  In this paper we present a novel framework for obtaining high-order numerical
methods for scalar conservation laws in one-space dimension for both the
homogeneous and non-homogeneous case. The numerical schemes for these two
settings are somewhat different in the presence of shocks, however at their
core they both rely heavily on the solution curve being represented
parametrically. By utilizing high-order parametric interpolation techniques we
succeed to obtain fifth order accuracy ( in space ) everywhere in the
computation domain, including the shock location itself. In the presence of
source terms a slight modification is required, yet the spatial order is
maintained but with an additional temporal error appearing. We provide a
detailed discussion of a sample scheme for non-homogeneous problems which
obtains fifth order in space and fourth order in time even in the presence of
shocks.
