In this paper, the discrete semigroup approximations to initial and initial-boundary value problems for first order PDEs in unbounded domains are presented. For the initial value problem, we obtain a modified problem posed on a bounded domain whose solution exactly coincides with the solution of the original problem on a smaller bounded domain. The numerical solution to the modified problem converges to the solution of the original problem on the smaller bounded domain. For the initial-boundary problem, we present the discrete semigroup approximations after decomposing it into two problems each of which gives rise to a semigroup.

semigroup, approximation, initial value problem, initial-boundary value problem, unbounded domain.