Convection Diffusion

Consider the convection-diffusion equation

With the boundary conditions

Part 1. Pure convection (α = 0)

Consider the following initial profile

The exact solution is

(1) Explicit Euler time advancement and second-order central difference for the spatial derivative.

Discretization:

From modified wave number analysis, Explicit Euler is unstable.

(2) Leapfrog time advancement and the second-order central difference for the spatial derivative.

Discretization:

From modified wave number analysis, Leapfrog is stable when

Part 2. Convection-diffusion

Let

(1) Explicit Euler time advancement and second-order central difference for the spatial derivative.

Discretization:

From Von-Neumann stability analysis, Explicit Euler is stable when

(2) Leapfrog time advancement and the second-order central difference for the spatial derivative.

Discretization:

From modified wave number analysis, Leapfrog is unstable.

Simulation Result

Let , ,and

Part 1. Pure convection (α = 0)

Part 2. Convection-diffusion