Construction of approximate entropy measure valued solutions for hyperbolic systems of conservation laws
This work challenges the foundational solution concept for conservation laws and offers a new paradigm, but its practical impact is limited to the specific domain of hyperbolic PDEs.
The paper argues that entropy solutions may not be unique or suitable for hyperbolic conservation laws in multiple dimensions, and proposes entropy measure valued solutions as a more appropriate framework. It provides a numerical procedure that constructs stable approximations guaranteed to converge to such solutions, supported by experiments.
Entropy solutions have been widely accepted as the suitable solution framework for systems of conservation laws in several space dimensions. However, recent results in \cite{CDL1,CDL2} have demonstrated that entropy solutions may not be unique. In this paper, we present numerical evidence that demonstrates that state of the art numerical schemes \emph{may not} necessarily converge to an entropy solution of systems of conservation laws as the mesh is refined. Combining these two facts, we argue that entropy solutions may not be suitable as a solution framework for systems of conservation laws, particularly in several space dimensions. Furthermore, we propose a more general notion, that of \emph{entropy measure valued solutions}, as an appropriate solution paradigm for systems of conservation laws. To this end, we present a detailed numerical procedure, which constructs stable approximations to entropy measure valued solutions and provide sufficient conditions that guarantee that these approximations converge to an entropy measure valued solution as the mesh is refined, thus providing a viable numerical framework for systems of conservation laws in several space dimensions. A large number of numerical experiments that illustrate the proposed schemes are presented and are utilized to examine several interesting properties of the computed entropy measure valued solutions.