Hybrid Godunov-Glimm Scheme

• Hybrid Godunov-Glimm Scheme is a numerical algorithm that interleaves Glimm's random-choice transport for exact contact preservation with Godunov's finite-volume correction for shock and rarefaction management.
• It leverages two distinct update mechanisms to resolve linearly degenerate contacts sharply while maintaining stability and TVD properties for genuinely nonlinear waves.
• The scheme is applied to traffic flow models based on Aw-Rascle dynamics, showing improvements in contact resolution with a reduction in numerical diffusion by up to 5–10 cell widths compared to pure methods.

The hybrid Godunov-Glimm scheme is a numerical algorithm constructed to approximate solutions of strictly hyperbolic systems of conservation laws, focusing on models that possess both linearly degenerate (contact discontinuity) and genuinely nonlinear wave families. In “A hybrid numerical method for a microscopic and macroscopic traffic flow model” (Wu et al., 20 Jan 2026), the scheme is formulated and benchmarked in the context of traffic models that reformulate the classical Aw-Rascle-Zhang (ARZ) and modified Aw-Rascle systems. The hybrid Godunov-Glimm approach interleaves two well-established update mechanisms: (i) random-choice (Glimm) transport for exact resolution of contact discontinuities, and (ii) Godunov finite-volume correction for robust integration of shocks and rarefactions. This division achieves superior sharpness for contact jumps while retaining the stability and TVD (total variation diminishing) performance of Godunov-type methods for nonlinear phenomena.

1. Macroscopic Model Structure and Motivating Principles

The representative macroscopic system targeted by the hybrid scheme is the RARZ model, given (in 1D) by: $$\rho_t + (\rho u)_x = 0, \qquad (\rho\,\tilde{v}p)_t + (\rho u\,\tilde{v}p)_x = 0$$ where $\rho$ is density, $u$ is velocity, $\tilde{v}=(1/u - 1/u^*)^{-1}$ (with $u^*$ the maximal velocity), and $p=(1/\rho - 1/\rho^*)^{-\gamma}$ (with $\rho^*$ the maximal density constraint).

The flux Jacobian possesses two eigenvalues:

• $\lambda_1 = u$ (linearly degenerate/contact discontinuity)
• $$\lambda_2 = u - \gamma u \rho^* (u^*-u)/[u^* (\rho^*-\rho)]$$ (genuinely nonlinear)

Correspondingly, shock and rarefaction curves are parameterized by invariance of $w = \tilde{v}p$, which is strictly advected.

Pure Godunov methods handle nonlinear waves efficiently but diffuse contacts. Pure Glimm exactly preserves contacts, but in the presence of shocks or rarefactions introduces noise and computational expense. The hybrid method exploits each scheme’s strengths in their respective regimes.

2. Hybrid Time-Stepping Algorithm in One Dimension

Discretizing $x \in \mathbb{R}$ into cells $C_j=[x_{j-\frac{1}{2}}, x_{j+\frac{1}{2}}]$ with step sizes $\Delta x$, the update cycle at time $t^n$ comprises:

Step 1: Glimm Contact Transport ($t^n \rightarrow t^{n+\frac{1}{2}}$):

• For each interface, solve the local Riemann problem to obtain the intermediate state $U^*_{j-\frac{1}{2}}$.
• Contacts propagate only from the left due to the sign of $\lambda_1=u$.
• A random sample $a_{n+1}$ from the van der Corput sequence determines if the cell is updated:

$$U_j^{n+\frac{1}{2}} = \begin{cases} U^*_{j-\frac{1}{2}}, & a_{n+1} \leq \frac{\Delta t}{\Delta x} u_j^n \ U_j^n, & \text{otherwise} \end{cases}$$

• This transport mechanism maintains zero numerical diffusion for contacts.

Step 2: Godunov Nonlinear Correction ($t^{n+\frac{1}{2}} \rightarrow t^{n+1}$):

• At each interface $(j+\frac{1}{2})$, use the Godunov flux with the half-step left state $U_j^{n+\frac{1}{2}}$ and right state $U_{j+1}^n$ to compute:

$$U_{j+\frac{1}{2},L}^{n+1} = U_j^{n+\frac{1}{2}} - \frac{2\Delta t}{\Delta x}\left[ F(U_r(0^-; U_j^{n+\frac{1}{2}}, U_{j+1}^n)) - F(U_j^{n+\frac{1}{2}}) \right]$$

• A symmetric formula applies for the left interface update $U_{j-\frac{1}{2},R}^{n+1}$.
• The final cell-average is taken as:

$$U_j^{n+1} = \frac{1}{2}\left( U_{j-\frac{1}{2},R}^{n+1} + U_{j+\frac{1}{2},L}^{n+1} \right)$$

This two-step update ensures robust integration of shocks and rarefactions while preserving sharpness at contacts.

