Methods that are usually adopted for time integration in transport phenomena problems are either:

Euler (explicit, first-order accurate)

$\frac{dY}{dt}=f(t,Y)$

$Y^{n+1}=Y^n+\Delta t f(t,Y^n)$

Backward (implicit, first-order accurate)

$Y^{n+1}=Y^n+\Delta t f(t+\Delta t,Y^{n+1})$

or Crank-Nicholson (implicit, second-order)

$Y^{n+1}=Y^n+\frac{\Delta t}{2}(f(t,Y^{n})+f(t+\Delta t,Y^{n+1})$

I am trying to use a higher order scheme so that larger time steps can be used at the same amount of error. I have recently been looking at the Gauss-Legendre Method that has a butcher tableau of this from the article here.

- Given the Butcher tableau for Gauss-Legendre, is it possible to formulate an expression similar to that above for Euler, Backward Euler, and CN? Right now I have

$Y^{n+1} = Y^n+\frac{1}{2}k_1+\frac{1}{2}k_2$

where

$k_1=\Delta tf(t+(\frac{1}{2}-\frac{1}{6}\sqrt(3)\Delta t,Y^n+\frac{1}{4}k_1+(\frac{1}{4}-\frac{1}{6}\sqrt{3})k_2)$ $k_2=\Delta tf(t+(\frac{1}{2}+\frac{1}{6}\sqrt(3)\Delta t,Y^n+(\frac{1}{4}+\frac{1}{6}\sqrt{3})k_1+\frac{1}{4}k_2)$

but I'm not sure where the $Y^{n+1}$ appears in this formulation on the RHS of $k_1$, $k_2$ or the first equation.

Are their any limitations to the transport equations? Ideally it is something like $f(t,C) = -\nabla\cdot(\vec{U}C)+D\nabla^2C$

Are there any other higher order implicit time integration schemes that are a) a lower number of stages and b) higher order of accuracy than 2?

Help is much appreciated and or advice on higher-order time integration schemes would be nice