An iteration free backward semi-Lagrangian scheme for solving incompressible Navier–Stokes equations

Piao, Xiangfan; Bu, Sunyoung; Bak, Soyoon; Kim, Philsu

doi:10.1016/j.jcp.2014.11.040

Cited by 20 publications

(14 citation statements)

References 38 publications

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“…ECM2 :

\begin{array}{l} π_{j} ((), t_{n - 1}) & \approx x_{j} - 2 Δ italictu ((),, t_{n}, x_{j}) + 2 Δ t \frac{u ((),, t_{n}, x_{j}) - u ((),, t_{n}, x_{j} - Δ italictu ((),, t_{n}, x_{j}))}{1 + Δ {tu}_{x} ((),, t_{n}, x_{j} - Δ italictu ((),, t_{n}, x_{j}))}, \\ π_{j} ((), t_{n}) & \approx \frac{1}{4} ((),, x_{i} + 3 π_{j} ((), t_{n - 1}) + 2 Δ italictu true(t_{n - 1}, π_{j} ((), t_{n - 1}) true)) . \end{array}

…”

Section: New Characteristic‐tracking Methodsunclassified

“…In this subsection, we introduce a new qth‐order ( q = 2, 3) iteration‐free method based on the error‐correction scheme and implicit type multistep methods to find the approximate values

π_{j}^{n + 1, n - κ}

of the departure points π ( x j , t n +1 ; t n − κ ) ( κ = 0, …, 1 − q ) by solving (8). The q th‐order ( q = 2, 3) scheme for (8) begins with the construction of a modified Euler's polygon y j ( t ) as follows:

\begin{array}{l} y_{j} ((), t) ≔ normalmin ({},, normalmax ({},, {\overset{true^}{y}}_{j} ((), t), x_{0}), x_{J}), ((), Dirichlet boundary case) \\ y_{j} ((), t) ≔ ({, \begin{array}{l} {\overset{true^}{y}}_{j} ((), t) & {\overset{true^}{y}}_{j} ((), t) \in ([],, x_{0}, x_{J}), \\ x_{J} - ((), x_{0} - {\overset{true^}{y}}_{j} ((), t)) & {\overset{true^}{y}}_{j} ((), t) < x_{0}, ((), Periodic boundary case) \\ x_{0} + ((), {\overset{true^}{y}}_{j} ((), t) - x_{J}) & {\overset{true^}{y}}_{j} ((), t) > x_{J}, \end{array}) \end{array}

where

{\overset{y}{true}}_{j} (t) ≔ x_{j} + (t - t_{n + 1}) U_{j}^{n} .

Hereafter, we represent y j ( t ) as

{\overset{y}{true}}_{j} (t)

for simplicity.…”

Section: New Characteristic‐tracking Methodsmentioning

confidence: 99%

“…In this subsection, we introduce a new qth-order (q = 2, 3) iteration-free method based on the error-correction scheme [4,14,21] and implicit type multistep methods to find the approximate values n+1,n− j of the departure points (x j , t n+1 ;t n− ) ( = 0, …, 1 − q) by solving (8). The qth-order (q = 2, 3) scheme for (8) begins with the construction of a modified Euler's polygon y j (t) as follows:…”

Section: Derivation Of the Proposed Characteristic-tracking Schemesmentioning

confidence: 99%

“…In particular, the backward semi-Lagrangian method (BSLM) has a good stability in implicitly solving problems the characteristic curves of particles in the opposite direction involving a large time-step size. Because of these advantages, BSLM has been widely used to deal with advection-diffusion type problems [3][4][5][6][7][8][9] as much as the pure advection problem which also includes an advection problem induced by the splitting method.…”

Section: Introductionmentioning

confidence: 99%

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High‐order characteristic‐tracking strategy for simulation of a nonlinear advection–diffusion equation

Bak

2019

Numerical Methods Partial

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In this study, new high‐order backward semi‐Lagrangian methods are developed to solve nonlinear advection–diffusion type problems, which are realized using high‐order characteristic‐tracking strategies. The proposed characteristic‐tracking strategies are second‐order L‐stable and third‐order L(α)‐stable methods, which are based on a classical implicit multistep method combined with a error‐correction method. We also use backward differentiation formulas and the fourth‐order finite‐difference scheme for diffusion problem discretization in the temporal and spatial domains, respectively. To demonstrate the adaptability and efficiency of these time‐discretization strategies, we apply these methods to nonlinear advection–diffusion type problems such as the viscous Burgers' equation. Through simulations, not only the temporal and spatial accuracies are numerically evaluated but also the proposed methods are shown to be superior to the compared existing characteristic‐tracking methods under the same rates of convergence in terms of accuracy and efficiency. Finally, we have shown that the proposed method well preserves the energy and mass when the viscosity coefficient becomes zero.

