Temporal splitting algorithms for non-stationary multiscale problems

Efendiev, Yalchin; Pun, Sai-Mang; Vabishchevich, Petr N.

doi:10.1016/j.jcp.2021.110375

Cited by 12 publications

(6 citation statements)

References 21 publications

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“…In these cases, the operator is decomposed based on physical processes. In our recent works, we have proposed several approaches for temporal splitting that uses multiscale spaces [18,17]. In [18], a general framework is proposed where the transition to simpler problems is carried out based on spatial decomposition of the solution.…”

Section: Introductionmentioning

confidence: 99%

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Contrast-independent partially explicit time discretizations for multiscale flow problems

Chung,

Efendiev,

Leung

et al. 2021

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Many multiscale problems have a high contrast, which is expressed as a very large ratio between the media properties. The contrast is known to introduce many challenges in the design of multiscale methods and domain decomposition approaches. These issues to some extend are analyzed in the design of spatial multiscale and domain decomposition approaches. However, some of these issues remain open for time dependent problems as the contrast affects the time scales, particularly, for explicit methods. For example, in parabolic equations, the time step is dt = H 2 /κ max , where κ max is the largest diffusivity. In this paper, we address this issue in the context of parabolic equation by designing a splitting algorithm. The proposed splitting algorithm treats dominant multiscale modes in the implicit fashion, while the rest in the explicit fashion. The unconditional stability of these algorithms require a special multiscale space design, which is the main purpose of the paper. We show that with an appropriate choice of multiscale spaces we can achieve an unconditional stability with respect to the contrast. This could provide computational savings as the time step in explicit methods is adversely affected by the contrast. We discuss some theoretical aspects of the proposed algorithms. Numerical results are presented.

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Section: Introductionmentioning

confidence: 99%

“…In [18], a general framework is proposed where the transition to simpler problems is carried out based on spatial decomposition of the solution. In [17], we combine the temporal splitting algorithms and spatial multiscale methods. We divide the spatial space into various components and use these subspaces in the temporal splitting.…”

Section: Introductionmentioning

confidence: 99%

Contrast-independent partially explicit time discretizations for multiscale flow problems

Chung,

Efendiev,

Leung

et al. 2021

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“…We remark that if the permeability tensor has the form κ = κ 11 (a ⊗ a) for some bounded function κ 11 and the conditions βκ 11 + ∇κ 11 • a ≥ 0 and a • n ∂Ω = 0 hold, where n ∂Ω is the unit outward normal vector to the boundary ∂Ω, then one can show that the continuous problem ( 7) is stable. See Appendix B for more details on the stability of the continuous problem (7). The stability analysis for the discretized convection-diffusion model with memory effects is challenging and will be one of our future works.…”

Section: Dememorizationmentioning

confidence: 99%

“…We consider two types of discretizations, namely implicit and partially explicit. The latter is designed for multiscale problems based on a solution decomposition strategy [7,8]. Because of the multiscale nature of the velocity and diffusion terms, one needs a very small time step when performing explicit discretization.…”

Section: Introductionmentioning

confidence: 99%

Nonlocal transport equations in multiscale media. Modeling, dememorization, and discretizations

Efendiev¹,

Leung²,

Li³

et al. 2022

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In this paper, we consider a class of convection-diffusion equations with memory effects. These equations arise as a result of homogenization or upscaling of linear transport equations in heterogeneous media and play an important role in many applications. First, we present a dememorization technique for these equations. We show that the convection-diffusion equations with memory effects can be written as a system of standard convection diffusion reaction equations. This allows removing the memory term and simplifying the computations. We consider a relation between dememorized equations and micro-scale equations, which do not contain memory terms. We note that dememorized equations differ from microscale equations and constitute a macroscopic model. Next, we consider both implicit and partially explicit methods. The latter is introduced for problems in multiscale media with high-contrast properties. Because of high-contrast, explicit methods are restrictive and require time steps that are very small (scales as the inverse of the contrast). We show that, by appropriately decomposing the space, we can treat only a few degrees of freedom implicitly and the remaining degrees of freedom explicitly. We present a stability analysis. Numerical results are presented that confirm our theoretical findings about partially explicit schemes applied to dememorized systems of equations.

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“…These algorithms are originally designed for multi-physics problems to separate various physics and reduce the computational cost. In recent works, we have proposed approaches for temporal splitting that uses multiscale spaces [21,20]. In [21], a general framework is proposed where the transition to simpler problems is carried out based on spatial decomposition of the solution.…”

Section: Introductionmentioning

confidence: 99%

Contrast-independent partially explicit time discretizations for multiscale wave problems

Chung,

Efendiev,

Leung

et al. 2021

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In this work, we design and investigate contrast-independent partially explicit time discretizations for wave equations in heterogeneous high-contrast media. We consider multiscale problems, where the spatial heterogeneities are at subgrid level and are not resolved. In our previous work [Chung, Efendiev, Leung, and Vabishchevich, Contrast-independent partially explicit time discretizations for multiscale flow problems, arXiv:2101.04863], we have introduced contrast-independent partially explicit time discretizations and applied to parabolic equations. The main idea of contrast-independent partially explicit time discretization is to split the spatial space into two components: contrast dependent (fast) and contrast independent (slow) spaces defined via multiscale space decomposition. Using this decomposition, our goal is further appropriately to introduce time splitting such that the resulting scheme is stable and can guarantee contrast-independent discretization under some suitable (reasonable) conditions. In this paper, we propose contrast-independent partially explicitly scheme for wave equations. The splitting requires a careful design. We prove that the proposed splitting is unconditionally stable under some suitable conditions formulated for the second space (slow). This condition requires some type of non-contrast dependent space and is easier to satisfy in the "slow" space. We present numerical results and show that the proposed methods provide results similar to implicit methods with the time step that is independent of the contrast.

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Temporal splitting algorithms for non-stationary multiscale problems

Cited by 12 publications

References 21 publications

Contrast-independent partially explicit time discretizations for multiscale flow problems

Contrast-independent partially explicit time discretizations for multiscale flow problems

Nonlocal transport equations in multiscale media. Modeling, dememorization, and discretizations

Contrast-independent partially explicit time discretizations for multiscale wave problems

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