  We introduce a new class of finite differences schemes to approximate one
dimensional dissipative semilinear hyperbolic systems with a BGK structure.
Using precise analytical time-decay estimates of the local truncation error, it
is possible to design schemes, based on the standard upwind approximation,
which are increasingly accurate for large times when approximating small
perturbations of constant asymptotic states. Numerical tests show their better
performances with respect to those of other schemes.
