E19 numerical methods for engineering applications spring

Question

E19: Numerical Methods for Engineering Applications Spring 2016 - HOMEWORK 13

Vectorized fluid simulation code

Download the (slow, explicitly looped) fluid simulation code from the course website. It contains a complete Python port of Jos Stam's method. Your job is to convert every explicit loop in Fluid.py over array indices (which appear in the program as i and j) into vectorized operations.

You will hopefully find the indexing_strategies.py script useful for this, as it gives a few hints about useful techniques. Here are some other miscellaneous tips for converting the various functions in Fluid.py:

• For set_bnd, each individual border (top/bottom/left/right) can be handled in one assignment statement.
• The looped version of solve uses the Gauss-Seidel method (https://en.wikipedia.org/wiki/Gauss-Seidel_method); for the vectorized version, it is far more natural to use the Jacobi method (https://en.wikipedia.org/wiki/Jacobi_method) instead. The former updates the solution element-by-element, in-place, whereas the latter updates the entire solution at once. You can keep the outermost loop over k.
• For advect, you can let i and j be arrays created from numpy.meshgrid. The max and min calls will have to be replaced with the functions numpy.maximum and numpy.minimum, respectively. The other subtlety here is that i₀ and j₀ should be arrays of integer type (e.g., i₀ = y.astype(int)).
• For project, looking at dx and dy in the indexing_strategies.py script may be helpful.
• It is best to vectorize one function at a time, and make sure you get identical results for each change you make (or near-identical, for Jacobi iteration vs. Gauss-Seidel).

You will find that a successful vectorized implementation is vastly faster than one with explicit loops. On my laptop, for N = 32, the looped code takes about 130 ms per update vs. 5 ms for vectorized. For N = 128, the respective runtimes are 2000 ms vs. 20 ms.

Submit your vectorized code via Moodle (with your name at the top of the file). Please do not change the interfaces of any of the functions - I want a drop-in replacement for Fluid.py that I can run against the code in sim.py.

Attachment:- fluid sim.zip