Two families of two-time level difference schemes are developed for the numerical solution of first order hyperbolic partial differential equations with one space variable. The space derivative is replaced by (i) a first order, (ii) a second order backward difference approximant and the resulting system of first order ordinary differential equations is solved using A0-stable and L0-stable methods.
The methods are tested on a number of problems from the literature involving wave-form solutions, increasing solutions with discontinuities in function values or first derivatives across a characteristic, and exponentially decaying solutions
