Quickstart¶

This page walks through a minimal working example: a 1D transient diffusion equation.

The problem¶

We solve the diffusion equation

\[\frac{\partial c}{\partial t} = D \frac{\partial^2 c}{\partial x^2}\]

on a 1D domain \([0, L]\) with a fixed concentration \(c = 1\) at the left boundary and a no-flux (closed) boundary on the right. The initial concentration is zero everywhere.

Code¶

import pyfvtool as pf

# Parameters
Nx = 20          # number of finite volume cells
Lx = 1.0         # [m] domain length
c_left = 1.0     # left boundary concentration
c_init = 0.0     # initial concentration
D_val = 1e-5     # [m²/s] diffusion coefficient
t_end = 7200.0   # [s] simulation time
dt = 60.0        # [s] time step

# 1. Define the mesh
mesh = pf.Grid1D(Nx, Lx)

# 2. Create the cell variable (concentration)
#    By default, no-flux boundary conditions are applied on all boundaries
c = pf.CellVariable(mesh, c_init)

# 3. Set the left boundary to a fixed (Dirichlet) condition
c.BCs.left.a = 0.0
c.BCs.left.b = 1.0
c.BCs.left.c = c_left

# 4. Assign diffusivity and compute face-averaged values
D_cell = pf.CellVariable(mesh, D_val)
D_face = pf.geometricMean(D_cell)

# 5. Time loop
t = 0
while t < t_end:
    eqn = [pf.transientTerm(c, dt, 1.0),
           -pf.diffusionTerm(D_face)]
    pf.solvePDE(c, eqn)
    t += dt

# 6. Visualize
pf.visualizeCells(c)

What each step does¶

Step	Function	Purpose
1	`Grid1D`	Creates a uniform 1D Cartesian mesh
2	`CellVariable`	Stores values at cell centres; holds boundary conditions
3	`BCs` attributes	Specifiy BCs. Here, fixed-value (Dirichlet) on the left.
4	`geometricMean`	Interpolates cell values to face values
5	`transientTerm`, `diffusionTerm`, `solvePDE`	Assembles and solves the linear system
6	`visualizeCells`	Plots the cell-centred values

Next steps¶

Browse the Examples for more complete use cases.
See Meshes for 2D and 3D grids, cylindrical, and spherical coordinates.
See Boundary conditions for Dirichlet, Neumann, Robin, and periodic BCs.