Parametric Interpolation Framework for Scalar Conservation Laws

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.