Advection–Diffusion Models with TVPs

• Advection–diffusion models with TVPs are advanced spatio-temporal frameworks that integrate transport, dispersion, and time-dependent coefficients to capture nonstationary phenomena.
• They employ stochastic PDE formulations and adaptive numerical methods—including spectral, finite element, and finite volume discretizations—to achieve efficient and robust inference.
• These models enable precise parameter estimation, uncertainty quantification, and reduced-order modeling, supporting real-time forecasting in diverse scientific applications.

Advection–diffusion models with time-varying parameters (TVPs) encompass a broad class of stochastic and deterministic partial differential equations (PDEs), statistical models, and numerical methods, all of which aim to describe the evolution of fields subject to simultaneous transport (advection), spreading (diffusion/dispersion), and temporally variable system coefficients. These models are foundational in the paper of environmental processes, geophysical dynamics, transport in porous media, climatology, ecology, and statistical postprocessing of high-frequency space–time data. The rigorous incorporation of TVPs allows such models to represent nonstationary, nonseparable, and often anisotropic or heterogeneous phenomena, while presenting significant theoretical and computational challenges. The following sections synthesize concepts, methodologies, and key results from the substantial literature on advection–diffusion models with TVPs, with an emphasis on recent developments in stochastic PDE formulations, adaptive discretizations, Bayesian inference, efficient simulation, and practical applications.

1. Stochastic Advection–Diffusion SPDEs with TVPs

The stochastic partial differential equation (SPDE) approach has become central to modeling spatio-temporal fields exhibiting transport, dispersion, damping, and stochastic excitation. In prototypical form, the latent process $u(s,t)$ is modeled via

$$\frac{\partial}{\partial t} u(s,t) + \mathcal{A}(s,t) u(s,t) = \tau(t)\,\frac{\partial}{\partial t} B(s,t),$$

where $\mathcal{A}(s,t)$ encodes dampening, anisotropic and possibly spatially and temporally varying diffusion, and advection fields (e.g., $$\mathcal{A}(s,t)u = \kappa(s,t) u - \nabla \cdot (\mathbf{H}(s,t) \nabla u) + \nabla \cdot (\omega(s,t) u)$$) (Berild et al., 5 Jun 2024, Sigrist et al., 2012, Clarotto et al., 2022). The innovation process $B(s,t)$ may be a Q-Wiener process determined by a spatial Whittle–Matérn covariance, and all coefficients ($\kappa, \mathbf{H}, \omega, \tau$) may vary in time, creating a fully nonstationary model.

The SPDE solution induces a Gaussian process with a nonseparable, often nonstationary space–time covariance structure, directly reflecting the physical mechanisms of transport and local dispersion. Model parameters (e.g., advection velocity $\omega(s,t)$, diffusion tensor $\mathbf{H}(s,t)$, and damping strength $\kappa(s,t)$) are physically interpretable and can be made time-dependent to represent transient, seasonal, or externally forced phenomena, as in environmental prediction or forecasting applications. Notably, these model structures support both spatial and temporal nonstationarity.

2. Numerical Discretization and Efficient Inference for Large-Scale TVP Models

With large datasets and nonseparable SPDEs, computational efficiency is a major concern. Modern approaches reduce high-dimensional inference and simulation to tractable computation via careful discretization, making use of both spectral (Fourier), finite element (FEM), and finite volume (FVM) methods. The choice of discretization is crucial:

• Spectral Methods: Truncated Fourier bases allow diagonalization of spatial operators, so that advection and diffusion become simple multiplications in the frequency domain. This yields vector autoregressive time evolution of the Fourier coefficients and, with FFT-based computation, scales efficiently as $O(N\log N)$ (Sigrist et al., 2012).
• Finite Element Methods: FEM (continuous Galerkin for smooth fields) is general but sensitive to advection dominance. Stabilized formulations such as Streamline Diffusion or SUPG (Streamline Upwind Petrov–Galerkin) are introduced to suppress spurious oscillations and maintain stability when advection dominates over diffusion (Clarotto et al., 2022, Veiga et al., 17 Oct 2024).
• Finite Volume Methods: FVM is favored for conservation law compliance and robustness in advection-dominated, nonstationary settings. Upwind schemes, as well as local grid adaptation, are often applied to accommodate spatial and temporal heterogeneity in the coefficients (Berild et al., 5 Jun 2024).

The discretized SPDE solution is represented as a high-dimensional Gaussian Markov random field (GMRF) whose precision matrix arises naturally from the discretized system. The temporal block-tridiagonal structure induced by Markovian dynamics and the localized spatial dependencies confer sparsity, supporting efficient maximum likelihood or Bayesian inference, conditional simulation, and prediction by kriging.

3. Adaptive and Robust Space–Time Solvers for Nonstationary Systems

Numerical solvers for advection–diffusion equations with TVPs demand discretization strategies that robustly resolve sharp propagating fronts, internal layers, and interfaces, particularly in heterogeneous and nonstationary settings. Recent developments include:

• Adaptive Discontinuous Galerkin (DG) Methods: These schemes employ time–space adaptive mesh refinement based on a posteriori error estimators derived from elliptic reconstruction techniques, such as residual-based indicators in robust ($L^\infty(L^2)+L^2(H^1)$)-type norms. This ensures tracking of moving fronts, adaptation to parameter changes, and rigorous control of errors even with strongly time-varying advection and diffusion coefficients (Karasözen et al., 2015).
• SUPG-Stabilized Virtual/FEM Elements with DG in Time: Such formulations provide inf-sup stability independent of the diffusion coefficient, ensuring optimal convergence in both diffusion- and advection-dominated regimes, with generality for complex domain geometry (Veiga et al., 17 Oct 2024, Diaz et al., 26 Jun 2025).
• Finite Volume/Markov Chain Approaches: Discrete-time, discrete-space random walk models derived from PDEs, with time-dependent transition probabilities corresponding to the current diffusion and advection fields, provide a flexible, particle-based framework for capturing randomness and parameter variability. Nonnegativity and stochastic consistency are ensured by constraining step sizes based on maximal parameter values (Carr, 13 Sep 2024).

