Solving the equation Advection

a simulation of advection equation u = (sin t, cos t) solenoidal.

the advection equation not simple solve numerically: system hyperbolic partial differential equation, , interest typically centers on discontinuous shock solutions (which notoriously difficult numerical schemes handle).

even 1 space dimension , constant velocity field, system remains difficult simulate. equation becomes

∂
ψ


∂
t



+

u

x





∂
ψ


∂
x



=
0


{\displaystyle {\frac {\partial \psi }{\partial t}}+u_{x}{\frac {\partial \psi }{\partial x}}=0}

where



ψ
=
ψ
(
x
,
t
)


{\displaystyle \psi =\psi (x,t)}

scalar field being advected ,




u

x




{\displaystyle u_{x}}





x


{\displaystyle x}

component of vector




u

=
(

u

x


,
0
,
0
)


{\displaystyle \mathbf {u} =(u_{x},0,0)}

.