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

A Parametric Interpolation Framework for 1D Scalar Conservation Laws using the Equal Area Principle