Editing Advection (section)

==Mathematical description==
The '''advection equation''' is a first-order [[hyperbolic partial differential equation]] that governs the motion of a conserved [[scalar field]] as it is advected by a known [[velocity field|velocity vector field]].{{sfn | LeVeque | 2002 | p=1}} It is derived using the scalar field's [[conservation law]], together with [[Gauss's theorem]], and taking the [[infinitesimal]] limit.

One easily  visualized example of advection is the transport of ink dumped into a river. As the river flows, ink will move downstream in a "pulse" via advection, as the water's movement itself transports the ink. If added to a lake without significant bulk water flow, the ink would simply disperse outwards from its source in a [[Diffusion|diffusive]] manner, which is not advection.  Note that as it moves downstream, the "pulse" of ink will also spread via diffusion.  The sum of these processes is called [[convection]].

===The advection equation===

The advection equation for a conserved quantity described by a [[scalar field]] <math>\psi(t,x,y,z)</math> is expressed by a [[Continuity_equation#Differential_form|continuity equation]]:
<math display="block"> \frac{\partial\psi}{\partial t} +\nabla\cdot\left( \psi{\mathbf u}\right) =0, </math>
where [[vector field]] <math>\mathbf{u} = (u_x, u_y, u_z)</math> is the [[flow velocity]] and <math>\nabla</math> is the [[del]] operator.<ref group=note>The subscripts denote the coordinates of the vector field; not to be confused with the [[Notation_for_differentiation#Partial_derivatives|notation for partial derivatives]].</ref>  If the flow is assumed to be [[incompressible flow|incompressible]] then <math>\mathbf{u}</math> is [[solenoidal]], that is, the [[Del#Divergence|divergence]] is zero:<math display="block">\nabla\cdot{\mathbf u} = 0,</math>and (by using a [[Del#Divergence|product rule associated with the divergence]]) the above equation reduces to

<math display="block"> \frac{\partial\psi}{\partial t} +{\mathbf u}\cdot\nabla\psi =0.</math>

In particular, if the flow is [[steady state|steady]], then{{sfn|LeVeque|2002|p=391}}<math display="block">{\mathbf u}\cdot\nabla \psi = 0,</math>which shows that <math>\psi</math> is constant (because <math display="inline">\nabla \psi = 0</math> for any vector <math display="inline">\mathbf u</math>) along a [[Streamlines, streaklines and pathlines|streamline]]. 

If a vector quantity <math>\mathbf{a}</math> (such as a [[magnetic field]]) is being advected by the [[solenoidal]] [[velocity field]] <math>\mathbf{u}</math>, then the advection equation above becomes:

<math display="block"> \frac{\partial{\mathbf a}}{\partial t} + \left( {\mathbf u} \cdot \nabla \right) {\mathbf a} =0. </math>

Here, <math>\mathbf{a}</math> is a [[vector field]] instead of the [[scalar field]] <math>\psi</math>.

===Solution===
{{see also|Method of characteristics}}

[[File:GaussianUpwind2D.gif|thumb|A simulation of the advection equation where {{math|1='''u''' = (sin ''t'', cos ''t'')}} is solenoidal.]]
Solutions to the advection equation can be approximated using [[Numerical_methods_for_partial_differential_equations|numerical methods]], where interest typically centers on [[Continuous function|discontinuous]] "shock" solutions and necessary conditions for convergence (e.g. the [[Courant–Friedrichs–Lewy_condition|CFL condition]]).{{sfn | LeVeque | 2002 | pp=4-6,68-69}} 

Numerical simulation can be aided by considering the [[Skew-symmetric matrix|skew-symmetric]] form of advection
<math display="block"> \tfrac12 {\mathbf u} \cdot \nabla {\mathbf u} + \tfrac12 \nabla ({\mathbf u} {\mathbf u}),</math>
where
<math display="block"> \nabla ({\mathbf u} {\mathbf u}) = \nabla \cdot[{\mathbf u} u_x,{\mathbf u} u_y,{\mathbf u} u_z].</math>

Since skew symmetry implies only [[Imaginary number|imaginary]] [[eigenvalues]], this form reduces the "blow up" and "spectral blocking" often experienced in numerical solutions with sharp discontinuities.{{sfn | Boyd | 2001 | p=213}}