High-Order Locally A-Stable Implicit Schemes for Linear ODEs
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High-Order Locally A-Stable Implicit Schemes for Linear ODEs Hélène Barucq1 · Marc Duruflé2,3 · Mamadou N’diaye1,4 Received: 23 July 2019 / Revised: 12 May 2020 / Accepted: 3 June 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Accurate simulations of wave propagation in complex media like Earth subsurface can be performed with a reasonable computational burden by using hybrid meshes stuffing fine and coarse cells. Locally implicit time discretizations are then of great interest. They indeed allow using unconditionally stable Schemes in the regions of computational domain covered by small cells. The receivable values of the time step are then increased which reduces the computational costs while limiting the dispersion effects. In this work we construct a method that combines optimized explicit Schemes and implicit Schemes to form locally implicit schemes for linear ODEs, including in particular semi-discretized wave problems that are considered herein for numerical experiments. Both the explicit and implicit schemes used are one-step methods constructed using their stability function. The stability function of the explicit schemes are computed by maximizing the time step that can be chosen. The implicit schemes used are unconditionally stable and do not necessary require the same number of stages as the explicit schemes. The performance assessment we provide shows a very good level of accuracy for locally implicit schemes. It also shows that a locally implicit scheme is a good compromise between purely explicit and purely implicit schemes in terms of computational time and memory usage. Keywords Time integration · Hybrid discontinuous Galerkin method · Hyperbolic problems · Wave equations
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Mamadou N’diaye [email protected]; [email protected] Hélène Barucq [email protected] Marc Duruflé [email protected]
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INRIA Magique-3D, E2S UPPA, CNRS, Avenue de l’Université, 64012 Pau, France
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Bordeaux INP, Institut de Mathématiques de Bordeaux UMR 5251, Bordeaux, France
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INRIA Magique-3D, E2S UPPA, CNRS, 200 Avenue de la Vieille Tour, 33405 Talence, France
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Department of Energy Resources Engineering, Stanford University, School of Earth, Energy & Environmental Sciences, 367 Panama Mall, Stanford, CA 94305, USA 0123456789().: V,-vol
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Page 2 of 33
Journal of Scientific Computing
(2020) 85:31
1 Introduction The interest of using high order finite elements to simulate wave problems has been demonstrated in many works as for example in [1–5]. When the propagation medium is complex, it may be necessary to refine the mesh locally to accurately track the discontinuities in the medium, whether geometric or constitutive. In this case, it might be necessary to locally couple high order elements with low order elements. The question then arises of choosing the most suitable time scheme so as not to destroy the quality of the approximation in space. Explicit time schemes [6,7] are very popular as a low-cost in memory and highly scalable integration process
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