Matrix-Free Spectral Implementation and Parallel Acceleration of BiCGStab and BiCGStab(ℓ) Methods for the Magnetic Induction Problem

Abstract

The paper studies parallel implementations of the BiCGStab and BiCGStab(ℓ) iterative methods for auxiliary linear problems arising in the computation of the magnetic α-effect. The magnetic induction equation is considered in a three-dimensional periodic domain. After spectral discretization, the problem leads to large nonsymmetric systems of linear algebraic equations whose full matrix is not assembled. The proposed implementation is based on a matrix-free application of the magnetic induction operator, FFT-based transitions between physical and spectral spaces, preconditioned Krylov subspace methods, and OpenMP/MPI parallelization. Numerical experiments for a grid with N = 96 show that the OpenMP BiCGStab implementation reduces the solution time by a factor of 13 with 32 threads. Optimization of the OpenMP BiCGStab(ℓ) implementation reduces the solution time at 32 threads by 62.0%. The MPI BiCGStab implementation with a manual 2D pencil decomposition reduces the solution time at 32 processes by 13.6% compared with the baseline FFTW-MPI implementation. Additional scans over the magnetic diffusivity show that increasing the polynomial degree ℓ extends the range of stable convergence for smaller values of η.