High Accuracy Schemes for Inviscid Traffic Models Eric Liu 18.336 Final Project

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

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

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

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

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

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

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

Ripples

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.

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

Questions?

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

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