Python finite-difference CFD project for rebuilding canonical incompressible-flow solvers from first principles. The focus is on numerical schemes, pressure-velocity coupling, boundary conditions, validation against analytical solutions, and visualisation of flow fields.
Currently implemented:
- 1D linear advection with upwind, leapfrog, Lax-Friedrichs, and Lax-Wendroff schemes
- 2D linear advection
- 1D nonlinear convection with upwind, Lax-Friedrichs, Richtmyer, Lax-Wendroff, and MacCormack schemes in non-conservative and conservative forms
- 2D nonlinear convection
- 1D and 2D diffusion
- 1D and 2D Burgers equation
- 2D Laplace equation
- 2D Poisson equation with configurable source terms
- 2D lid-driven cavity flow using a pressure Poisson solve
- 2D pressure-driven channel flow with periodic x-boundaries
- uniform grid generation
- hat-function, Heaviside-style step, and Cole-Hopf initial conditions
- explicit finite-difference time-marching solvers
- iterative pressure/potential solves using L1 convergence or fixed iteration limits
- contour, surface, quiver, and animation visualizations
core/config.py simulation dataclasses
core/setup/ grids, time steps, initial conditions
core/numerics/ finite-difference solvers
core/analytical/ analytical reference solutions
post_processing/ contour, surface, quiver, and animation helpers
run_*.py executable examples
The current solvers model:
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the 1D linear advection equation:
du/dt + c du/dx = 0
using selectable explicit finite-difference schemes: upwind, leapfrog, Lax-Friedrichs, and Lax-Wendroff.
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the 2D linear advection equation:
du/dt + c du/dx + c du/dy = 0
using an explicit upwind finite-difference scheme.
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the 1D nonlinear convection equation:
du/dt + u du/dx = 0
and its conservative form:
du/dt + d(u^2 / 2)/dx = 0
using selectable explicit schemes: upwind, Lax-Friedrichs, Richtmyer, Lax-Wendroff, and MacCormack. Conservative variants are included for flux-form shock propagation.
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the 2D nonlinear convection equations:
du/dt + u du/dx + v du/dy = 0
dv/dt + u dv/dx + v dv/dy = 0
using an explicit upwind finite-difference scheme.
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the 1D diffusion equation:
du/dt = ν d²u/dx²
using an explicit central finite-difference scheme.
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the 2D diffusion equation:
du/dt = ν (d²u/dx² + d²u/dy²)
using an explicit central finite-difference scheme.
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the 1D Burgers equation:
du/dt + u du/dx = ν d²u/dx²
using an explicit upwind scheme for the convective term and a central scheme for the diffusive term.
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the 2D Burgers equations:
du/dt + u du/dx + v du/dy = ν (d²u/dx² + d²u/dy²)
dv/dt + u dv/dx + v dv/dy = ν (d²u/dx² + d²u/dy²)
using an explicit upwind scheme for the convective terms and a central scheme for the diffusive terms.
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the 2D Laplace equation:
d²p/dx² + d²p/dy² = 0
solved iteratively with finite differences until an L1 target is reached.
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the 2D Poisson equation:
d²p/dx² + d²p/dy² = b
solved iteratively with configurable positive and negative source terms.
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the 2D incompressible lid-driven cavity flow problem:
du/dt + u du/dx + v du/dy = -1/rho dp/dx + nu (d^2u/dx^2 + d^2u/dy^2)
dv/dt + u dv/dx + v dv/dy = -1/rho dp/dy + nu (d^2v/dx^2 + d^2v/dy^2)
d^2p/dx^2 + d^2p/dy^2 = b
using explicit finite differences for the velocity equations and an iterative pressure Poisson solve.
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the 2D incompressible pressure-driven channel flow problem:
du/dt + u du/dx + v du/dy = -1/rho dp/dx + nu (d^2u/dx^2 + d^2u/dy^2) + F
dv/dt + u dv/dx + v dv/dy = -1/rho dp/dy + nu (d^2v/dx^2 + d^2v/dy^2)
d^2p/dx^2 + d^2p/dy^2 = b
using explicit finite differences for the velocity equations, an iterative pressure Poisson solve, periodic boundary conditions in x, and no-slip walls at y = 0 and y = Ly.
This project includes validation workflows that compare numerical finite-difference solutions with analytical reference solutions.
For the 1D diffusion equation, the numerical solver is validated against a Fourier-based analytical solution of the heat equation.
For the 1D Burgers equation, the numerical solver is validated against the analytical Cole-Hopf solution.
The 1D convection scheme comparison script visualizes conservative and non-conservative schemes on a Heaviside step problem, comparing grid resolution and CFL number effects. This highlights numerical diffusion, dispersive oscillations near shocks, and conservative-form shock propagation.
These comparisons are used to assess solver correctness and visualize agreement between numerical and analytical results.
- Benchmark lid-driven cavity profiles against reference data.
- Validate pressure-driven channel flow against analytical Poiseuille behaviour.
- Add regression tests for boundary conditions and pressure-source terms.
- Add convergence studies for grid spacing and time-step sensitivity.
- Add limiters or artificial viscosity for oscillation control near shocks.
- Add regression tests for 1D convection scheme updates.
python run_advection_1d.py
python run_advection_1d_scheme_comparison.py
python run_advection_2d.py
python run_convection_1d.py
python run_convection_1d_scheme_comparison.py
python run_convection_2d.py
python run_diffusion_1d.py
python run_diffusion_1d_vs_heat.py
python run_diffusion_2d.py
python run_burgers_equation_1d.py
python run_burgers_equation_1d_vs_cole_hopf.py
python run_burgers_equation_2d.py
python run_laplace_2d.py
python run_poisson_2d.py
python run_cavity_flow.py
python run_channel_flow.py