3 research outputs found
The Regionally-Implicit Discontinuous Galerkin Method: Improving the Stability of DG-FEM
Discontinuous Galerkin (DG) methods for hyperbolic partial differential
equations (PDEs) with explicit time-stepping schemes, such as strong
stability-preserving Runge-Kutta (SSP-RK), suffer from time-step restrictions
that are significantly worse than what a simple Courant-Friedrichs-Lewy (CFL)
argument requires. In particular, the maximum stable time-step scales inversely
with the highest degree in the DG polynomial approximation space and becomes
progressively smaller with each added spatial dimension. In this work we
introduce a novel approach that we have dubbed the regionally implicit
discontinuous Galerkin (RIDG) method to overcome these small time-step
restrictions. The RIDG method is based on an extension of the Lax-Wendroff DG
(LxW-DG) method, which previously had been shown to be equivalent to a
predictor-corrector approach, where the predictor is a locally implicit
spacetime method (i.e., the predictor is something like a block-Jacobi update
for a fully implicit spacetime DG method). The corrector is an explicit method
that uses the spacetime reconstructed solution from the predictor step. In this
work we modify the predictor to include not just local information, but also
neighboring information. With this modification we show that the stability is
greatly enhanced; in particular, we show that we are able to remove the
polynomial degree dependence of the maximum time-step and show how this extends
to multiple spatial dimensions. A semi-analytic von Neumann analysis is
presented to theoretically justify the stability claims. Convergence and
efficiency studies for linear and nonlinear problems in multiple dimensions are
accomplished using a MATLAB code that can be freely downloaded.Comment: 26 pages, 4 figures, 8 table
Parallel Algorithms for Successive Convolution
In this work, we consider alternative discretizations for PDEs which use
expansions involving integral operators to approximate spatial derivatives.
These constructions use explicit information within the integral terms, but
treat boundary data implicitly, which contributes to the overall speed of the
method. This approach is provably unconditionally stable for linear problems
and stability has been demonstrated experimentally for nonlinear problems.
Additionally, it is matrix-free in the sense that it is not necessary to invert
linear systems and iteration is not required for nonlinear terms. Moreover, the
scheme employs a fast summation algorithm that yields a method with a
computational complexity of , where is the number of mesh
points along a direction. While much work has been done to explore the theory
behind these methods, their practicality in large scale computing environments
is a largely unexplored topic. In this work, we explore the performance of
these methods by developing a domain decomposition algorithm suitable for
distributed memory systems along with shared memory algorithms. As a first
pass, we derive an artificial CFL condition that enforces a nearest-neighbor
communication pattern and briefly discuss possible generalizations. We also
analyze several approaches for implementing the parallel algorithms by
optimizing predominant loop structures and maximizing data reuse. Using a
hybrid design that employs MPI and Kokkos for the distributed and shared memory
components of the algorithms, respectively, we show that our methods are
efficient and can sustain an update rate DOF/node/s. We provide
results that demonstrate the scalability and versatility of our algorithms
using several different PDE test problems, including a nonlinear example, which
employs an adaptive time-stepping rule.Comment: 36 pages, 12 figures, 2 table
Parallel Scaling of the Regionally-Implicit Discontinuous Galerkin Method with Quasi-Quadrature-Free Matrix Assembly
In this work we investigate the parallel scalability of the numerical method
developed in Guthrey and Rossmanith [The regionally implicit discontinuous
Galerkin method: Improving the stability of DG-FEM, SIAM J. Numer. Anal.
(2019)]. We develop an implementation of the regionally-implicit discontinuous
Galerkin (RIDG) method in DoGPack, which is an open source C++ software package
for discontinuous Galerkin methods. Specifically, we develop and test a hybrid
OpenMP and MPI parallelized implementation of DoGPack with the goal of
exploring the efficiency and scalability of RIDG in comparison to the popular
strong stability-preserving Runge-Kutta discontinuous Galerkin (SSP-RKDG)
method. We demonstrate that RIDG methods are able to hide communication latency
associated with distributed memory parallelism, due to the fact that almost all
of the work involved in the method is highly localized to each element,
producing a localized prediction for each region. We demonstrate the enhanced
efficiency and scalability of the of the RIDG method and compare it to SSP-RKDG
methods and show extensibility to very high order schemes. The two-dimensional
scaling study is performed on machines at the Institute for Cyber-Enabled
Research at Michigan State University, using up to 1440 total cores on Intel(R)
Xeon(R) Gold 6148 CPU @ 2.40GHz CPUs. The three dimensional scaling study is
performed on Livermore Computing clusters at at Lawrence Livermore National
Laboratory, using up to 28672 total cores on Intel Xeon CLX-8276L CPUs with
Omni-Path interconnects.Comment: 26 pages, 2 figures, 6 table