I am reading up some models and comparing/contrasting/modifying with my own method for linear advection and Burgers test suites (will summarise it soon).Take any (intro) book on numerical schemes for conservative laws. You can go with Roe, HLLC or FCT and you are done.

Essentially,

Somewhere you (Monsieur JeanLeM mentioned Godounov scheme but couldn't find it. A related scheme (?) for the pde

[$]\frac{\partial u}{\partial t} +a\frac{\partial u}{\partial x} = 0 [$] on the interval [$](0,1)[$] (1)

is

[$](u_i^{n+1} - u_i^{n})/k + a(u_i^{n+1} - u_{i-1}^{n})/h = 0[$]

here

[$] a > 0 [$] constant

[$]h,k[$] step sizes in [$]x[$] and [$]t[$] directions, respectively.