PDE-Solver

This project implements a numerical solver for the incompressible Navier–Stokes equations coupled with the heat (advection–diffusion) equation, enabling realistic simulation of fluid flow with thermal transport, using the pytorch library.

$$\frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla)\mathbf{u} = -\nabla p + \nu \nabla^2 \mathbf{u} + \mathbf{f}(T) $$

$$ \frac{\partial T}{\partial t} + \mathbf{u} \cdot \nabla T = \alpha \nabla^2 T $$

The solver iteratively applies the given differential equations while maintaining the incompressibility constraint $\nabla \cdot \mathbf{u} = 0 $.

• Time integration: Semi-implicit scheme with explicit advection
• Pressure solve: Poisson equation solved via iterative methods
• Advection: Upwind / semi-Lagrangian hybrid
• Diffusion: Central finite differences
• Boundary conditions: No-slip walls, inflow/outflow, and thermal Dirichlet or Neumann conditions

More information about the implementation can be found on the github page.