An energy-stable generalized-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="mml42" display="inline" overflow="scroll" altimg="si42.gif"><mml:mi>α</mml:mi></mml:math> method for the Swift–Hohenberg equation

Sarmiento, A. F.; Espath, Luis; Vignal, Philippe; Dalcín, Lisandro; Parsani, Matteo; Calo, Victor M.

doi:10.1016/j.cam.2017.11.004

Cited by 27 publications

(11 citation statements)

References 24 publications

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“…[30]). Several numerical methods have been developed to alleviate the time step restriction while still keeping the energy dissipation, related contributions include the fully implicit operator splitting finite difference method [6,5], the semi-analytical Fourier spectral method [18], the unconditionally energy stable method [12] derived from an integration quadrature formula, the large time-stepping method [32] based on the use of an extra artificial stabilized term, and the energy stable generalized-α method [24]. However, these methods generally require the use of an iteration in solving the fully discrete nonlinear systems.…”

Section: Introductionmentioning

confidence: 99%

Unconditionally Energy Stable DG Schemes for the Swift–Hohenberg Equation

Liu

Yin

2019

J Sci Comput

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The Swift-Hohenberg equation as a central nonlinear model in modern physics has a gradient flow structure. Here we introduce fully discrete discontinuous Galerkin (DG) schemes for a class of fourth order gradient flow problems, including the nonlinear Swift-Hohenberg equation, to produce free-energy-decaying discrete solutions, irrespective of the time step and the mesh size. We exploit and extend the mixed DG method introduced in [H. Liu and P. Yin, J. Sci. Comput., 77: 467-501, 2018] for the spatial discretization, and the "Invariant Energy Quadratization" method for the time discretization. The resulting IEQ-DG algorithms are linear, thus they can be efficiently solved without resorting to any iteration method. We actually prove that these schemes are unconditionally energy stable. We present several numerical examples that support our theoretical results and illustrate the efficiency, accuracy and energy stability of our new algorithm. The numerical results on two dimensional pattern formation problems indicate that the method is able to deliver comparable patterns of high accuracy.1991 Mathematics Subject Classification. 65N12, 65N30, 35K35.

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Section: Introductionmentioning

confidence: 99%

Unconditionally Energy Stable DG Schemes for the Swift–Hohenberg Equation

Liu

Yin

2019

J Sci Comput

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“…In these models, the phase-field or orderparameter varies smoothly (i.e., continuously) over these thin layers, with a key point being that the energy of these models needs to be dissipated as time progresses. The existence of a Lyapunov functional for these diffuseinterface problems implies strong energy stability [60], a property which can be lost if inadequate algorithms and/or spatio-temporal resolutions are used to solve the partial differential equation [24,48,55,60]. This work addresses the nonlinear stability issue for both conserved and non-conserved phase-field variables, and presents a simple process that relies on Taylor series to handle the nonlinear terms present in the partial differential equation.…”

Section: Energy Stability Of Phase-field Modelsmentioning

confidence: 99%

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

Vignal

Collier

Dalcín

et al. 2017

Computer Methods in Applied Mechanics and Engineering

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We introduce a provably energy-stable time-integration method for general classes of phase-field models with polynomial potentials. We demonstrate how Taylor series expansions of the nonlinear terms present in the partial differential equations of these models can lead to expressions that guarantee energy-stability implicitly, which are second-order accurate in time. The spatial discretization relies on a mixed finite element formulation and isogeometric analysis. We also propose an adaptive time-stepping discretization that relies on a first-order backward approximation to give an error-estimator. This error estimator is accurate, robust, and does not require the computation of extra solutions to estimate the error. This methodology can be applied to any second-order accurate time-integration scheme. We present numerical examples in two and three spatial dimensions, which confirm the stability and robustness of the method. The implementation of the numerical schemes is done in PetIGA, a high-performance isogeometric analysis framework.

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“…We discretize the model problem using PetIGA [4], a high-performance isogeometric analysis solver built on PETSc (portable extensible toolkit for scientific computation) [60]. PetIGA has been utilized in many scientific and engineering applications (see, e.g., [10,11,16,20,21,23,25,26,[61][62][63][64]). It allows us to investigate both IGA and rIGA discretizations on test cases with different numbers of elements in 2D and 3D, different polynomial degrees of the B-spline spaces, and different partitioning levels of the mesh.…”

Section: Implementation Detailsmentioning

confidence: 99%

Refined isogeometric analysis for generalized Hermitian eigenproblems

Hashemian

Pardo

Calo

2021

Computer Methods in Applied Mechanics and Engineering

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We use refined isogeometric analysis (rIGA) to solve generalized Hermitian eigenproblems (Ku = λMu). rIGA conserves the desirable properties of maximum-continuity isogeometric analysis (IGA) while it reduces the solution cost by adding zero-continuity basis functions, which decrease the matrix connectivity. As a result, rIGA enriches the approximation space and reduces the interconnection between degrees of freedom. We compare computational costs of rIGA versus those of IGA when employing a Lanczos eigensolver with a shift-and-invert spectral transformation. When all eigenpairs within a given interval [λ s , λ e ] are of interest, we select several shifts σ k ∈ [λ s , λ e ] using a spectrum slicing technique. For each shift σ k , the factorization cost of the spectral transformation matrix K − σ k M controls the total computational cost of the eigensolution. Several multiplications of the operator matrix (K − σ k M) −1 M by vectors follow this factorization. Let p be the polynomial degree of the basis functions and assume that IGA has maximum continuity of p − 1. When using rIGA, we introduce C 0 separators at certain element interfaces to minimize the factorization cost. For this setup, our theoretical estimates predict computational savings to compute a fixed number of eigenpairs of up to O(p 2 ) in the asymptotic regime, that is, large problem sizes. Yet, our numerical tests show that for moderate-size eigenproblems, the total observed computational cost reduction is O(p). In addition, rIGA improves the accuracy of every eigenpair of the first N 0 eigenvalues and eigenfunctions, where N 0 is the total number of modes of the original maximum-continuity IGA discretization.

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

Cited by 27 publications

References 24 publications

Unconditionally Energy Stable DG Schemes for the Swift–Hohenberg Equation

Unconditionally Energy Stable DG Schemes for the Swift–Hohenberg Equation

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

Refined isogeometric analysis for generalized Hermitian eigenproblems

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