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“…ECM2 :

\begin{array}{l} π_{j} ((), t_{n - 1}) & \approx x_{j} - 2 Δ italictu ((),, t_{n}, x_{j}) + 2 Δ t \frac{u ((),, t_{n}, x_{j}) - u ((),, t_{n}, x_{j} - Δ italictu ((),, t_{n}, x_{j}))}{1 + Δ {tu}_{x} ((),, t_{n}, x_{j} - Δ italictu ((),, t_{n}, x_{j}))}, \\ π_{j} ((), t_{n}) & \approx \frac{1}{4} ((),, x_{i} + 3 π_{j} ((), t_{n - 1}) + 2 Δ italictu true(t_{n - 1}, π_{j} ((), t_{n - 1}) true)) . \end{array}

…”

Section: New Characteristic‐tracking Methodsunclassified

“…In this subsection, we introduce a new qth‐order ( q = 2, 3) iteration‐free method based on the error‐correction scheme and implicit type multistep methods to find the approximate values

π_{j}^{n + 1, n - κ}

\begin{array}{l} y_{j} ((), t) ≔ normalmin ({},, normalmax ({},, {\overset{true^}{y}}_{j} ((), t), x_{0}), x_{J}), ((), Dirichlet boundary case) \\ y_{j} ((), t) ≔ ({, \begin{array}{l} {\overset{true^}{y}}_{j} ((), t) & {\overset{true^}{y}}_{j} ((), t) \in ([],, x_{0}, x_{J}), \\ x_{J} - ((), x_{0} - {\overset{true^}{y}}_{j} ((), t)) & {\overset{true^}{y}}_{j} ((), t) < x_{0}, ((), Periodic boundary case) \\ x_{0} + ((), {\overset{true^}{y}}_{j} ((), t) - x_{J}) & {\overset{true^}{y}}_{j} ((), t) > x_{J}, \end{array}) \end{array}

where

{\overset{y}{true}}_{j} (t) ≔ x_{j} + (t - t_{n + 1}) U_{j}^{n} .

Hereafter, we represent y j ( t ) as

{\overset{y}{true}}_{j} (t)

for simplicity.…”

Section: New Characteristic‐tracking Methodsmentioning

confidence: 99%

Section: Derivation Of the Proposed Characteristic-tracking Schemesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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High‐order characteristic‐tracking strategy for simulation of a nonlinear advection–diffusion equation

Bak

2019

Numerical Methods Partial

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show abstract

“…Among them, the backward semi-Lagrangian method (BSL) describes the movement of particles in the Lagrangian description and resets the positions of the particles to the Eulerian grid point at each time level to reduce the accumulation and collision of particles. Moreover, it is well known that BSL exhibits good stability, allowing the use of larger temporal steps than the spatial grid size by solving equations implicitly along characteristic curves of particles in the reverse direction to time evolution [3,5,30,31]. Therefore, this paper proposes a robust high-order BSL algorithm for solving nonlinear advection-diffusion-dispersion equations.…”

Section: Introductionmentioning

confidence: 99%

Simulation of advection–diffusion–dispersion equations based on a composite time discretization scheme

Bak

2020

Adv Differ Equ

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In this work, we develop a high-order composite time discretization scheme based on classical collocation and integral deferred correction methods in a backward semi-Lagrangian framework (BSL) to simulate nonlinear advection-diffusion-dispersion problems. The third-order backward differentiation formula and fourth-order finite difference schemes are used in temporal and spatial discretizations, respectively. Additionally, to evaluate function values at non-grid points in BSL, the constrained interpolation profile method is used. Several numerical experiments demonstrate the efficiency of the proposed techniques in terms of accuracy and computation costs, compare with existing departure traceback schemes.

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A mixed approximate method to simulate generalized Hirota–Satsuma coupled KdV equations

Bak

2022

Comp. Appl. Math.

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An iteration free backward semi-Lagrangian scheme for solving incompressible Navier–Stokes equations

Cited by 20 publications

References 38 publications

High‐order characteristic‐tracking strategy for simulation of a nonlinear advection–diffusion equation

High‐order characteristic‐tracking strategy for simulation of a nonlinear advection–diffusion equation

Simulation of advection–diffusion–dispersion equations based on a composite time discretization scheme

A mixed approximate method to simulate generalized Hirota–Satsuma coupled KdV equations

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