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 = ...