A conservative, second order, unconditionally stable artificial compression method

DeCaria, Victor; Layton, William; McLaughlin, M. R.

doi:10.1016/j.cma.2017.07.033

Cited by 47 publications

(43 citation statements)

References 22 publications

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“…These waves decrease solution accuracy, degrade stability and generate nonphysical solution behavior. This report gives an analysis of adding the Robert–Asselin (RA) time filter to a new artificial compression method presented in . The analysis herein, confirmed by numerical tests, shows that time filters increase accuracy and improve stability by reducing high frequency waves in p and ∇ · u .…”

Section: Introductionmentioning

confidence: 82%

“…In , it was shown that the method AC Step without RA filtering satisfies the following theorem.Theorem Consider the method AC Step without the filter step, that is, n ,

u^{n} = {\overset{u}{true}}^{n}

and

p^{n} = {\overset{p}{true}}^{n}

. Then, for any N ∈ ℕ and timestep k, the energy equality

E_{italicAC}^{N + 1 / 2} + italick ν {false\sum}_{n = 1}^{N} {(‖‖, \nabla ((), u^{n + 1} + u^{n - 1}))}^{2} = E_{italicAC}^{1 / 2} + 2 k {false\sum}_{n = 1}^{N} (f^{n}, u^{n + 1} + u^{n - 1})

is satisfied.…”

Section: Notation and Preliminariesmentioning

confidence: 98%

“…The specific artificial compressibility method from analyzed herein is an approximation to the continuum model

(1 - 2 β ∇∇ \cdot) u_{t} + u \cdot \nabla u + \frac{1}{2} (\nabla \cdot u) u - ν Δ u + \nabla p = f

2 α k^{2} p_{t} + \nabla \cdot u = 0,

where α , β are

O (1)

parameters. The equation of acoustic perturbations about a rest state ( u = 0, p = const ) of is obtained in the standard way: Taking ∇ · of the momentum equation,

\frac{\partial}{\partial t}

of the continuity equation, and simplifying gives

2 α k^{2} (1 - 2 β Δ) p_{italictt} - 2 α ν k^{2} Δ p_{t} - Δ p = \nabla \cdot (u \cdot \nabla u + \frac{1}{2} (\nabla \cdot u) u) .

…”

Section: Introductionmentioning

confidence: 99%

“…The acoustic wavespeed is then

c = {((), \sqrt{2 α} k)}^{- 1} \to \infty as k \to 0 .

The method, (AC Step) in Algorithm below, is a conservative, second‐order, unconditionally stable time‐discretization of , introduced in and derived from applied to the NSE.…”

Section: Introductionmentioning

confidence: 99%

“…Suppressing the subsidiary spatial discretization, we analyze the following algorithm. It is a combination of the artificial compression method of with the Robert–Asselin time filter applied to both the velocity and pressure.Algorithm (AC + RA). Let k > 0, t n = nk , and parameters α , β > 0.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

An analysis of the Robert–Asselin time filter for the correction of nonphysical acoustics in an artificial compression method

DeCaria

Layton

McLaughlin

2018

Numerical Methods Partial

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Section: Introductionmentioning

confidence: 82%

“…In , it was shown that the method AC Step without RA filtering satisfies the following theorem.Theorem Consider the method AC Step without the filter step, that is, n ,

u^{n} = {\overset{u}{true}}^{n}

and

p^{n} = {\overset{p}{true}}^{n}

. Then, for any N ∈ ℕ and timestep k, the energy equality

E_{italicAC}^{N + 1 / 2} + italick ν {false\sum}_{n = 1}^{N} {(‖‖, \nabla ((), u^{n + 1} + u^{n - 1}))}^{2} = E_{italicAC}^{1 / 2} + 2 k {false\sum}_{n = 1}^{N} (f^{n}, u^{n + 1} + u^{n - 1})

is satisfied.…”

Section: Notation and Preliminariesmentioning

confidence: 98%

“…The specific artificial compressibility method from analyzed herein is an approximation to the continuum model

(1 - 2 β ∇∇ \cdot) u_{t} + u \cdot \nabla u + \frac{1}{2} (\nabla \cdot u) u - ν Δ u + \nabla p = f

2 α k^{2} p_{t} + \nabla \cdot u = 0,

where α , β are

O (1)

parameters. The equation of acoustic perturbations about a rest state ( u = 0, p = const ) of is obtained in the standard way: Taking ∇ · of the momentum equation,

\frac{\partial}{\partial t}

of the continuity equation, and simplifying gives

2 α k^{2} (1 - 2 β Δ) p_{italictt} - 2 α ν k^{2} Δ p_{t} - Δ p = \nabla \cdot (u \cdot \nabla u + \frac{1}{2} (\nabla \cdot u) u) .

…”

Section: Introductionmentioning

confidence: 99%

“…The acoustic wavespeed is then

c = {((), \sqrt{2 α} k)}^{- 1} \to \infty as k \to 0 .

The method, (AC Step) in Algorithm below, is a conservative, second‐order, unconditionally stable time‐discretization of , introduced in and derived from applied to the NSE.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

An analysis of the Robert–Asselin time filter for the correction of nonphysical acoustics in an artificial compression method

DeCaria

Layton

McLaughlin

2018

Numerical Methods Partial

Self Cite

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An artificial compressibility ensemble algorithm for a stochastic Stokes‐Darcy model with random hydraulic conductivity and interface conditions

Jiang

Qiu

2019

Numerical Meth Engineering

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We propose and analyze an efficient ensemble algorithm with artificial compressibility for fast decoupled computation of multiple realizations of the stochastic Stokes-Darcy model with random hydraulic conductivity (including the one in the interface conditions), source terms, and initial conditions. The solutions are found by solving three smaller decoupled subproblems with two common time-independent coefficient matrices for all realizations, which significantly improves the efficiency for both assembling and solving the matrix systems. The fully coupled Stokes-Darcy system can be first decoupled into two smaller sub-physics problems by the idea of the partitioned time stepping, which reduces the size of the linear systems and allows parallel computing for each sub-physics problem. The artificial compressibility further decouples the velocity and pressure which further reduces storage requirements and improves computational efficiency. We prove the long time stability and the convergence for this new ensemble method. Three numerical examples are presented to support the theoretical results and illustrate the features of the algorithm, including the convergence, stability, efficiency and applicability.

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Conservative unconditionally stable decoupled numerical schemes for the Cahn–Hilliard–Navier–Stokes–Darcy–Boussinesq system

Chen

Han

Wang

et al. 2021

Numerical Methods Partial

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We propose two mass and heat energy conservative, unconditionally stable, decoupled numerical algorithms for solving the Cahn-Hilliard-Navier-Stokes-Darcy-Boussinesq system that models thermal convection of two-phase flows in superposed free flow and porous media. The schemes totally decouple the computation of the Cahn-Hilliard equation, the Darcy equations, the heat equation, the Navier-Stokes equations at each time step, and thus significantly reducing the computational cost. We rigorously show that the schemes are conservative and energy-law preserving. Numerical results are presented to demonstrate the accuracy and stability of the algorithms.

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A conservative, second order, unconditionally stable artificial compression method

Cited by 47 publications

References 22 publications

An analysis of the Robert–Asselin time filter for the correction of nonphysical acoustics in an artificial compression method

An analysis of the Robert–Asselin time filter for the correction of nonphysical acoustics in an artificial compression method

An artificial compressibility ensemble algorithm for a stochastic Stokes‐Darcy model with random hydraulic conductivity and interface conditions

Conservative unconditionally stable decoupled numerical schemes for the Cahn–Hilliard–Navier–Stokes–Darcy–Boussinesq system

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