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High order linearly implicit methods for evolution equations: How to solve an ODE by inverting only linear systems

Guillaume Dujardin 1, 2 Ingrid Lacroix-Violet 1, 3
2 Paradyse
LPP - Laboratoire Paul Painlevé - UMR 8524, Inria Lille - Nord Europe
3 RAPSODI - Reliable numerical approximations of dissipative systems
LPP - Laboratoire Paul Painlevé - UMR 8524, Inria Lille - Nord Europe
Abstract : This paper introduces a new class of numerical methods for the time integration of evolution equations set as Cauchy problems of ODEs or PDEs. The systematic design of these methods mixes the Runge-Kutta collocation formalism with collocation techniques, in such a way that the methods are linearly implicit and have high order. The fact that these methods are implicit allows to avoid CFL conditions when the large systems to integrate come from the space discretization of evolution PDEs. Moreover, these methods are expected to be efficient since they only require to solve one linear system of equations at each time step, and efficient techniques from the literature can be used to do so. After the introduction of the methods, we set suitable definitions of consistency and stability for these methods. This allows for a proof that arbitrarily high order linearly implicit methods exist and converge when applied to ODEs. Eventually, we perform numerical experiments on ODEs and PDEs that illustrate our theoretical results for ODEs, and compare our methods with standard methods for several evolution PDEs.
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https://hal.inria.fr/hal-02361814
Contributor : Guillaume Dujardin <>
Submitted on : Thursday, October 22, 2020 - 2:36:47 PM
Last modification on : Friday, November 27, 2020 - 2:18:03 PM

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  • HAL Id : hal-02361814, version 2
  • ARXIV : 1911.06016

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Guillaume Dujardin, Ingrid Lacroix-Violet. High order linearly implicit methods for evolution equations: How to solve an ODE by inverting only linear systems. 2019. ⟨hal-02361814v2⟩

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