1. High Accuracy Schemes for Inviscid Traffic Models
Eric Liu
Final Project

2. The Payne-Whitham Model
Models the density and velocity of cars
Conservation of cars
Drivers change their speed with finite acceleration: 1) according to traffic conditions ahead, 2) to maintain a steady velocity (e.g. speed limit)
No movement at road’s max density
Linear velocity source term; logarithmic pressure with a singularity at rho = rho_max

3. Numerical Approach: Overview
Roe’s Approximate Flux Solver
Monotone Upstream-centered Schemes for Conservation Laws (MUSCL)
Linear (2nd order) spatial discretization
Quadratic (3rd order)
Flux-Limiters
van Leer, superbee, minmod, van Albada
Time-Stepping
RK4 (standard method)
SSP-RK3 (Strong Stability Preserving)
Note:
1)mention that only van Leer (for linear MUSCL) and modified van Albada (qudratic MUSCL) is shown in the results because performance did not change significantly with different limiters.
2) RK4 time stepping not shown because performance is no better than with the cheaper SSP-RK3 method. The latter also provides stronger guarantees on solution behavior around shocks.
3) Quadratic results not shown b/c of time constraints

4. Linear MUSCL Uses linear reconstruction inside each cell
Result uses the difference between successive cell averages and the ratio of successive slopes:

5. Linear MUSCL Resulting semi-discretization:
Better resolution of sharp (shock) features than diffusive first order schemes
2nd order accurate in space (smooth solution)
Not TVD: requires flux-limiters to avoid spurious oscillations around shocks

6. SSP-RK3 Advantage of SSP methods: Downsides:
Guarantees that time-step restrictions will be no worse than with Forward Euler
Holds all oscillation diminishing properties of Forward Euler + TVD schemes
Downsides:
Formulations do not exist for arbitrarily high orders of convergence
Methods beyond 3rd order are expensive

7. Convergence Rates No source term Source term
Similar to the Shallow Water Equations, except with a logarithmic pressure (instead of linear)
Source term

8. Ripples?
Despite the use of TVD spatial discretizations and SSP time stepping, ripples (high frequency oscillations) were sometimes observed.
Only occurs under 1) backward traveling shocks
Observed for higher order AND first order schemes!
Ripple train width decreases with:
increasing grid resolution
increasing solution order
Ripple train maximum amplitude does not decrease significantly with more resolution

9. Ripples

10. Conclusion: Methods
Higher order schemes work well on the Payne-Whitham model
TVD algorithms in space
SSP time stepping
Linear reconstruction leads to sharper shock resolution than quadratic methods (tend to be more dispersive)
Tend to overshoot more than quadratic methods at lower grid resolution
Quadratic performance likely limited by inadequate limiters
Recent limiters by Colella have improved performance; these were not implemented.

11. Conclusion: Ripples Very surprising observation
Analytic results by Flynn et al. imply that the ripples are numerical artifacts
Exact solution may be unstable, so disturbances associated with floating point arithmetic induce the ripples
Ripple train width may decrease with increasing resolution due to better resolution of the ripples (i.e. less downstream pollution)
Is consistent with the ripples’ non-diminishing amplitude

12. Questions?

13. Roe’s Approximate Flux
Solving a nonlinear Riemann problem is generally hard
Interpolates the flux Jacobian linearly between the left/right interface states
Selects average flux Jacobian by “Roe-Averaging” the left/right states.
Exact solution if the interfacial jump is a single discontinuous wave

14. Roe’s Approximate Flux
Roe (rho?)-Average:
Computing the Roe Flux:
Uses F (physical flux) and , the flux Jacobian