3. Riemann Solver and Characteristic Analysis

The hybrid scheme requires an exact Riemann solver at each interface, both for contact transport (Glimm) and flux calculation (Godunov). Key aspects include:

• The characteristic decomposition identifies which waves are linearly degenerate ($w_1=u$) and which are genuinely nonlinear ($w_2=\ln(u/(u^*-u))+\gamma\rho/(\rho^*-\rho)$).
• The value $w=\tilde{v}p$ acts as a conserved invariant across material characteristics and is the variable tracked during Glimm updates.
• The solver generates intermediate states for both contact jumps and nonlinear wave propagation.

4. Coupling with Microscopic Dynamics

The RARZ macroscopic model is systematically derived from a microscopic follow-the-leader rule with quadratic acceleration, scaled up to a continuum:

• Individual vehicles are indexed with density $\rho_i$ and spacing $\tau_i=1/\rho_i-1/\rho^*$.
• The microscopic advected variable $w_i = \tilde{v}_i p_i$ is conserved along each trajectory ($\dot{w}_i=0$).
• In the macroscopic limit, this property persists, yielding $(\rho\,\tilde{v}p)_t + (\rho u\,\tilde{v}p)_x = 0$.
• This tight coupling drives the hybrid numerical scheme’s choice of variables and invariant advection.

5. Stability, CFL Condition, and Convergence Properties

The hybrid update sequence imposes a classical CFL-type stability restriction: $$\Delta t \leq \text{CFL} \cdot \frac{\Delta x}{\max_j \max(|\lambda_1(U_j)|, |\lambda_2(U_j)|)}, \quad \text{typically}\ \text{CFL}\approx 0.5$$ Under this condition, the Glimm step remains stable in the random-choice sense, while the Godunov step is TVD and stable for shocks and rarefactions. Prior analyses (e.g., Betancourt et al., 2018; Chalons & Goatin, 2007) indicate first-order $L^1$ convergence as $\Delta x,\Delta t \to 0$. The algorithm requires no source-term splitting in $1$D, as the conservative structure is maintained.

6. Two-Dimensional Extension via Operator Splitting

For the fully two-dimensional case, the system adopts three conserved quantities $(\rho,\rho\,\tilde{v}p,\rho\,\tilde{v}_v p)$. The update proceeds via Strang splitting for alternating direction steps, with the hybrid scheme or HLL applied in each:

• Each 1D substep utilizes the hybrid update described above.
• Example tests (quadrant-wise Riemann problems) demonstrate that the hybrid approach produces sharper contact curves and clear interaction regions compared to HLL, which tends to smear contacts and introduce intermediate diffusion.

7. Computational Complexity and Comparative Performance

Each time step in the hybrid scheme involves, per cell interface:

• Two local Riemann solves (one in Glimm, one in Godunov)
• A single random sample (van der Corput sequence)

This results in $O(N)$ work per step, identical to pure Godunov in scaling, with minor overhead (one additional Riemann solve and randomization) and significant improvement in contact sharpness. “Pure” Glimm is less robust for nonlinear waves and incurs higher noise for shocks; pure Godunov diffuses contact jumps. Hybrid Godunov-Glimm achieves zero-diffusion at contact discontinuities—reducing contact smearing by approximately a factor of 5–10 in cell-widths compared to pure Godunov or HLL—without sacrificing shock/rarefaction accuracy.

8. Numerical Experimentation and Results

Benchmark Riemann test problems verify the scheme’s performance:

• Shock + contact: The hybrid method tracks the exact shock and contact position; Godunov distributes the contact across several cells.
• Rarefaction + contact: Contacts remain localized to a single cell, rarefaction fans are accurately resolved.
• Pure contact jumps: Resolved in one cell by hybrid, smeared across many cells by Godunov.

In two-dimensional settings, quadrant-based Riemann problems confirm that hybrid Godunov-Glimm delineates contact regions sharply and eliminates spurious intermediate states observed in pure HLL solutions.

In summary, the hybrid Godunov-Glimm scheme merges random-choice transport for the advected invariant $\tilde{v}p$ (capturing contacts with zero diffusion) and classical finite-volume Godunov corrections (ensuring stability for shocks and rarefactions). The algorithm is straightforward to implement, robust, and offers demonstrably superior contact resolution within the prescribed CFL conditions (Wu et al., 20 Jan 2026).