Discretization at the boundaries for the Burgers’ eqn.

More on the Burgers’ equation! The process of discretization mentioned earlier works at the interior points in the grid. However, at the boundaries, the as-presented format fails to perform a numerical approximation. Revisiting the discretized form, 

 

The highlighted term in the above representation clearly does not work at i=1, since the term u(i-1) does not exist! To avoid this contradiction, the forward difference formulation is used at i=1, while the backward difference is used at i=N.

The forward difference formulation becomes

And the backward difference formulation becomes

 

The above reformulations facilitate a complete ODE representation of the Burgers’ PDE. From a computational perspective, however, one is better off treating the boundaries as algebraic representations with the actual values for the independent variable as opposed to the discrete formulations mentioned above. Where these forward and backward representations come in handy is when boundary conditions themselves are a partial differential equation. A good example is adiabatic boundary conditions for the heat equation.