Toward error estimates for general space-time discretizations of the advection equation
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PINT 2019
Toward error estimates for general space-time discretizations of the advection equation Martin J. Gander1 · Thibaut Lunet1 Received: 8 December 2019 / Accepted: 8 June 2020 © The Author(s) 2020
Abstract We develop new error estimates for the one-dimensional advection equation, considering general space-time discretization schemes based on Runge–Kutta methods and finite difference discretizations. We then derive conditions on the number of points per wavelength for a given error tolerance from these new estimates. Our analysis also shows the existence of synergistic space-time discretization methods that permit to gain one order of accuracy at a given CFL number. Our new error estimates can be used to analyze the choice of space-time discretizations considered when testing Parallel-in-Time methods. Keywords Advection equation · Space-time discretization · Error estimates · Parallel-in-time integration · Parareal
1 Introduction Over the last decade, Parallel-in-Time (PinT) methods have received sustained attention in the numerical analysis research community, because of the new scientific challenges coming with the ever growing parallel super-computing architectures; for a review, see [12]. Many PinT algorithms are known to work well for parabolic problems (see, e.g, [20,30,37]), but the PinT community still struggles with hyperbolic problems. This motivated many contributions in the literature, some of which considered solutions based on Parareal, a well known PinT algorithm proposed in [28], see for instance [3,4,6,29,32,35]. A geometric multigrid interpretation of Parareal was given in [20], followed by the genesis of the MGRIT algorithm [10], based on algebraic multigrid methods. MGRIT was first presented during a student paper competition at the Sixteenth Copper Mountain Conference on Multigrid Methods, from which [10] originates. A journal version of this work was also published later in [9]. In contrast to [20], in the first paper for MGRIT the authors consider a Full Approximation Scheme (FAS) interpretation of Parareal, and based on this introduce a variant of MGRIT, with additional coarse and fine relaxCommunicated by Robert Speck.
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Thibaut Lunet [email protected] Section de Mathématiques, University of Geneva, 2-4 rue du Liévre, Case postale 64, 1211 Genéve 4, Switzerland
ation steps (FCF-relaxation, in contrast with F-relaxation only for the basic MGRIT method). It was then proved in [19] that this variant with FCF-relaxation is identical to Parareal with overlap, where each additional C F relaxation adds one coarse time interval more in the overlap in Parareal. Many recent contributions focused on applying MGRIT to hyperbolic problems, see e.g. [5,25,26]. There are however also other PinT algorithms especially designed for hyperbolic problems, see e.g. ParaExp [13,14], the diagonalization technique [18], and waveform relaxation [16,17,39,40], and combinations thereof, see e.g. [21], based on earlier work in [43]. The advection equation has proven to be a critical test problem for
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