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A Semidiscrete Lagrangian-Eulerian scheme for the LWR traffic model with discontinuous flux

Eduardo Abreu, Maria Teresa Chiri, Richard De la cruz, Juan Juajibioy, Wanderson Lambert

Abstract

In this work, we present a semi-discrete scheme to approximate solutions to the scalar LWR traffic model with spatially discontinuous flux, described by the equation $u_t + (k(x)u(1-u))_x = 0$. This approach is based on the Lagrangian-Eulerian method proposed by E. Abreu, J. Francois, W. Lambert, and J. Perez [J. Comp. Appl. Math. 406 (2022) 114011] for scalar conservation laws. We derive a non-uniform bound on the growth rate of the total variation for approximate solutions. Since the total variation can explode only at $x=0$, we can provide a convergence proof for our scheme in $BV_{loc}(\mathbb{R}\setminus \lbrace 0 \rbrace)$ by using Helly's compactness theorem.

A Semidiscrete Lagrangian-Eulerian scheme for the LWR traffic model with discontinuous flux

Abstract

In this work, we present a semi-discrete scheme to approximate solutions to the scalar LWR traffic model with spatially discontinuous flux, described by the equation . This approach is based on the Lagrangian-Eulerian method proposed by E. Abreu, J. Francois, W. Lambert, and J. Perez [J. Comp. Appl. Math. 406 (2022) 114011] for scalar conservation laws. We derive a non-uniform bound on the growth rate of the total variation for approximate solutions. Since the total variation can explode only at , we can provide a convergence proof for our scheme in by using Helly's compactness theorem.

Paper Structure

This paper contains 12 sections, 6 theorems, 127 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. A semidiscrete Lagrangian-Eulerian approach
  3. A semi-discrete formulation
  4. Convergence analysis of the semidiscrete Lagrangian-Eulerian
  5. $L^{\infty}$-estimates
  6. $L^1$-estimates
  7. Non-uniform TV-estimates
  8. Temporal continuity
  9. Convergence to entropic solutions
  10. Numerical examples
  11. Examples of local flux with a discontinuous flux
  12. Concluding Remarks and Future Directions

Key Result

Theorem 3.1

Let $\Delta x > 0$ be a spatial discretization parameter. Suppose that $u^{\Delta x}_0 \in L^1(\mathbb{R}) \cap BV(\mathbb{R})$. Then, there exists a time horizon $T > 0$ such that the initial value problem Cauchy-1-Cauchy-Data has a unique solution in $C^{1}([0,T]; L^{1}(\mathbb{R}))$.

Figures (6)

  • Figure 1: Behavior of no-flow curves satifaying $|\sigma_{j}(t)|<\frac{1}{2}\frac{\Delta x}{\Delta t}$.
  • Figure 2: Geometric description of calculus of approximate $u^{n+1}_{j+\frac{1}{2}}$.
  • Figure 3: Graphical description of the jumps of the function $L(x)$ at $x=x_j$.
  • Figure 4: Illustration comparing the exact solution of the Cauchy problem \ref{['1-1']} with datum \ref{['Dat-ex1']}. The exact solution is given by \ref{['Ex1-sol']} with the approximate solutions. a) and b) at time $t=0.5$ considering 1025 and 2049 cells. c) and d) results al time $t=1$ with 1025 and 2049 cells.
  • Figure 5: Illustration comparing the exact solution of the Cauchy problem \ref{['1-1']} with datum \ref{['Dataex-2']}. The exact solution is given by \ref{['Ex2-sol']}
  • ...and 1 more figures

Theorems & Definitions (21)

  • Definition 1.1: Weak solutions
  • Definition 1.2: Entropy solution
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • ...and 11 more