My question is not related to any particular problem, rather, I am looking at the equations of the form $$u_t+c(t,x)u_x=0$$ and attempt to solve it numerically. According to http://en.wikipedia.org/wiki/Flux_limiter I can apply Flux limiter method, say with Van Leer limiter function, and obtain a numerical method for the equation.
Flux limiter technique is for the discretization in space, I have to make a choice for the time. I compare explicit, implicit, and central difference. I would think explicit would have have some sort of stability/CFL condition which I don't like. Thus, I can use the other two. The latter is preferred due to a bettor error of second order. Thus, my first question is what type of time discretization is used in practice and why? Some basic intuition is sufficient to me.
Second, in either one of the implicit/central case, if I write the iteration explicitly for each time step, it requires the performance of Newton's iterations, simply because the equation is not linear anymore. However, in some cases there are only a few iterations are sufficient to obtain a solution provided a good initial guess. Since this is a nonlinear method the complexity is larger relative to a linear method, say implicit Euler in time and one sided difference in space. But, if there are only a few iterations to solve the nonlinear system to be performed, isn't comparable to the linear numerical methods in complexity? Thus, I want to know how complex those limiter methods and can they compete with linear methods in complexity? Or there is something else apart from Newton's iterations that make flux limiter methods more complex? Why one would like to have a linear method to solve an equation compare to non-linear?