An energy-stable time-integrator for phase-field models

Vignal, Philippe; Collier, Nathan; Dalcín, Lisandro; Brown, Donald L.; Calo, Victor M.

doi:10.1016/j.cma.2016.12.017

Cited by 28 publications

(45 citation statements)

References 56 publications

(134 reference statements)

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“…In this work, we construct a variational formulation of explicit Runge-Kutta methods for parabolic problems. Such formulation could be applied to design explicit-in-time (and consequently cheaper) goal-oriented adaptive algorithms, to build new time-stepping schemes and also to extend the existing space discretizations like IGA [38][39][40], DPG [41] and Trefftz [42], to time domain problems.…”

Section: Introductionmentioning

confidence: 99%

Variational formulations for explicit Runge-Kutta Methods

Muñoz‐Matute

Pardo

Calo

et al. 2019

Finite Elements in Analysis and Design

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Variational space-time formulations for Partial Differential Equations have been of great interest in the last decades. While it is known that implicit time marching schemes have variational structure, the Galerkin formulation of explicit methods in time remains elusive. In this work, we prove that the explicit Runge-Kutta methods can be expressed as discontinuous Petrov-Galerkin methods both in space and time. We build trial and test spaces for the linear diffusion equation that lead to one, two, and general stage explicit Runge-Kutta methods. This approach enables us to design explicit time-domain (goal-oriented) adaptive algorithms.

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Section: Introductionmentioning

confidence: 99%

Variational formulations for explicit Runge-Kutta Methods

Muñoz‐Matute

Pardo

Calo

et al. 2019

Finite Elements in Analysis and Design

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“…where z h 2 (0) is the approximated solution of the classical dual problem (7) at t = 0. Selecting (21), formula (15) becomes…”

Section: Hybrid Algorithmmentioning

confidence: 99%

“…Most authors select discontinuous‐in‐time test functions to discretize both primal and dual problems. This selection decouples the resulting systems and we can solve them as time‐marching schemes . However, as the dual problem runs backward in time, the goal‐oriented adaptive process involves solving two problems running in opposite directions in time.…”

Section: Introductionmentioning

confidence: 99%

“…This selection decouples the resulting systems and we can solve them as time-marching schemes. [13][14][15] However, as the dual problem runs backward in time, the goal-oriented adaptive process involves solving two problems running in opposite directions in time. This fact implies that we need to store all the information coming from the primal and dual problems to estimate the error contributions and perform the adaptivity.…”

Section: Introductionmentioning

confidence: 99%

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Forward‐in‐time goal‐oriented adaptivity

Muñoz‐Matute

Pardo

Calo

et al. 2019

Numerical Meth Engineering

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Summary In goal‐oriented adaptive algorithms for partial differential equations, we adapt the finite element mesh to reduce the error of the solution in some quantity of interest. In time‐dependent problems, this adaptive algorithm involves solving a dual problem that runs backward in time. This process is, in general, computationally expensive in terms of memory storage. In this work, we define a pseudo‐dual problem that runs forward in time. We also describe a forward‐in‐time adaptive algorithm that works for some specific problems. Although it is not possible to define a general dual problem running forwards in time that provides information about future states, we provide numerical evidence via one‐dimensional problems in space to illustrate the efficiency of our algorithm as well as its limitations. Finally, we propose a hybrid algorithm that employs the classical backward‐in‐time dual problem once and then performs the adaptive process forwards in time.

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“…Mixed formulations for phase-field equations are common [6,21], given that they allow the use of discretization spaces with lower polynomial order and reduced continuity. This leads to cheaper solution schemes [3,4].…”

Section: Primal and Mixed Weak Formsmentioning

confidence: 99%

An energy-stable generalized-α method for the Swift–Hohenberg equation

Sarmiento

Espath

Vignal³

et al. 2018

Journal of Computational and Applied Mathematics

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This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. Highlights • We present a second-order energy-stable time-integration method for the Swift-Hohenberg equation that suppresses numerical instabilities. • Based on the generalized-α method, numerical dissipation is controlled via the spectral radius. • A detailed energy-stability proof and an estimate for the stabilization parameter are provided. • The proposed stabilization vanishes for sufficiently small time step sizes, recovering the original weak form of the equation. • Numerical results for a pattern-formation example are used to test the convergence and energy-stability. • We compare the primal and mixed formulation in terms of wall-clock time and approximation error.

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An energy-stable time-integrator for phase-field models

Cited by 28 publications

References 56 publications

Variational formulations for explicit Runge-Kutta Methods

Variational formulations for explicit Runge-Kutta Methods

Forward‐in‐time goal‐oriented adaptivity

An energy-stable generalized-α method for the Swift–Hohenberg equation

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