  In this paper we develop a novel framework for numerically solving scalar
conservation laws in one space dimension. Utilizing the method of
characteristics in conjunction with the equal area principle we develop an
approach where the weak solution is obtained purely as the solution of a
parametric interpolation problem. As this framework hinges on the validity of
the equal area principle, we provide a rigorous discussion of the equal area
principle and show that, indeed, the equal area principle is equivalent to the
Rankine-Hugoniot condition, within the specific context studied in this paper.
Combining these results with properties of the characteristic equations yields
the desired setting to define the equivalent parametric interpolation problem.
We conclude by applying this framework to Burgers' equation and show how one
obtains machine precision in the shock position when the initial condition can
be represented exactly in the chosen space of parametric polynomials.

Numerical Analysis