SYLLABUS Previous: 1.3.1 Advection
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Slide : [ diffusion - Green function || VIDEO login]
where is the diffusion coefficient. For a homogeneous medium, the combined advection-diffusion equation
can be solved analytically in terms of the Green function (exercise 1.03)
A numerical solution is generally required to solve the equation in an inhomogeneous medium, where ,
Below is an example of a numerical solution describing the diffusion of an initial box function computed using the finite element method from chapter 3.
A harmonic ansatz can be used to analyze the diffusion equation and leads to the dispersion relation
This shows that short wavelengths (large ) are more strongly damped (large negative imaginary ) than long wavelengths, with a decay rate proportional to . This explains the numerical experiments above.
Note that the heat equation, which describes the evolution of the temperature in a medium with a heat conductivity in the presence of heat sources and sinks
is one particular application of the diffusion equation. It can easily be reduced to (1.3.2#eq.1) by substituting the differential operator in cartesian coordinates for a 1D slab.
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