Coupled Continuous-Discontinuous Galerkin Finite Element Solver for Compound Flood Simulations

This paper presents a locally conservative coupled Continuous-Discontinuous Galerkin discretization of the shallow water equations integrated into the ADCIRC model to accurately simulate compound floods by accounting for the complex interactions between storm surge and rainfall-induced runoff, as validated through laboratory tests and Hurricane Harvey simulations.

The Gist

Imagine you are trying to predict how a city will flood during a massive hurricane. In the past, scientists had to look at two separate problems: the ocean water crashing onto the shore (storm surge) and the rain falling from the sky (runoff). They would calculate these separately and then just add the numbers together, like adding the price of a burger to the price of fries.

But here's the problem: Floods don't work like math homework. When heavy rain meets a rising ocean, they interact in messy, chaotic ways. The rain can't drain into the ocean because the ocean is already full, so the water piles up higher than the sum of its parts. This is called a "Compound Flood."

This paper introduces a new, smarter computer program designed to simulate these messy, real-world interactions. Here is how they did it, explained simply:

1. The Old Way vs. The New Way

Think of the computer model as a giant jigsaw puzzle made of thousands of tiny triangles.

• The Old Model (ADCIRC): This model was great at moving water around (momentum), but it treated the puzzle pieces like they were glued together with a sticky, continuous sheet. It was fast, but it sometimes "leaked" water between the pieces, making the math slightly inaccurate when you added rain.
• The New Model (DG-CG): The authors built a hybrid engine.
  • For the Rain (Continuity): They used a method called Discontinuous Galerkin (DG). Imagine each puzzle piece is now a separate bucket. If you pour rain into one bucket, it stays exactly in that bucket. It doesn't leak. This ensures that every single drop of rain is accounted for perfectly.
  • For the Flow (Momentum): They kept the old, fast method (Continuous Galerkin) for how the water moves and swirls. This keeps the computer running fast so they can simulate huge areas like the entire Gulf of Mexico.

The Analogy: It's like hiring a team of strict accountants (DG) to count every penny of rain, while hiring a team of fast runners (CG) to carry the water buckets around the city. The accountants ensure no money is lost; the runners ensure the delivery is fast.

2. The "Dry Land" Problem

One of the hardest things to simulate is a coastline. Some parts of the map are deep ocean, and some are dry land.

• The Challenge: When a storm hits, dry land gets wet. When the tide goes out, wet land gets dry again. In computer land, this is a nightmare. If the math tries to calculate water flow on a dry patch of land, the numbers can explode and crash the program.
• The Solution: The authors created a "traffic cop" system. They have a special rule that says, "If a bucket is empty, stop calculating the flow for that bucket." They also figured out how to let a dry bucket suddenly fill up just because it rained, without needing a river to flow into it first. This allows the model to simulate a city turning from dry to flooded purely due to rain.

3. Testing the Engine

The team didn't just build it; they stress-tested it:

• The "Lake" Test: They simulated a small hill with a constant rain pouring down. They checked if the total amount of water in the box matched the rain they poured in. It did perfectly.
• The "Hurricane" Test: They simulated Hurricane Harvey (2017). This was the ultimate test because Harvey was a "compound flood" disaster—massive rain combined with a storm surge.
  • The Result: The old model (ADCIRC) underestimated the flood levels because it didn't handle the rain interaction well. The new model (DG-CG) predicted the water levels much closer to reality, showing that the rain and the ocean surge were indeed amplifying each other.

4. Why This Matters

This isn't just about better math; it's about saving lives and property.

• Better Forecasts: By accurately modeling how rain and storm surges combine, emergency managers can predict which neighborhoods will flood before the storm hits.
• Efficiency: Because they kept the "fast runner" part of the code, they can run these complex simulations on supercomputers without waiting weeks for the results.

The Bottom Line

The authors built a hybrid super-solver. It combines the precision of a strict accountant (to track every drop of rain) with the speed of a race car (to move the water). This allows scientists to finally simulate the true, chaotic nature of compound floods, helping us understand and prepare for the next big storm.

Technical Summary

1. Problem Statement

Compound flooding occurs when multiple flood drivers interact, specifically the combination of storm surge, river discharge, and rainfall runoff. Recent tropical cyclones, such as Hurricane Harvey (2017), have demonstrated that the interaction between these sources creates flood magnitudes that cannot be accurately predicted by simply superimposing individual flood sources.

Existing numerical models face a trade-off:

• Continuous Galerkin (CG) methods (e.g., the widely used ADCIRC model) are computationally efficient but do not guarantee local mass conservation. They often rely on a surrogate equation (Generalized Wave Continuity Equation, GWCE) which can struggle with adding source terms like rainfall and handling highly advective flows.
• Discontinuous Galerkin (DG) methods (e.g., DG-SWEM) offer exact local mass conservation and handle discontinuities well but are computationally expensive, particularly for the momentum equations.

The challenge is to develop a solver that combines the exact local mass conservation of DG (essential for modeling rainfall and runoff) with the computational efficiency of CG (essential for large-scale momentum simulations), all within a single framework capable of handling wetting/drying and complex coastal geometries.

