Upwind
First order: smooth but strongly diffusive. At t = 15 s the pulse peak has dropped to 0.22.
MacCormack (textbook)
Second order: keeps the peak (0.73 at t = 15 s) but adds trailing wiggles from dispersion.
Fortran · Python · CFD
Upwind vs MacCormack, side by side.
Two classic finite-difference schemes for the 1-D linear wave equation: the original Fortran studies, a Python port tested against their output, and an error comparison against the exact solution.
3platforms: Fortran, Python, CFD
MITopen source licence
Why it exists
Two schemes, one pulse, an honest comparison.
The grid is 301 nodes with a spacing of 0.01 and a time step of 0.01, so the Courant number is 0.09.
This repo merges two earlier repos, Upwind-Differencing-Scheme and Explicit-Maccormack-Method.
Screens
See more of it.
What's inside
What the comparison shows.
Exact solution
Every scheme is scored by RMS error against the exactly translated pulse.
Original Fortran
fortran/upwind.f90 and fortran/maccormack.f90 compile with any Fortran 90 compiler and write Gnuplot-ready .plt files.
Tested Python port
schemes.py reproduces the Fortran output; the tests check both ports against the committed results within 1e-5.
Two findings
Snapshots are at t = 1, 3, 6, 10, 15 ... 55 s, not 1 to 10 s, and the old MacCormack results came from a backward-difference predictor.
Run it yourself
Run the comparison.
From the README’s “Run it” section. Full options, requirements and troubleshooting are there too.
Open the README$ pip install numpy matplotlib
$ python schemes.py # error table + comparison.png
$ python tests/test_schemes.py # port vs Fortran output checksUnder the hood
How it's built.
Built with
Language
FortranPython
Output
Tecplot / Gnuplot data
Testing
Automated tests
37 files in the repo · last updated 8 Nov 2026.
Browse the codeQuestions
Good questions.
What does it run on?
Fortran, Python, CFD.
How do I get it running?
The commands in the “Run it” box above come straight from the README. Paste them into a terminal; the README covers options and troubleshooting.
Is it free?
The source is public on GitHub under the MIT licence.
Who made it?
Juwen Lie, an R&D mechanical engineer in Batam, Indonesia, as an independent personal project.
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Read the code and the numbers.
Upwind vs MacCormack, side by side.