Shadow wave tracking procedure and initial data problem for pressureless gas model

TL;DR

This paper introduces a novel wave tracking procedure for pressureless gas dynamics that effectively handles initial data with Dirac delta functions, extending the wave front tracking method to include singular solutions.

Contribution

The paper develops a new algorithm incorporating shadow waves to manage delta functions in pressureless gas models, enabling better handling of singular initial data.

Findings

The approximated solutions converge to a signed Radon measure.

The new method simplifies the interaction problem involving delta functions.

Extension of wave front tracking algorithm to include singular solutions.

Abstract

In this paper the new procedure for a construction of an approximated solution to initial data problem for one-dimensional pressureless gas dynamics system is introduced. The procedure is based on solving the Riemann problems and tracking singular wave interactions. For that system the new problem with initial data containing Dirac delta function is solved whenever two waves interact. Use of the shadow waves as singular solutions to such problems enables us to easily solve the interaction problems. That permits us to make a simple extension of the well known Wave Front Tracking algorithm. A non-standard part of the new algorithm is dealing with delta functions as a part of a solution. In the final part of the paper we show that the approximated solution has a subsequence converging to a signed Radon measure.