2. Methodology

The authors propose a hybrid DG-CG finite element solver integrated into the ADCIRC framework. The core innovation is the decoupling of the governing equations based on the physical properties best suited for each numerical scheme.

Governing Equations

The model solves the depth-averaged 2D Shallow Water Equations (SWE):

1. Continuity Equation: Modified to include a spatially and temporally variable rainfall source term ($R$).
2. Momentum Equations: Standard conservation of momentum including Coriolis, bottom friction, and wind stress.

Numerical Discretization Strategy

• Continuity Equation (DG Scheme):
  • Discretized using a Discontinuous Galerkin (DG) method with linear polynomials ($P^1$).
  • Uses the primitive continuity equation (not the GWCE surrogate).
  • Employs a Local Lax-Friedrichs (LLF) numerical flux to handle inter-element communication.
  • Key Benefit: Guarantees exact local mass conservation, allowing rainfall to be added as a source term without inducing numerical instability or mass loss.
• Momentum Equations (CG Scheme):
  • Discretized using the existing Continuous Galerkin (CG) formulation from ADCIRC.
  • Uses continuous piecewise linear basis functions.
  • Key Benefit: Maintains the computational efficiency and low cost of the standard ADCIRC solver for the momentum update.

Coupling and Implementation

• Variable Exchange: The solution involves a continuous water elevation ($\zeta$) in the CG space and a discontinuous elevation in the DG space. The authors implement a projection routine where the DG elevation is averaged to nodal values to update the CG momentum solver, and CG velocities are projected to modal DG representations for the continuity solver.
• Wetting and Drying: A unified wetting/drying algorithm is developed. It primarily relies on the CG nodal state but incorporates DG-specific mass redistribution logic to ensure that elements remain physically consistent (mass conservation) when transitioning between wet and dry states.
• Rainfall Integration: Rainfall is treated as a source term in the continuity equation. The model supports:
  • Parametric models: Specifically the R-CLIPER model, which generates rainfall fields based on storm center and wind speed parameters.
  • Observed Data: Interpolation of GRIB2 format data for hindcasting.

3. Key Contributions

1. Hybrid DG-CG Formulation: The first successful integration of a locally conservative DG continuity solver with a standard CG momentum solver within the ADCIRC ecosystem.
2. Exact Local Mass Conservation: By solving the primitive continuity equation via DG, the model ensures that rainfall input is perfectly conserved locally, a critical feature for compound flood modeling that previous CG-based models lacked.
3. Unified Wetting/Drying Scheme: A robust algorithm that manages the transition between wet and dry states across the coupled DG/CG spaces, preventing instability while allowing dry land to become wet solely through rainfall.
4. Modular Integration: The solver is implemented as a modular extension to ADCIRC, minimizing code restructuring while enabling advanced compound flood capabilities.

4. Results and Validation

The authors validated the solver through a hierarchy of numerical experiments:

• Convergence Tests (Lynch & Gray): Demonstrated optimal convergence rates ($O(h^2)$ for elevation) comparable to pure CG and DG methods, confirming numerical accuracy.
• Mass Conservation (Rain on a Hill): A test case with constant rainfall on a dry domain showed a relative mass error of only 0.07%, proving the solver's ability to conserve mass and transition dry land to wet solely via source terms.
• Neches River (Hurricane Harvey):
  • Compared against ADCIRC (CG) and observed data.
  • The DG-CG solver reduced underprediction of water levels compared to standard ADCIRC, particularly in areas dominated by river runoff and rainfall.
  • It better captured peak water levels in high-advection zones.
• Hurricane Ike (2008) & Harvey (2017):
  • Ike (Surge-dominated): DG-CG results were nearly identical to CG for peak surge, confirming that the hybrid approach does not degrade performance for surge-only events.
  • Harvey (Compound Event): The inclusion of rainfall via the DG scheme significantly improved the simulation of water levels in the aftermath of the storm surge. The model successfully captured the secondary peak in water levels caused by rainfall runoff, which the standard CG model missed.
  • High Water Marks: Comparison with observed high water marks showed a marked improvement in DG-CG accuracy over CG in the Harvey landfall region.

5. Significance and Performance

• Scientific Impact: This work provides a robust numerical foundation for compound flood forecasting. It resolves the long-standing issue of how to accurately model the non-linear interaction between storm surge and rainfall runoff in a single, mass-conserving framework.
• Computational Performance:
  • Serial: The DG-CG solver is approximately 18% slower than pure CG due to the overhead of the DG continuity solver (specifically edge integration and slope limiting). However, optimizations (vectorization, loop reordering) reduced the DG solver's runtime by nearly 50%.
  • Parallel: The code scales well on supercomputers (Frontera), though communication overhead is slightly higher than pure CG due to the exchange of additional variables (wet/dry flags, modal coefficients).
• Future Outlook: The authors identify that while the current model handles rainfall and runoff well, future work must address other hydrological processes (infiltration, evapotranspiration) and further refine the wetting/drying scheme to handle highly refined meshes without limiting fluxes.

In conclusion, this paper presents a significant advancement in coastal engineering modeling, offering a tool that is both computationally efficient and physically rigorous for simulating the complex, compound flooding events that are becoming more frequent due to climate change.