Methods of Numerical Mathematics

Marchuk, G. I.

doi:10.1007/978-1-4615-9988-3

Cited by 293 publications

(235 citation statements)

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“…By Equation (50), we have shown the absolute stability of scheme Equation (21) in the sense of Neumann [25].…”

Section: <1mentioning

confidence: 96%

“…So, for sufficiently smooth functions, we make use of some general results (see, e.g. Reference [25] …”

Section: Difference Operatorsmentioning

confidence: 99%

“…Following a coinage from [25], we call this kind of difference operators antisymmetric (or conservative). The approximation of the nonlinear terms is the other formidable obstacle on the way of constructing of a successful numerical schemes for N-S equations.…”

Section: Difference Operatorsmentioning

confidence: 99%

“…Now, following Reference [25] we introduce a useful norm · H defined for any grid function ∈ which means ( ij ) is defined on the grids Equation (4) and satisfies the boundary conditions Equation (24), namely…”

Section: Kellogg's Lemmamentioning

confidence: 99%

“…The analytical properties and numerical treatments of the operator A are exhaustively discussed in Reference [25] for the case when (*a/*x) + (*b/*y) = 0.…”

mentioning

confidence: 99%

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An operator splitting scheme for the stream‐function formulation of unsteady Navier–Stokes equations

Christov

Tang

2006

Numerical Methods in Fluids

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SUMMARYA fictitious time is introduced into the unsteady equation of the stream function rendering it into a higher-order ultra-parabolic equation. The convergence with respect to the fictitious time (we call the latter 'internal iterations') allows one to obtain fully implicit nonlinear scheme in full time steps for the physical-time variable. For particular choice of the artificial time increment, the scheme in full time steps is of second-order of approximation. For the solution of the internal iteration, a fractional-step scheme is proposed based on the splitting of the combination of the Laplace, bi-harmonic and advection operators. A judicious choice for the time staggering of the different parts of the nonlinear advective terms allows us to prove that the internal iterations are unconditionally stable and convergent. We assess the number of operations needed per time step and show computational effectiveness of the proposed scheme. We prove that when the internal iterations converge, the scheme is second-order in physical time and space, nonlinear, implicit and absolutely stable. The performance of the scheme is demonstrated for the flow created by oscillatory motion of the lid of a square cavity. All theoretical findings are demonstrated practically.

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“…By Equation (50), we have shown the absolute stability of scheme Equation (21) in the sense of Neumann [25].…”

Section: <1mentioning

confidence: 96%

“…So, for sufficiently smooth functions, we make use of some general results (see, e.g. Reference [25] …”

Section: Difference Operatorsmentioning

confidence: 99%

Section: Difference Operatorsmentioning

confidence: 99%

Section: Kellogg's Lemmamentioning

confidence: 99%

“…The analytical properties and numerical treatments of the operator A are exhaustively discussed in Reference [25] for the case when (*a/*x) + (*b/*y) = 0.…”

mentioning

confidence: 99%

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An operator splitting scheme for the stream‐function formulation of unsteady Navier–Stokes equations

Christov

Tang

2006

Numerical Methods in Fluids

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Investigating the Behaviour of Two-Dimensional Finite Element Models of Compound Channel Flow

Bates

Anderson

Hervouet

et al. 1997

Earth Surf. Process. Landforms

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This paper describes results from a recent study concerning the numerical modelling of compound channel flow using two generalized two-dimensional finite element codes specifically adapted to floodplain studies: RMA-2 and TELEMAC-2D. By application to an 11km reach of the River Culum, Devon, UK, simulations are developed to investigate the impact of numerical technique, mesh resolution and topographic parameterization on model results. The research is shown to raise a number of issues concerning the construction, calibration and validation of two-dimensional finite element models for this flow problem.

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A Symmetrical Streamline Stabilization scheme for high advective transport

Wendland

Schmid

2000

Int. J. Numer. Anal. Meth. Geomech.

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SUMMARYAn overview of numerical techniques and previous investigations related to the solution of advectiondominated transport processes is presented. In addition a new Symmetrical Streamline Stabilization (S) scheme is introduced. The basis of the technique is to treat the transport equation in two steps. In the "rst step the dispersion part is approximated by a standard Galerkin approach, while in the second step the advection is approximated by a least-squares method. The two parts are reassembled, resulting in one system of equations. The resulting coe$cients' matrix is symmetric. Only half of a sparse matrix needs to be stored. Robust iterative algorithms for symmetrical systems of equations such as the preconditioned conjugate gradient method (PCG) can be successfully used. The new method leads to an implicit introduction of an &arti"cial di!usion' term. Solute transport with high Peclet and Courant numbers does not lead to oscillations due to an inherent upwind damping. The upwind e!ect acts only in #ow direction. The e$ciency of the new formulation in terms of accuracy and computation time is shown in comparison with the Galerkin approach for mesh parallel and mesh oblique high advective solute transport.

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Methods of Numerical Mathematics

Cited by 293 publications

References 0 publications

An operator splitting scheme for the stream‐function formulation of unsteady Navier–Stokes equations

An operator splitting scheme for the stream‐function formulation of unsteady Navier–Stokes equations

Investigating the Behaviour of Two-Dimensional Finite Element Models of Compound Channel Flow

A Symmetrical Streamline Stabilization scheme for high advective transport

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