I've been wrestling for a number of days with the following scheme of the one dimensional first-order hyperbolic linear convection equation,
$$ u_t+cu_x=0 $$
Introduce a set of points $x_j = jh$ for $j = 0,1,\dots, N$ with $h = 1/N$, and $t_n = nk$ for $n = 0,1,\dots, M$. We introduce also $x_{j+1 / 2}=x_{j}+\frac{h}{2}$, $t_{n+1 / 2}=t_{n}+\frac{k}{2}$, and denote $v= ck/h$. The numerical method has three steps:
Step 1: $u_{j+1 / 2}^{(1)}=\frac{1}{2}\left(u_{j+1}^{n}+u_{j}^{n}\right)-\frac{1}{3} v\left(u_{j+1}^{n}-u_{j}^{n}\right)$
Step 2: $u_{j}^{(2)}=u_{j}^{n}-\frac{2}{3} v\left(u_{j+1 / 2}^{(1)}-u_{j-1 / 2}^{(1)}\right)$
Step 3: $u_{j}^{n+1}=u_{j}^{n}- \frac{v}{24}\left(-2 u_{j+2}^{n}+7 u_{j+1}^{n}-7 u_{j-1}^{n}+2 u_{j-2}^{n}\right) -\frac{3}{8} v\left(u_{j+1}^{(2)}-u_{j-1}^{(2)}\right) -\frac{\omega}{24}\left(u_{j+2}^{n}-4 u_{j+1}^{n}+6 u_{j}^{n}-4 u_{j-1}^{n}+u_{j-2}^{n}\right) $
According to these notes and this paper, it's called the third-order Rusanov method (Burstein-Mirin). The parameter $\omega$ is added for stability purposes, which is ensured if $|v|\leq 1$ and $4v^2-v^4\leq\omega<3$.
Please, how I can derive this scheme using a finite difference method? I'm in need of a nudge in the right direction
The scheme is derived in (1). For the evolution operator $u_t = F(u,x,t,u_x)$ with $F(u,x,t,u_x) = -cu_x$, the three steps correspond to the following Runge-Kutta scheme, represented by the Butcher tableau $$ \begin{array}{c|cc} 0 & & \\ \tfrac13 & \tfrac13 & \\ \tfrac23 & 0 & \tfrac23 \\ \hline & \tfrac14 & 0 & \tfrac34 \end{array} $$ (third-order Heun's method). The operator $F$ is evaluated by using appropriate centered finite differences, and third-order accuracy is satisfied. The term with coefficient $\omega$ -- added for stability reasons -- is a fourth-order-accurate finite difference (see articles for details).
(1) Burstein, S. Z., & Mirin, A. A. (1970). Third order difference methods for hyperbolic equations. J. Comput. Phys. 5(3), 547–571. doi:10.1016/0021-9991(70)90080-x
(2) Rusanov, V. (1970). On difference schemes of third order accuracy for nonlinear hyperbolic systems. J. Comput. Phys. 5(3), 507–516. doi:10.1016/0021-9991(70)90077-x