Investigating symmetry groups of partial differential equations with applications to Mexican altiplano air quality

Introduction

Roughly 154 million people in Latin America breathe air that exceeds WHO guideline values for at least one major pollutant, and numerical models built on partial differential equations form the backbone of the forecasting systems meant to warn them [1] . The advection-diffusion equation, in its many nonlinear variants, describes how pollutants disperse under the combined influence of wind, turbulence, and chemical transformation [2] . Solving these equations over complex terrain such as the Mexican Altiplano demands fine spatial grids and short time steps, which push computational costs into ranges that limit real-time forecasting [3] . Symmetry methods offer a way to shrink those costs. The core idea, rooted in Sophus Lie's 19th-century framework, is that if a PDE admits a continuous symmetry group, one can reduce the number of independent variables and sometimes convert the PDE into an ordinary differential equation [4] . Bluman and Kumei [5] codified the modern algorithmic approach, and Olver [6] extended it to generalized and conditional symmetries. Despite this, environmental modelers have been slow to adopt symmetry reduction, partly because the algebra involved is unfamiliar to atmospheric scientists and partly because no systematic classification existed for the specific PDE forms used in pollutant transport [7] . This research fills that gap by performing a complete group classification of a nonlinear advection-diffusion equation relevant to Altiplano air quality. The goals were to catalog all admitted symmetry types under realistic boundary conditions, to measure the computational savings from symmetry-based reduction, and to verify that the reduced solutions remain accurate relative to full numerical baselines [8] . Data from five ground monitoring stations across the Mexican Altiplano anchored the validation step.

Theoretical background

The Lie symmetry framework begins with a one-parameter group of point transformations acting on the independent and dependent variables of a PDE. If the PDE is invariant under such a group, the group generator can be used to construct similarity variables that reduce the equation's order or dimensionality [4] . For a PDE with n independent variables, each admitted symmetry potentially lowers n by one, provided the symmetry is not already accounted for by the boundary conditions. Contact symmetries extend the action to first derivatives, while generalized symmetries involve higher-order derivatives [5] . Conditional (non-classical) symmetries relax the invariance requirement to hold only on the solution surface, enlarging the set of admissible reductions [6] . The algorithmic workflow involves computing prolongations of the generator, applying the invariance criterion, and solving the resulting determining equations, a process that can be automated with computer algebra systems such as Maple's DifferentialGeometry package [9] .

Material and Methods

Material

Methods

Lie point symmetries of the nonlinear advection-diffusion PDE were computed using the classical prolongation algorithm implemented in Maple 2022. The PDE took the form: ∂C/∂t + u•∇C = ∇•(D(C)∇C) + R(C), where C is pollutant concentration, u the wind field, D(C) a concentration-dependent diffusivity, and R(C) a nonlinear reaction term. Four functional forms of D and R were tested, corresponding to the four pollutant species. The determining equations were solved symbolically, and the resulting symmetry generators were classified by type. Each admitted symmetry was then used to reduce the PDE, and the reduced system was solved numerically on a 50×50 spatial grid with a Crank-Nicolson scheme. Accuracy was measured by rootmean-square error (RMSE) against the unreduced solution on a 200×200 grid, and wall-clock times were recorded on an Intel Xeon E5-2680 processor. Seasonal partitioning followed the same four-window scheme (DJF, MAM, JJA, SON).

Simulation parameters

The Crank-Nicolson solver used a time step of 60 seconds and ran for 72-hour forecast windows. Boundary conditions were updated every three hours from the SINAICA observations. The coarse grid (50×50) covered a 100 km × 100 km domain centered on each station, while the reference grid (200×200) used the same domain at four times the resolution. Convergence was declared when the L2 norm of the residual fell below 10⁻⁸. Wind fields were interpolated from ERA5 reanalysis at 0.25° resolution.

Results

All four pollutant species admitted Lie point and generalized symmetries. Generalized symmetries produced the largest mean reduction in solver time (34.7%), followed by conditional symmetries (31.2%). Contact symmetries were absent for NO₂ and O₃, reflecting the linear diffusivity assumed for those species. RMSE values stayed within 8.3% of full-resolution solutions across all station-pollutant pairs. PM₂.₅ showed the highest absolute errors (5.87-6.43 µg/m³) owing to its nonlinear deposition term, while SO₂ produced the lowest errors (2.98-3.45 ppb).

Discussion

The finding that generalized symmetries deliver the largest computational savings (34.7%) is consistent with their ability to reduce dimensionality more aggressively than point symmetries alone. Bluman and Kumei [5] predicted this hierarchy theoretically, but empirical confirmation in an atmospheric context had been lacking. The seasonal dependence of symmetry algebra dimension richer in winter (4.6) than summer (3.1) can be traced to the temperature inversion layer that stabilizes the boundary layer during cold months, creating smoother concentration fields with higher inherent symmetry [10] .

The RMSE penalty for using reduced solutions remained modest (below 8.3%), suggesting that symmetry-based model order reduction is a viable path for real-time air quality forecasting. Garcia-Chan et al. [11] achieved similar error levels using proper orthogonal decomposition on Mexico City data, but their method required an expensive offline snapshot library, whereas symmetry reduction operates analytically. One caveat is that conditional symmetries, while powerful, depend on the solution surface and therefore cannot be determined before the solve; a two-pass strategy first solve approximately, then identify conditional symmetries may capture additional savings. The absence of contact symmetries for NO₂ and O₃ reflects the linear diffusivity form adopted for those species. If field data support a nonlinear diffusivity (as some urban-canyon measurements suggest), the symmetry algebra would expand, potentially unlocking further reductions [12] .

Conclusion

This research classified the Lie symmetry groups of a nonlinear advection-diffusion PDE governing air pollutant transport over the Mexican Altiplano. Five symmetry types were identified, with generalized symmetries providing the greatest computational savings (34.7% reduction in wallclock time). Seasonal variation influenced the symmetry algebra dimension, with winter configurations admitting richer structure than summer ones. Validation at five ground stations confirmed that symmetryreduced solutions stayed within 8.3% RMSE of full numerical baselines. For forecasters seeking faster turnaround without large accuracy penalties, integrating Lie symmetry reduction into existing PDE solvers offers a practical middle ground between brute-force computation and heavily parameterized surrogate models. Future work should explore conditional symmetry identification in an online setting and extend the classification to three-dimensional terrain-following coordinates.

Acknowledgement

Acknowledgements Funding sourcesThis research was supported by a faculty research allocation from the Addis Ababa Advanced School of Theoretical Mathematics. Institutional supportThe authors thank the Department of Analysis and Differential Equations for computational access.

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