4. Estimation and Uncertainty Quantification for Time-Varying Parameters

Estimation of TVPs within advection-diffusion models is a nontrivial inverse problem, especially when parameters affect the system nonlinearly (e.g., in source terms, boundary conditions, or through the coefficients of the PDEs). Recent advances include:

• Sequential Bayesian Filtering: Systematic particle filtering algorithms, in an offline–online decoupling strategy, enable online tracking of TVPs and their associated drift variance parameters, even under partial and noisy observation. The state-space representation incorporates both system evolution and parameter dynamics as latent stochastic processes, with resampling and adaptive variance shrinkage to maintain robustness and minimize degeneracy (Arnold, 16 Aug 2025).
• Hierarchical Bayesian and MCMC Strategies: In spatio-temporal models, Bayesian hierarchical frameworks encode priors on Gaussian process parameters, and MCMC–Metropolis–Hastings updates allow for simultaneous estimation of advection, diffusion, and damping parameters, as well as latent states, leveraging spectral or GMRF likelihood structures for scalability (Sigrist et al., 2012).

These estimation techniques provide joint uncertainty quantification for the entire set of unknowns (states and TVPs), supporting data-driven prediction and postprocessing in high-dimensional systems.

5. Model Order Reduction and Real-Time Control with TVPs

Construction of reduced-order models (ROMs) for advection–diffusion(-reaction) systems with TVPs is essential where real-time forecasting and operational control are required (e.g., water quality monitoring, process control in chemical reactors). A leading approach is:

• Projection-Based H₂-Optimal Reduction: The large-scale discretized system is projected onto a lower-dimensional subspace using double-sided Petrov–Galerkin projections with bases optimized to minimize H₂-norm error between the full and reduced models. This optimally preserves system structure, including the parametric dependence on time-varying coefficients, and supports orders-of-magnitude speedup with negligible loss in fidelity (e.g., NMSE < 2.3% in water quality forecasting) (Elkhashap et al., 2022).

Such ROMs are not only computationally efficient but also structurally robust, accommodating time-varying inputs and model parameters without recourse to extensive empirical data or retraining.

6. Applications and Impact: From Environmental Science to Biological Systems

Advection–diffusion models with TVPs have been successfully applied in diverse scientific and engineering domains:

• Environmental Prediction: Emulation of oceanographic models with nonstationary advection and diffusion, enabling real-time prediction and adaptive sampling from AUV (autonomous underwater vehicle) data streams, with performance benefits for nowcasting and forecasting unobserved locations (Berild et al., 5 Jun 2024).
• Atmospheric and Solar Irradiance Modeling: Statistical postprocessing of meteorological forecasts or radiation fields by modeling spatio-temporal error structures with advection–diffusion SPDEs yields calibrated, uncertainty-aware predictions (Sigrist et al., 2012, Clarotto et al., 2022).
• Ecological and Biological Dynamics: Models incorporating TVPs capture responses of animal populations to time-dependent stimuli or landscape features (e.g., movement in enclosures with changing incentive cues) (Tennenbaum et al., 2019). In tissues, variable- or fractional-order advection–diffusion models represent heterogeneous tumor growth or anomalous cell transport (Sadhukhan et al., 2019). Models with memory effects or state-dependent mobility (diffusion coefficients depending on past history or biochemical state) capture phenomena such as trapping, aggregation, or pathological tissue transitions (Burtea et al., 2023).
• Turbulent Mixing and Multiscale Physics: RG and operator-product-expansion studies of passive scalar and vector fields advected by turbulent, compressible flows with time-varying coefficients reveal rich scaling and universality properties (Antonov et al., 2019).

The models’ flexibility, physical interpretability, and readiness for scalable computation and real-time inference have broadened their applicability to scenarios that require both predictive power and robust uncertainty quantification under nonstationary, data-rich conditions.

7. Challenges and Future Directions

Despite these advances, several technical and theoretical challenges remain:

• Identifiability and Estimability: In high-dimensional multiscale settings, parameter identifiability and reliable estimation under partial observation or overlapping nonstationarities remain open problems, particularly as TVPs grow in number or become highly nonlinear in their effects.
• Model Generalization and Physical Consistency: Extending frameworks to fully nonstationary or non-Gaussian noise, developing models that maintain physical conservation under arbitrary discretization (especially in non-rectangular or discontinuous domains), and handling strong nonlinearities (in reaction, boundary, or memory terms) represent active research areas.
• Integration and Unification: Hybrid techniques that blend particle filtering for parameter inference, adaptive mesh refinement for numerical accuracy, model order reduction for real-time forecasting, and rigorous stochastic modeling are under development to address the most complex, real-time, and data-assimilative scenarios.

Advances in these areas are expected to further enhance the applicability of advection–diffusion models with TVPs in operational forecasting, large-scale environmental analysis, and complex multi-physics systems.