We note that along any characteristic line $$\frac{dx}{dt}=c$$ and $$\frac{D\phi}{Dt}=0$$ i.e. $$\phi$$ is conserved along characteristic lines. Thus we can draw the characteristic line from grid point $$(x_i,t_k)$$ to $$(x_i-c\Delta x,t_{k-1})$$. First $$x_{i-1}$$ is determined as shown in the figure to the right.
Since $$\phi(x_i,t_{k-1}), i=0,...,N_x$$ is known, we use interpolation to compute $$\phi(x_i-c\Delta x,t_{k-1})$$.
In simple cases linear interpolation can be used. In more complicated cases, a three-step procedure is used to calculate $$x_i-c\Delta t$$ and cubic interpolation ensures accurate solutions. Interpolation ensures that the CFL condition is satisfied.