Convergence of the implicit-explicit Euler scheme applied to perturbed dissipative evolution equations

Hansen, Eskil; Stillfjord, Tony

doi:10.1090/s0025-5718-2013-02702-0

Cited by 8 publications

(3 citation statements)

References 20 publications

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“…Furthermore, the considered problem can be split into a series of linear subproblems, both computational size and storage requirements are reduced. We only mention [7] for the dissipative evolution equations, [13,24,33] for the Navier-Stokes equations, [32] for the Cahn-Hilliard equations and the references therein. However, the stability of numerical solutions in implicit/explicit scheme holds under some restrictions on the time steps [9,11,33].…”

Section: Introductionmentioning

confidence: 99%

Unconditional Stability and Optimal Error Estimates of Euler Implicit/Explicit-SAV Scheme for the Navier–Stokes Equations

Zhang

Yuan

2021

J Sci Comput

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The unconditional stability and convergence analysis of the Euler implicit/explicit scheme with finite element discretization are studied for the incompressible time-dependent Navier-Stokes equations based on the scalar auxiliary variable approach. Firstly, a corresponding equivalent system of the Navier-Stokes equations with three variables is formulated, the stable finite element spaces are adopted to approximate these variables and the corresponding theoretical analysis results are provided. Secondly, a fully discrete scheme based on the backward Euler method is developed, the temporal treatment is based on the Euler implicit/explicit scheme, which is implicit for the linear terms and explicit for the nonlinear term. Hence, a constant coefficient algebraic system is formed and it can be solved efficiently. The discrete unconditional energy dissipation and stability of numerical solutions in various norms are established with any restriction on the time step, optimal error estimates are also provided. Finally, some numerical results are provided to illustrate the performances of the considered numerical scheme.

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Section: Introductionmentioning

confidence: 99%

Unconditional Stability and Optimal Error Estimates of Euler Implicit/Explicit-SAV Scheme for the Navier–Stokes Equations

Zhang

Yuan

2021

J Sci Comput

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“…However, a restriction on time step

normalΔ italict

is needed, such as For the incompressible flow, He and his co‐authors [15–19] gave the following stability and convergence condition of implicit/explicit scheme

normalΔ italict \leq C,

where

C

and

C_{i}, i = 1, 2, \dots

are some general positive constants depending on

ν, μ, σ, T, {boldu}_{0}, {boldH}_{0}, f, J

, which may take different values at different places. For the perturbed dissipative evolution equations and convection‐diffusion problems, please refer to References [20–23] the condition (1.3) is needed to establish the stability and error estimates. For the gradient flow, the condition (1.3) is required to preserve positivity and stability of numerical schemes, we can refer to References [24, 25] and reference therein. Among above literatures, the condition (1.3) seems to be a necessary condition of the implicit/explicit scheme. In order to bypass the restriction (1.3), Sun and his co‐authors [26, 27] developed the error splitting technique to establish the unconditional stability and convergence for nonlinear problems, and obtained a series of meaningful results.…”

Section: Introductionmentioning

confidence: 99%

“…• For the perturbed dissipative evolution equations and convection-diffusion problems, please refer to References [20][21][22][23] the condition (1.3) is needed to establish the stability and error estimates. • For the gradient flow, the condition (1.3) is required to preserve positivity and stability of numerical schemes, we can refer to References [24,25] and reference therein.…”

Section: Introductionmentioning

confidence: 99%

Decoupled and linearized scalar auxiliary variable finite element method for the time‐dependent incompressible magnetohydrodynamic equations: Unconditional stability and convergence analysis

Zhang

Yang

2021

Numerical Methods Partial

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This paper considers the stability and convergence of Euler implicit/explicit scheme for the incompressible magnetohydrodynamic (MHD) equations by the scalar auxiliary variable approach. The linear and nonlinear terms are treated by the implicit and explicit schemes, respectively, then the considered problem is split into two constant coefficient linear algebraic equations plus a nonlinear algebraic equation. Compared with the original discrete coefficient matrix, the computational size reduce and we can solve these subproblems conveniently and efficiently. First, an equivalent form of the MHD problem with four variables is developed, the corresponding stability and convergence results of spatial discrete scheme are presented. Second, the decoupled and linearized auxiliary variable finite element method is constructed, the discrete unconditional energy dissipation and stability of u n h and H n h in various norms are established. The optimal error estimates of numerical solutions in L 2 and H-norms are also developed by the energy method and Gronwall lemma. Finally, some numerical examples are presented to verify the established theoretical findings and show the performances of the considered numerical algorithm.

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A randomized operator splitting scheme inspired by stochastic optimization methods

Eisenmann,

Stillfjord

2024

Numer. Math.

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In this paper, we combine the operator splitting methodology for abstract evolution equations with that of stochastic methods for large-scale optimization problems. The combination results in a randomized splitting scheme, which in a given time step does not necessarily use all the parts of the split operator. This is in contrast to deterministic splitting schemes which always use every part at least once, and often several times. As a result, the computational cost can be significantly decreased in comparison to such methods. We rigorously define a randomized operator splitting scheme in an abstract setting and provide an error analysis where we prove that the temporal convergence order of the scheme is at least 1/2. We illustrate the theory by numerical experiments on both linear and quasilinear diffusion problems, using a randomized domain decomposition approach. We conclude that choosing the randomization in certain ways may improve the order to 1. This is as accurate as applying e.g. backward (implicit) Euler to the full problem, without splitting.

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Convergence of the implicit-explicit Euler scheme applied to perturbed dissipative evolution equations

Cited by 8 publications

References 20 publications

Unconditional Stability and Optimal Error Estimates of Euler Implicit/Explicit-SAV Scheme for the Navier–Stokes Equations

Unconditional Stability and Optimal Error Estimates of Euler Implicit/Explicit-SAV Scheme for the Navier–Stokes Equations

Decoupled and linearized scalar auxiliary variable finite element method for the time‐dependent incompressible magnetohydrodynamic equations: Unconditional stability and convergence analysis

A randomized operator splitting scheme inspired by stochastic optimization methods

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