A novel, high-order, parallel-in-time integration procedure for complex fluid flows

• Applicant: Dr. Jonas Zeifang
• Subject Area: Mathematics ; Fluid Mechanics
• Term: from 2021 to 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 457811052

• Final Report Year: 2022

Final Report Abstract

In this project, a high-order numerical method in space and time for the simulation of fluid flows has been developed and investigated. The main focus of this project has been the development of a novel time discretization for a discontinuous Galerkin spatial discretization, offering the possibility of a temporal parallelization. First, the basis has been set by introducing a class of implicit parallel-in-time integrators for ODEs. Furthermore, we have introduced a two-derivative discontinuous Galerkin method for fluid flow simulations. However, the assembly of both building blocks into an efficient parallel-intime PDE discretization required some rework of the numerical methods. As we have observed poor stability properties of the novel integrator class, an improvement for the novel time discretization class had to be developed, which has been achieved. A redesign of the temporal parallelization procedure became necessary to obtain good parallel speedups for fluid flow simulations. These modifications and some illustrative simulations on more than 1000 processors highlighting the good parallel performance of the novel method has been shown. We have shown that temporal parallelization can leverage further speedups of complex simulations when the speedup of the spatial parallelization already saturates.

Publications

• A data-driven high order sub-cell artificial viscosity for the discontinuous Galerkin spectral element method. Journal of Computational Physics, 441, 110475.
  Zeifang, Jonas & Beck, Andrea
  
• A Low Mach Number IMEX Flux Splitting for the Level Set Ghost Fluid Method. Communications on Applied Mathematics and Computation, 5(2), 722-750.
  Zeifang, Jonas & Beck, Andrea
  
• Parallel-in-Time High-Order Multiderivative IMEX Solvers. Journal of Scientific Computing, 90(1).
  Schütz, Jochen; Seal, David C. & Zeifang, Jonas
  
• Implicit two-derivative deferred correction time discretization for the discontinuous Galerkin method. Journal of Computational Physics, 464, 111353.
  Zeifang, Jonas & Schütz, Jochen
  
• Jacobian-Free Explicit Multiderivative Runge–Kutta Methods for Hyperbolic Conservation Laws. Journal of Scientific Computing, 90(3).
  Chouchoulis, Jeremy; Schütz, Jochen & Zeifang, Jonas
  
• Stability of implicit multiderivative deferred correction methods. BIT Numerical Mathematics, 62(4), 1487-1503.
  Zeifang, Jonas; Schütz, Jochen & Seal, David C.
  
• A narrow band-based dynamic load balancing scheme for the level-set ghost-fluid method. High Performance Computing in Science and Engineering '21 (2023), 305-320. American Geophysical Union (AGU).
  Appel, Daniel; Jöns, Steven; Keim, Jens; Müller, Christoph; Zeifang, Jonas & Munz, Claus-Dieter
  
• Time parallelism and Newton-adaptivity of the two-derivative deferred correction discontinuous Galerkin method. Applied Mathematics and Computation, 457 (2023, 11), 128198.
  Zeifang, Jonas; Thenery, Manikantan Arjun & Schütz, Jochen