7. The Advection Equation to Higher order and Higher Dimension

• Higher-order schemes. Numerical Recipes pages are here.

• Exercise 7.1: The Lax-Wendroff method.  [Solution]

• Combined code for the 1-D advection problem.

• The wave equation. Numerical Recipes pages are here.

• Exercise 7.2: Lax-Wendroff for the wave equation.  [Set-up]  [Solution]  [Numpy solution]

• Problems with discontinuous initial conditions (DEMO).

  Solution by smoothing the ends of the range. Here is the approach adopted in ex7.2_v2.py.

• Exercise 7.3: A simple physics application.

  • 1-D interference.  Create two gaussian disturbances of the form u(x,0) = A exp[-(x-xc)²/w²] at the ends of the grid, one of width w = 2 and amplitude A = 1 for -10 < x < 0, centered on xc = -7.5 and moving to the right, the other of width 2 and amplitude 0.5 for 0 < x < 10, centered on xc = 7.5 and moving to the left.

    As the waves meet they interfere. Determine the maximum and minimum values of the field u on the grid during the simulation. Do they agree with your expectations?

    Now using the smoothing procedure just described, repeat the interference calculation using two sinusoidal wave segments, each of wavelength 2 and 1 period in length. The left-hand segment is centered on xc = -8, has amplitude 1, and moves to the right. The right-hand segment is centered on xc = 8, has amplitude 0.5, and moves to the left. Smooth the edges of each pulse using the functions in mysmooth.py, with n = 3 and delx = 0.4.