Weak* Solutions III: A Convergent Front Tracking Scheme

TL;DR

This work introduces a modified Front Tracking (mFT) scheme for one-dimensional hyperbolic systems that accommodates arbitrarily large waves by solving a generalized Riemann problem (gRP) and its discretized version (dgRS). It enforces exact adjacency of states, uses virtual widths to distinguish shocks from simple waves, and employs mechanisms like multi-rarefactions and composite waves to bound the wave count and control the residual, enabling convergence to a weak* solution under BV bounds. The authors establish convergence results and rate under mild structural assumptions and a nonlocal Glimm potential in the p-system, and they apply the framework to gas dynamics, proving equivalence between Lagrangian and Eulerian frames and detailing initialization, reconstruction, and long-time continuation. The approach provides a pathway to global existence results for large amplitude, large BV data by focusing on BV bounds and residual control, with concrete schemes for isentropic and full gas dynamics. The work thus advances a rigorous, flexible, wave-based method for simulating highly nonlinear hyperbolic systems beyond the small BV regime, with potential implications for global BV existence proofs in gas dynamics.

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.

Figures (14)

• Figure 1: Generalized (left) and discretized (right) Riemann solutions
• Figure 2: Interactions of waves in the same family (left) and different families (right)
• Figure 3: Avoiding overlap: reduction of widths
• Figure 4: Multi-rarefaction
• Figure 5: Wave patterns with accumulation of interactions

Theorems & Definitions (61)

• Theorem 1.1: Theorem \ref{['thm:hatU']}
• Theorem 1.2: Theorem \ref{['thm:noaccum']}
• Theorem 1.3: Theorem \ref{['thm:conv']}
• Theorem 1.4: Theorem \ref{['thm:noacc']}
• Definition 2.1
• Definition 2.2
• Lemma 2.3
• proof
• Lemma 3.1
• proof