Weak* Solutions III: A Convergent Front Tracking Scheme

TL;DR

This paper introduces a modified Front Tracking scheme for hyperbolic conservation laws that handles large nonlinear waves exactly, proves convergence to weak* solutions, and applies it to gas dynamics equations.

Contribution

It develops a convergent Front Tracking scheme using generalized Riemann problems that accurately captures large waves and proves convergence to weak* solutions.

Findings

The scheme converges to weak* solutions under reasonable assumptions.

Exact solutions of generalized Riemann problems are used for large waves.

Application to Euler equations demonstrates the scheme's effectiveness.

Abstract

We present a modified Front Tracking (mFT) scheme for hyperbolic systems of conservation laws in one space dimension, in which we allow arbitrarily large nonlinear waves. We build the scheme by introducing and solving a ``generalized Riemann Problem'', which yields exact solutions for finite times. This allows us to treat the states adjacent to all waves exactly, and approximate compressive simple waves in addition to rarefactions, contacts and shocks. In particular, we require exact expression of the various wave curves and avoid the use of Taylor expansions. After construction of the scheme, under reasonable assumptions, we show that the mFT approximations converge to a weak* solution of the system. This essentially reduces existence of solutions with large amplitude data to obtaining uniform bounds on the total variation of the approximations. We then apply the scheme to the Euler equations of gas dynamics, for which we exactly solve the generalized Riemann Problem and define the scheme for both 3x3 and 2x2 systems, and prove the equivalence of Eulerian and Lagrangian frames. For the $p$-system, modeling isentropic gas dynamics in a Lagrangian frame, we show that there is no finite accumulation of interaction times. This means that the last remaining obstacle to global existence of large data, large amplitude solutions is the construction of a decreasing Glimm potential.