Advection equation

From Academic Kids

The advection equation is the partial differential equation that governs the motion of a conserved scalar as it is advected by a known velocity field. It is derived using the scalar's conservation law, together with Gauss's theorem, and taking the infinitesimal limit.

Perhaps the best image to have in mind is the transport of dissolved salt in water.

The advection equation expressed mathematically is:

<math>

\frac{\partial\psi}{\partial t} +\nabla\cdot\left( \psi{\bold u}\right) =0 <math> where ∇ is the divergence. Frequently, it is assumed that the velocity field is solenoidal, that is, that <math>\nabla\cdot{\bold u}=0<math>. If this is so, the above equation reduces to

<math>

\frac{\partial\psi}{\partial t} +{\bold u}\cdot\nabla\psi=0. <math>

In particular, if the flow is steady, <math>{\bold u}\cdot\nabla\psi=0<math> which shows that <math>\psi<math> is constant along a streamline.

The advection equation is not simple to solve numerically: the system is a hyperbolic partial differential equation, and interest typically centers on discontinuous "shock" solutions (which are notoriously difficult for numerical schemes to handle).

Even in one space dimension and constant velocity, the system remains difficult to simulate (it is a standard test for advection schemes known as the pigpen problem). The equation becomes

<math>

\frac{\partial\psi}{\partial t}+u\frac{\partial\psi}{\partial x}=0 <math>

where <math>\psi=\psi(x,t)<math>.

According to Zang [2], numerical simulation can be aided by considering the skew symmetric form for the advection operator.

<math>

\frac{1}{2} {\bold u} \cdot \nabla {\bold u} + \frac{1}{2} \nabla ({\bold u} {\bold u}) <math>

where <math> \nabla ({\bold u} {\bold u}) <math> is a vector with components <math>[\nabla ({\bold u} u_x),\nabla ({\bold u} u_y),\nabla ({\bold u} u_z)]<math> and the notation <math> {\bold u} = [u_x,u_y,u_z]<math> has been used.

Since skew symmetry implies only complex eigenvalues, this form reduces the "blow up" and "spectral blocking" often experienced in numerical solutions with sharp discontinuities (see Boyd [1] pp. 213).

References

[1] Boyd, J.P.: 2000, Chebyshev and Fourier Spectral Methods 2nd edition (http://www-personal.engin.umich.edu/~jpboyd/BOOK_Spectral2000.html), Dover, New York

[2] Zang, T: 1991, On the rotation and skew-symmetric forms for incompressible flow simulations, Applied Numerical Mathematics,7,27-40

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