Editing Advection (section)

===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}}