Parallel Implicit-Explicit General Linear Methods
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Parallel Implicit‑Explicit General Linear Methods Steven Roberts1 · Arash Sarshar1 · Adrian Sandu1 Received: 1 February 2020 / Revised: 21 April 2020 / Accepted: 12 June 2020 © Shanghai University 2020
Abstract High-order discretizations of partial differential equations (PDEs) necessitate high-order time integration schemes capable of handling both stiff and nonstiff operators in an efficient manner. Implicit-explicit (IMEX) integration based on general linear methods (GLMs) offers an attractive solution due to their high stage and method order, as well as excellent stability properties. The IMEX characteristic allows stiff terms to be treated implicitly and nonstiff terms to be efficiently integrated explicitly. This work develops two systematic approaches for the development of IMEX GLMs of arbitrary order with stages that can be solved in parallel. The first approach is based on diagonally implicit multi-stage integration methods (DIMSIMs) of types 3 and 4. The second is a parallel generalization of IMEX Euler and has the interesting feature that the linear stability is independent of the order of accuracy. Numerical experiments confirm the theoretical rates of convergence and reveal that the new schemes are more efficient than serial IMEX GLMs and IMEX Runge–Kutta methods. Keywords Parallel · Time integration · IMEX methods · General linear methods Mathematics Subject Classification 65L05 · 65L20 · 65L80
1 Introduction In this work, we consider the autonomous, additively partitioned system of ordinary differential equations (ODEs)
This work was funded by awards NSF CCF1613905, NSF ACI1709727, AFOSR DDDAS FA9550-171-0015, and by the Computational Science Laboratory at Virginia Tech. * Steven Roberts [email protected] Arash Sarshar [email protected] Adrian Sandu [email protected] 1
Computational Science Laboratory, Virginia Polytechnic Institute and State University, Blacksburg, VA 24060, USA
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Vol.:(0123456789)
Communications on Applied Mathematics and Computation
y� = f (y) + g(y),
y(t0 ) = y0 ,
t0 ⩽ t ⩽ tF ,
(1.1)
where f is nonstiff, g is stiff, and y ∈ ℝd . Such systems frequently arise from applying the methods of lines to semidiscretize a partial differential equation (PDE). For example, processes such as diffusion, advection, and reaction all have different stiffnesses, CFL conditions, and optimal integration schemes. Implicit-explicit (IMEX) methods offer a specialized approach for solving Eq. (1.1) by treating f with an inexpensive explicit method and limiting the application of an implicit method, which is generally more expensive, to g. The IMEX strategy has a relatively long history in the context of Runge–Kutta methods [2, 4, 17, 25, 28] and linear multistep methods [3, 19, 21]. Zhang et al. proposed IMEX schemes based on two-step Runge–Kutta (TSRK) and general linear methods (GLM) [33, 34, 36] with further developments reported in [5, 6, 12–14, 22, 24, 35]. Similarly, peer methods, a subclass of GLMs, have been utilized for IMEX integration in the literature such as [18, 27, 31,
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