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Automated GPU Acceleration of Stabilized Shallow Water Solvers with FEniCSx

Benjamin Pachev, Ph.D Candidate, Oden Institute

Abstract

 The shallow-water equations are a set of nonlinear partial differential equations that describe the dynamics of an incompressible fluid for which horizontal length scales dominate vertical length scales. Important physical phenomena described by the shallow water equations include storm surge from tropical cyclones, tides, the circulation of debris or pollutants in the ocean, flooding from a dam failure, landslides, and watershed management. Due to their importance in applications, a large number of numerical methods have been developed to solve the shallow-water equations. While these include finite-volume and finite-differencing schemes, practical applications are dominated by finite element methods. This primarily is due to the ability of finite element methods to easily handle complex geometries and complicated boundary conditions which arise in applications. This work explores several aspects
   of finite element solvers for the shallow water equations: stabilization, acceleration, and practicality.
   Stabilization: It is well known that under a classical Galerkin discretization, the shallow water equations develop instabilities. Consequently, they must be solved with a stabilized method. Popular stabilization approaches for the shallow water equations include Stablized Upwind Petrov-Galerkin (SUPG), Discontinuous Galerkin (DG), and Hybridizable Discontinuous Galerkin (HDG). In this work, we compare these established stabilization techniques with the Discontinuous Petrov-Galerkin method (DPG), a recent development in the theory of stabilized methods that has yet to be applied to the shallow water equations. The DPG methodology has been successfully applied to a number of linear and nonlinear equations, and possesses several attractive features. These include the potential for unconditional stability, hp-adaptivity, no reliance on hand-tuned stabilization parameters, and theoretically optimal convergence rates.
   Acceleration: While most solvers for the shallow-water equations are parallelized traditionally via OpenMP or MPI, a number of works in recent years have proposed GPU-accelerated methods. Graphics processing units (GPUs), have revolutionized fields such as machine learning by enabling speedups that outstrip what is possible with traditional parallelization. Despite recent work that demonstrates the feasibility and advantages of GPU acceleration for finite element methods, it has yet to become the norm for finite element solvers, of which shallow water solvers are a special case. This is due in part to the difficulty of modifying existing code to support GPU acceleration, and to the relatively young state of the literature on GPU acceleration for finite element solvers. A major contribution of this thesis is programming work to make GPU acceleration capabilities in the popular finite element framework FEniCSx available for use to the public. This provides the scientific community a convenient option for developing GPU-accelerated finite element solvers. It also directly enables the development of the GPU-accelerated shallow water solvers examined in this work.
   Practicality: While numerical stability and performance are important, a practical solver additionally needs to handle real-world problems, and should ideally be easy to use. While most solvers for the shallow water equations in the literature possess at least baseline levels of stability and efficiency, the majority apply only to toy problems, and have little use in applications. Those that support the full range of boundary conditions and physical forcing required for application problems are typically difficult to use and maintain. An advantage of the FEniCSx shallow water solvers studied in this work is the combination of ease-of-use with support for real-world test cases of actual interest. Another contribution of this work is the enhancement of preexisting FEniCSx shallow water solvers to more easily support a wider range of application use cases.
 

Biography

Benjamin Pachev is a Ph.D. candidate in Computational Science, Engineering, and Mathematics at the University of Texas at Austin. His dissertation advisor is Professor Clint Dawson, with Dr Eirik Valseth as co-advisor. His research focuses on GPU acceleration of the finite element method, automation of high-performance computing workflows, and surrogate models of hurricane storm surge. Prior to his Ph.D. he earned an M.S and B.S. in Mathematics at Brigham Young University.