I want to discretize the LWR(Lighthill-Whitham-Richards)-v partial differential equation (as shown in 1) $$ \frac{\partial u}{\partial t} + \frac{\partial R(u)}{\partial x} = 0 \quad \text{i.e.,} \quad \frac{\partial u}{\partial t} + \frac{dR}{du}\,\frac{\partial u}{\partial x} = 0 \quad \dots\dots \quad \text{(1)} $$ In the equation $u(x,t) \equiv u_x^t$ is the speed at time $t$ at section $x$ and $R(u)$ is given by Greenshield’s speed-density relationship $$ R(u) = u^2 - u\,u_f \quad \implies \quad \frac{dR}{du} = 2u - u_f =: a(u) $$ where $u$ is the stream speed and $u_f$ is the free flow speed or the maximum speed in the section.
I want to use an implicit finite difference scheme (backward time, back ward space as shown in (2) to discretize equation (1): $$ \frac{u_x^{t+1}-u_x^t}{\Delta t} + a(u_x^{t+1})\,\frac{u_x^{t+1}-u_{x-1}^{t+1}}{\Delta x} = 0 \quad \dots\dots \quad \text{(2)} $$
In literature I could find solutions for explicit schemes alone like upwind scheme, downwind scheme, Godunov scheme etc. Can anyone suggest a good discretization scheme or some references in literature for the implicit case shown above in equation (2)?
Thanks for the help!
