A $\mu$-mode integrator for solving evolution equations in Kronecker form

Abstract

In this paper, we propose a $\mu$-mode integrator for computing the solution of stiff evolution equations. The integrator is based on a $d$-dimensional splitting approach and uses exact (usually precomputed) one-dimensional matrix exponentials. We show that the action of the exponentials, i.e. the corresponding batched matrix-vector products, can be implemented efficiently on modern computer systems. We further explain how $\mu$-mode products can be used to compute spectral transforms efficiently even if no fast transform is available. We illustrate the performance of the new integrator by solving, among the others, three-dimensional linear and nonlinear Schrödinger equations, and we show that the $\mu$-mode integrator can significantly outperform numerical methods well established in the field. We also discuss how to efficiently implement this integrator on both multi-core CPUs and GPUs. Finally, the numerical experiments show that using GPUs results in performance improvements between a factor of $10$ and $20$, depending on the problem.

References in corpus (15)

• NVIDIA Tensor Core Programmability, Performance & Precision
• A Krylov subspace algorithm for evaluating the phi-functions appearing in exponential integrators
• What is a post-Lie algebra and why is it useful in geometric integration
• KIOPS: A fast adaptive Krylov subspace solver for exponential integrators
• The Leja method revisited: backward error analysis for the matrix exponential
• Overcoming order reduction in diffusion-reaction splitting. Part 1: Dirichlet boundary conditions
• Overcoming order reduction in diffusion-reaction splitting. Part 2: oblique boundary conditions
• Reproducibility, accuracy and performance of the Feltor code and library on parallel computer architectures
• Exponential methods for solving hyperbolic problems with application to kinetic equations
• Magnus integrators on multicore CPUs and GPUs
• Semi-Lagrangian Vlasov simulation on GPUs
• An exponential integrator for the drift-kinetic model
• Efficient boundary corrected Strang splitting
• Evaluation of the Intel Xeon Phi 7120 and NVIDIA K80 as accelerators for two-dimensional panel codes
• ADI type preconditioners for the steady state inhomogeneous Vlasov equation

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