# Convection equation expanded in Legendre polynomials

In some areas of physics, for example radiation transport in plasmas, it is beneficial to re-express the convection equation in terms of a Legendre polynomial expansion. This results in a set of coupled equations for the expansion coefficients.

So, if the standard convection equation for a function f is: $$\partial f/\partial t=-v_x \partial f/\partial x$$, after expansion, the first two of the coupled equations are

$$\partial f_0/\partial t=-a v\partial f_1/\partial x$$

and

$$\partial f_1/\partial t=-b v\partial f_0/\partial x$$

where $$a$$ and $$b$$ are constants and $$v$$ is the magnitude of $$v_x$$.

Since these equations describe convection, their numerical solution suffers from the usual problems of fluid convection, for example spurious short scale oscillations, problems at sharp interfaces, false extrema etc. These problems have been solved for the standard convection equation (e.g. WENO, Van-Leer-type flux limiters, artificial diffusion/viscosity, Riemann solvers etc). Are there equivalents of these for the coupled equations?

One approach I have tried is to re-express these coupled equations in terms of new function g and h like this:

$$\partial g/\partial t=-c v\partial g/\partial x$$

and

$$\partial h/\partial t=-k v\partial h/\partial x$$

which are in the usual convection form, but for other reasons I need to work directly with the functions $$f_0$$ and $$f_1$$ so this doesn't help.