arXiv subjects

Andrew Welter

Publications and source records attributed to Andrew Welter.

2 recordsLinked to original sources

arXiv2026

A Massively Parallel Hybridizable Discontinuous Galerkin Solver for Direct Numerical Simulation of Compressible Flows on GPUs

Direct numerical simulation (DNS) of compressible transitional and turbulent flows requires numerical methods that combine high-order accuracy, robustness, and computational efficiency to resolve a broad range of spatial and temporal scales. This paper presents a massively parallel hybridizable discontinuous Galerkin (HDG) solver for DNS of the compressible Navier-Stokes equations on GPU-accelerated high-performance computing systems. The proposed solver combines high-order HDG discretization with robust shock capturing, diagonally implicit Runge-Kutta (DIRK) time integration, and an efficient Newton-GMRES solution strategy accelerated by additive Schwarz preconditioning and reduced-basis approximation. A distributed implementation of these methods based on GPU-aware MPI, Kokkos, and CUDA/HIP libraries enables scalable execution on heterogeneous computing platforms. The solver is demonstrated on three canonical benchmark problems covering a wide range of Mach-number flow regimes: subsonic transitional flow over the Eppler 387 airfoil, the supersonic Taylor-Green vortex, and hypersonic boundary-layer transition. Numerical results are compared with available experimental measurements and published DNS data, showing good agreement across distinct flow regimes. The results demonstrate the ability of the proposed solver to resolve laminar-turbulent transition, strong compressibility effects, shock-associated flow structures, and fully three-dimensional turbulent dynamics.

math.NA↗

arXiv2025

Preconditioning Techniques for Hybridizable Discontinuous Galerkin Discretizations on GPU Architectures

We present scalable iterative solvers and preconditioning strategies for Hybridizable Discontinuous Galerkin (HDG) discretizations of partial differential equations (PDEs) on graphics processing units (GPUs). The HDG method is implemented using GPU-tailored algorithms in which local element degrees of freedom are eliminated in parallel, and the globally condensed system is assembled directly on the device using dense-block operations. The global matrix is stored in a block format that reflects the natural HDG structure, enabling all iterative solver kernels to be executed with strided batched dense matrix-vector multiplications. This implementation avoids sparse data structures, increases arithmetic intensity, and sustains high memory throughput across a range of meshes and polynomial orders. The nonlinear solver combines Newton's method with preconditioned GMRES, integrating scalable preconditioners such as block-Jacobi, additive Schwarz domain decomposition, and polynomial smoothers. All preconditioners are implemented in batched form with architecture-aware optimizations--including dense linear algebra kernels, memory-coalesced vector operations, and shared-memory acceleration--to minimize memory traffic and maximize parallel occupancy. Comprehensive studies are conducted for a variety of PDEs (including Poisson equation, Burgers equation, linear and nonlinear elasticity, Euler equations, Navier-Stokes equations, and Reynolds-Averaged Navier-Stokes equations) using structured and unstructured meshes with different element types and polynomial orders on both NVIDIA and AMD GPU architectures.

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