BTCS Scheme

The BTCS scheme (Backward-Time Centred-Space) is an implicit finite difference method for the Heat equation.

Discretisation

Using a backward difference in time and a centred second difference in space:

$$u_{i+1,n}(1+2c)-c{u}_{i+1,n+1}-c{u}_{i+1,n-1}=u_{i,n}$$

where

$$c=\frac{\alpha dt}{d{x}^2}$$

Because the unknowns $u_{i+1,n}$ appear on the left-hand side, this gives a tridiagonal linear system at each time step:

$$A{\mathbf{u}}_{i+1}={\mathbf{u}}_i$$

The matrix $A$ is tridiagonal with $1+2c$ on the diagonal and $-c$ on the off-diagonals. It can be inverted using Gaussian elimination or np.linalg.inv. The Thomas algorithm (a specialised $O(N)$ solver for tridiagonal systems) also exists but is not covered in this module.

Stability

The BTCS scheme is unconditionally stable - there is no restriction on the step size $dt$. This makes it suitable for stiff problems where the FTCS stability condition would force an impractically small $dt$.

Accuracy

• First-order in time: $O(dt)$
• Second-order in space: $O(d{x}^2)$

Same order as FTCS scheme but without the stability constraint.

FTCS scheme | Heat equation | Finite differences | Implicit Euler method | Boundary conditions | Diagonal dominance