A Fully Coupled Solver for Incompressible Navier–Stokes Equations using Operator Splitting

Marinova, Rossitza S.; Christov, Christo I.; Marinov, Tchavdar

doi:10.1080/1061856031000114300

Cited by 17 publications

(16 citation statements)

References 25 publications

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“…Since the unknown grid function is a vector of dimension K = (M − 1) × (N − 1), the size of the matrix is K × K which renders impractical the straightforward implicit scheme Equation (21). Although the matrix is sparse with only nine nontrivial diagonals, the actual width of the band is 4(N + 3) + 1 and that is what matters when treating the system as a system with a banded matrix.…”

Section: Splitting Scheme For the Internal Iterationsmentioning

confidence: 99%

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

Section: <1mentioning

confidence: 99%

“…Application to stationary N-S equation of the splitting method with artificial time is implemented in Reference [19], whereas featuring example used is the well known lid-driven cavity flow. The artificial-time approach with splitting for solving stationary N-S equations was developed further for the primitive variables and highly accurate and stable results were obtained for the lid-driven flow [20] and the flow behind a backward-facing step [21].…”

Section: Introductionmentioning

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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Section: Splitting Scheme For the Internal Iterationsmentioning

confidence: 99%

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

Section: <1mentioning

confidence: 99%

Section: Introductionmentioning

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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“…The operator-splitting algorithm (BRUSDAL et al, 1998;MARINOVA et al, 2003) is utilized to solve these Navier-Stokes (NS) equations. This numerical computing scheme uses different methods to solve different types of partial differential equations, respectively.…”

Section: Vol 165 2008 Modeling and Visualization Of Tsunamismentioning

confidence: 99%

Modeling and Visualization of Tsunamis

Zhang

Shi

Yuen

et al. 2008

Pure appl. geophys.

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Abstract-Modeling tsunami wave propagation is a very challenging numerical task, because it involves many facets: Such as the formation of various types of waves and the impingement of these waves on the coast. We will discuss the different levels of approximations made in numerical modeling of 2-D and 3-D tsunami waves and their relative difficulties. In this paper new attempts are proposed to evaluate the hazards of tsunami's and visualization of large-scale numerical results generated from tsunami simulations. Specialized low-level computer language, based on a parallel computing environment, is also employed here for generating FORTRAN source code for finite elements. This code can then be run very efficiently in parallel on distributed computing systems. We will also discuss the need to study tsunami waves with modern software and visualization hardware.

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“…A broad range of problems has been addressed by such methods, including the Navier-Stokes equation [23,24], the Hamilton-Jacobi equation [25,26], and advection-diffusion problems [27,28]. In these methods, the spatial differential operator is split into a sum of sub-operators that have simpler forms and can be handled easier.…”

mentioning

confidence: 99%

Advanced operator splitting-based semi-implicit spectral method to solve the binary phase-field crystal equations with variable coefficients

Tegze

Bansel

Tóth

et al. 2009

Journal of Computational Physics

101

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We present an efficient method to solve numerically the equations of dissipative dynamics of the binary phase-field crystal model proposed by Elder et al. [Phys. Rev. B 75, 064107 (2007)] characterized by variable coefficients. Using the operator splitting method, the problem has been decomposed into sub-problems that can be solved more efficiently. A combination of non-trivial splitting with spectral semi-implicit solution leads to sets of algebraic equations of diagonal matrix form. Extensive testing of the method has been carried out to find the optimum balance among errors associated with time integration, spatial discretization, and splitting. We show that our method speeds up the computations by orders of magnitude relative to the conventional explicit finite difference scheme, while the costs of the pointwise implicit solution per timestep remains low. Also we show that due to its numerical dissipation, finite differencing can not compete with spectral differencing in terms of accuracy. In addition, we demonstrate that our method can efficiently be parallelized for distributed memory systems, where an excellent scalability with the number of CPUs is observed.

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A Fully Coupled Solver for Incompressible Navier–Stokes Equations using Operator Splitting

Cited by 17 publications

References 25 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

Modeling and Visualization of Tsunamis

Advanced operator splitting-based semi-implicit spectral method to solve the binary phase-field crystal equations with variable coefficients

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