Remarks on the Asymptotic Behavior of Scalar Auxiliary Variable (Sav) Schemes for Gradient-Like Flows

Bouchriti, Anass; Pierre, Morgan; Alaa, Nour Eddine

doi:10.11948/20190373

Cited by 8 publications

(8 citation statements)

References 30 publications

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“…By elliptic regularity [68], (u n+1 ) is bounded in H 2 (Ω). Since H 2 (Ω) is compactly embedded in V [57], this shows that the set {u n : n ∈ N} is precompact in V. It implies that ω((u n ) n∈N ) is a nonempty compact subset of V. Moreover, u n+1 − u n = −τ ∆ N ẇn+1 tends to 0 in V , and by compactness, also in V. Thus, ω((u n ) n∈N ) is also a connected subset of V (see, e.g., [22,Lemma 3.1]). Now, let u ∞ ∈ ω((u n ) n∈N ) and let (u n k ) be a subsequence such that u n k → u ∞ in V as k → +∞.…”

Section: Finite Difference Approximationmentioning

confidence: 85%

“…Since Ū = U ∞ satisfies ( 16) on [0, 1], we have AW ∞ = 0. This shows that (U ∞ , W ∞ ) solves (22), and so (U ∞ , W ∞ ) is a steady state.…”

Section: Matthieu Brachet Philippe Parnaudeau and Morgan Pierrementioning

confidence: 86%

“…The asymptotic behaviour of sequences generated by (73) was studied in [22]. We say that (u , w , r ) ∈ V × V × R is a steady state for (73) if…”

Section: Finite Difference Approximationmentioning

confidence: 99%

See 2 more Smart Citations

Convergence to equilibrium for time and space discretizations of the Cahn-Hilliard equation

Brachet

Parnaudeau

Pierre

2022

DCDS-S

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<p style='text-indent:20px;'>We review space and time discretizations of the Cahn-Hilliard equation which are energy stable. In many cases, we prove that a solution converges to a steady state as time goes to infinity. The proof is based on Lyapunov theory and on a Lojasiewicz type inequality. In a few cases, the convergence result is only partial and this raises some interesting questions. Numerical simulations in two and three space dimensions illustrate the theoretical results. Several perspectives are discussed.</p>

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Section: Finite Difference Approximationmentioning

confidence: 85%

“…Since Ū = U ∞ satisfies ( 16) on [0, 1], we have AW ∞ = 0. This shows that (U ∞ , W ∞ ) solves (22), and so (U ∞ , W ∞ ) is a steady state.…”

Section: Matthieu Brachet Philippe Parnaudeau and Morgan Pierrementioning

confidence: 86%

See 1 more Smart Citation

Convergence to equilibrium for time and space discretizations of the Cahn-Hilliard equation

Brachet

Parnaudeau

Pierre

2022

DCDS-S

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“…This is due to the discretization of the equation for the scalar variable. In a recent work of Bouchriti et al [6], the authors showed that the use of the SAV method for the damped wave equation and the Cahn-Hilliard equation leads to the convergence to modified steady states as well.…”

Section: Introductionmentioning

confidence: 99%

Scalar auxiliary variable finite element scheme for the parabolic-parabolic Keller-Segel model

Poulain¹

2020

Preprint

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We describe and analyze a finite element numerical scheme for the parabolic-parabolic Keller-Segel model. The scalar auxiliary variable method is used to retrieve the monotonic decay of the energy associated with the system at the discrete level. This method relies on the interpretation of the Keller-Segel model as a gradient flow. The resulting numerical scheme is efficient and easy to implement. We show the existence of a unique non-negative solution and that a modified discrete energy is obtained due to the use of the SAV method. We also prove the convergence of the discrete solutions to the ones of the weak form of the continuous Keller-Segel model.

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“…Thanks to their simplicity, efficiency, accuracy and generality, the SAV and GSAV approaches have received much attention recently, they, along with their various variations/extensions, have been used to construct unconditionally energy stable schemes for a large class of nonlinear systems, including various gradient flows (see, for instance, [5,6,27,23,33,40,39]), gradient flows with other global constraints (see, for instance, [8]), Navier-Stokes equations and related systems (see, for instance, [25,22,15,24]), time fractional PDEs [17], conservative or Hamiltonian systems (see, for instance, [3,42,2,14]), and many more. However, in the original SAV approach and its various variants, the discrete value of the SAV is not directly linked to its definition at the continuous level, and this may lead to a loss of accuracy when the time step is not sufficiently small.…”

mentioning

confidence: 99%

A generalized SAV approach with relaxation for dissipative systems

Zhang,

Shen

2022

Preprint

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The scalar auxiliary variable (SAV) approach [30] and its generalized version GSAV proposed in [19] are very popular methods to construct efficient and accurate energy stable schemes for nonlinear dissipative systems. However, the discrete value of the SAV is not directly linked to the free energy of the dissipative system, and may lead to inaccurate solutions if the time step is not sufficiently small. Inspired by the relaxed SAV method proposed in [20] for gradient flows, we propose in this paper a generalized SAV approach with relaxation (R-GSAV) for general dissipative systems. The R-GSAV approach preserves all the advantages of the GSAV appraoch, in addition, it dissipates a modified energy that is directly linked to the original free energy. We prove that the k-th order implicit-explicit (IMEX) schemes based on R-GSAV are unconditionally energy stable, and we carry out a rigorous error analysis for k = 1, 2, 3, 4, 5. We present ample numerical results to demonstrate the improved accuracy and effectiveness of the R-GSAV approach.

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Remarks on the Asymptotic Behavior of Scalar Auxiliary Variable (Sav) Schemes for Gradient-Like Flows

Cited by 8 publications

References 30 publications

Convergence to equilibrium for time and space discretizations of the Cahn-Hilliard equation

Convergence to equilibrium for time and space discretizations of the Cahn-Hilliard equation

Scalar auxiliary variable finite element scheme for the parabolic-parabolic Keller-Segel model

A generalized SAV approach with relaxation for dissipative systems

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