A data-driven ODE–PDE model for ramp impact on traffic flow on King Fahd Road in Riyadh

Yasser Almoteri; Abdulaziz Alsamil; Yasser Almoteri; Abdulaziz Alsamil

doi:10.3934/math.20261072

AIMS Mathematics

2026, Volume 11, Issue 8: 26720-26738. doi: 10.3934/math.20261072

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Research article

A data-driven ODE–PDE model for ramp impact on traffic flow on King Fahd Road in Riyadh

Yasser Almoteri ,
Abdulaziz Alsamil ^,

Department of Mathematics and Statistics, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, Saudi Arabia

Received: 29 April 2026 Revised: 03 August 2026 Accepted: 19 August 2026 Published: 25 August 2026
MSC : 35K57, 65M60, 65Z05, 92D25

On-ramp inflows cause congestion on urban arterials. We present a coupled ODE–PDE model for traffic density on King Fahd Road, a 13 km corridor in Riyadh. The ODE represents diurnal ramp demand, calibrated from an independent feeder-road speed signal, and couples to the LWR conservation law via a localized source with a merge-admittance closure that limits injection by available supply. We introduce the ramp pressure number $ R_p = L\alpha_0/(\rho_{\max}v_{\max}^2) $, a dimensionless control parameter, and non-dimensionalize the system. A conservative finite-volume method with characteristic-respecting boundary conditions solves the PDE; grid convergence and mass conservation confirm the numerics. Counterfactual simulations assess ramp metering, an extended acceleration lane, and merge capacity drop. Comparisons with independent mainline observations show the model captures the diurnal timing ($ r = 0.88 $) but under-predicts amplitude ($ R^2 = 0.34 $), which we attribute to the absence of a downstream bottleneck. The non-dimensional density stays within $ [0, 1] $ and profiles show wave relaxation.
- traffic flow,
- LWR model,
- ramp demand,
- ODE-PDE coupling,
- finite volume,
- King Fahd Road,
- Riyadh
Citation: Yasser Almoteri, Abdulaziz Alsamil. A data-driven ODE–PDE model for ramp impact on traffic flow on King Fahd Road in Riyadh[J]. AIMS Mathematics, 2026, 11(8): 26720-26738. doi: 10.3934/math.20261072

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Abstract

On-ramp inflows cause congestion on urban arterials. We present a coupled ODE–PDE model for traffic density on King Fahd Road, a 13 km corridor in Riyadh. The ODE represents diurnal ramp demand, calibrated from an independent feeder-road speed signal, and couples to the LWR conservation law via a localized source with a merge-admittance closure that limits injection by available supply. We introduce the ramp pressure number $ R_p = L\alpha_0/(\rho_{\max}v_{\max}^2) $, a dimensionless control parameter, and non-dimensionalize the system. A conservative finite-volume method with characteristic-respecting boundary conditions solves the PDE; grid convergence and mass conservation confirm the numerics. Counterfactual simulations assess ramp metering, an extended acceleration lane, and merge capacity drop. Comparisons with independent mainline observations show the model captures the diurnal timing ($ r = 0.88 $) but under-predicts amplitude ($ R^2 = 0.34 $), which we attribute to the absence of a downstream bottleneck. The non-dimensional density stays within $ [0, 1] $ and profiles show wave relaxation.

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