High-order splitting methods for the incompressible Navier-Stokes equations

Karniadakis, George Em; Israeli, M.; Orszag, Steven A.

doi:10.1016/0021-9991(91)90007-8

Cited by 1,180 publications

(1,030 citation statements)

References 15 publications

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“…We employ the semi-implicit high-order fractional step method, which for the standard deterministic Navier-Stokes equations ( (25) and (26)) has the form [28]:…”

Section: Temporal Discretizationmentioning

confidence: 99%

Modeling uncertainty in flow simulations via generalized polynomial chaos

Xiu

Karniadakis

2003

Journal of Computational Physics

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1,304

729

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“…We employ the semi-implicit high-order fractional step method, which for the standard deterministic Navier-Stokes equations ( (25) and (26)) has the form [28]:…”

Section: Temporal Discretizationmentioning

confidence: 99%

Modeling uncertainty in flow simulations via generalized polynomial chaos

Xiu

Karniadakis

2003

Journal of Computational Physics

Self Cite

1,304

729

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“…The scheme is an adaptation of the sti✏y-stable velocity-correction projection scheme originally developed for solving the incompressible Navier-Stokes equations (Karniadakis et al, 1991). In this method, the velocity and pressure matrices are decoupled.…”

Section: High-order Splitting Schemementioning

confidence: 99%

A combined numerical and experimental framework for determining permeability properties of the arterial media

Comerford

Chooi

Nowak

et al. 2014

Biomech Model Mechanobiol

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The medial layer of the arterial wall may play an important role in the regulation of water and solute transport across the wall. In particular, a high medial resistance to transport could cause accumulation of lipid carrying molecules in the inner wall. In this study, the water transport properties of medial tissue were characterised in a numerical model utilising experimentally obtained data for the medial microstructure and the relative permeability of di↵erent constituents. For the model, a new solver for flow in porous materials, based on a high-order splitting scheme, was implemented in the spectral/hp element library nektar++ and validated. The data were obtained by immersing excised aortic bifurcations in a solution of fluorescent protein tracer and subsequently imaging them with a confocal microscope. Cuboidal regions of interest were selected in which the microstructure and relative permeability of di↵erent structures were transformed to a computational mesh. Impermeable objects were treated fictitiously in the numerical scheme. On this cube, a pressure drop was applied in the three coordinate directions and the principal components of the permeability tensor were determined. The reconstructed images demonstrated the arrangement of elastic lamellae and interspersed smooth muscle cells in rat aortic media; the distribution and alignment of the smooth muscle cells varied spatially within the extracellular matrix. The numerical simulations high-

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“…The linear stability of the high-dimensional cIIF methods can be analyzed by an approach similar to that for the one-dimensional system [2,13,8]. We test the linear stability on the the following linear equation (35) where q 1 and q 2 represent diffusions in the x and y directions, respectively.…”

Section: Stability Analysis Of Ciif Methodsmentioning

confidence: 99%

“…To apply the integration factor technique to the compact discretization form (10), we multiply (10) by exponential matrix e −At from the left, and e −Bt from the right to obtain (11) Integration of (11) over one time step from t n to t n+1 ≡ t n + Δt, where Δt is the time step, leads to (12) To construct a scheme of rth order truncation error, we approximate the integrand in (12), (13) using a (r − 1)th order Lagrange polynomial at a set of interpolation points t n+1 , t n ,…, t n+2−r : (14) where (15) The specific form of the polynomial (15) at low orders is listed in Table 1. In terms of (τ), (12) takes the form, (16) So the new r-th order implicit schemes are (17) where α 1 , α 0 , α −1 ,···, α −r+2 are coefficients calculated from the integrals of the polynomial in (τ), (18) In Table 2, the value of coefficients, α −j , for schemes of order up to four are listed.…”

Section: Two-dimensionsmentioning

confidence: 99%

Compact integration factor methods in high spatial dimensions

Nie

Wan

Zhang

et al. 2008

Journal of Computational Physics

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The dominant cost for integration factor (IF) or exponential time differencing (ETD) methods is the repeated vector-matrix multiplications involving exponentials of discretization matrices of differential operators. Although the discretization matrices usually are sparse, their exponentials are not, unless the discretization matrices are diagonal. For example, a two-dimensional system of N × N spatial points, the exponential matrix is of a size of N 2 × N 2 based on direct representations. The vector-matrix multiplication is of O(N 4 ), and the storage of such matrix is usually prohibitive even for a moderate size N. In this paper, we introduce a compact representation of the discretized differential operators for the IF and ETD methods in both two-and three-dimensions. In this approach, the storage and CPU cost are significantly reduced for both IF and ETD methods such that the use of this type of methods becomes possible and attractive for two-or three-dimensional systems. For the case of two-dimensional systems, the required storage and CPU cost are reduced to O(N 2 ) and O(N 3 ), respectively. The improvement on three-dimensional systems is even more significant. We analyze and apply this technique to a class of semi-implicit integration factor method recently developed for stiff reaction-diffusion equations. Direct simulations on test equations along with applications to a morphogen system in two-dimensions and an intra-cellular signaling system in three-dimensions demonstrate an excellent efficiency of the new approach.

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High-order splitting methods for the incompressible Navier-Stokes equations

Cited by 1,180 publications

References 15 publications

Modeling uncertainty in flow simulations via generalized polynomial chaos

Modeling uncertainty in flow simulations via generalized polynomial chaos

A combined numerical and experimental framework for determining permeability properties of the arterial media

Compact integration factor methods in high spatial dimensions

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