluctuate between -1 and 0. Thus, $$\frac{v^2 (\Delta t)^2}{(\Delta x)^2} \lt 2$$

(e) Because the real term is the same for both mode amplitudes, they both decay at the same rate.

(f)

(g)

(h)

##### Problem 8.2

(a) Below is the code for the 1D diffusion problem at different time step sizes of 1, 0.5, and 0.1.


import matplotlib.pyplot as plt
import numpy as np

L = 500
delta_x = 1.0
Nx = int(L/delta_x)
T = 100
delta_t = [1.0, 0.5, 0.1]

for del_t in delta_t:
Nt = int(T/del_t)
D = delta_x
x = np.linspace(0, L, Nx+1)    # mesh points in space
dx = x - x
t = np.linspace(0, int(T), Nt+1)    # mesh points in time
dt = t - t
F = D*dt/dx**2
u   = np.zeros(Nx+1)           # unknown u at new time level
u_1 = np.zeros(Nx+1)           # u at the previous time level
# Set initial condition u(x,0) = I(x)
u_1[int(len(u)/2)] = 1.0
print()
print("For delta_t = ", del_t, "Nt = ", Nt)
for n in range(0, Nt):
# Compute u at inner mesh points
for i in range(1, Nx):
u[i] = u_1[i] + F*(u_1[i-1] - 2*u_1[i] + u_1[i+1])
# Insert boundary conditions
u = 0;  u[Nx] = 0
# Update u_1 before next step
u_1[:]= u
print("Final output of u: ")
print(u)
print()


The numbers blow up at t = 1, then start to calm down as the step size decreases.