A semi-implicit augmented IIM for Navier–Stokes equations with open, traction, or free boundary conditions

Li, Zhilin; Xiao, Li; Cai, Qiong; Zhao, Hongkai; Luo, Ray

doi:10.1016/j.jcp.2015.05.003

Cited by 10 publications

(11 citation statements)

References 47 publications

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“…During the numerical solution, the procedure to advance from a time step t k to the next one t k+1 includes two steps [20]. The first is to solve (15) with (16) for the velocity using AIIM.…”

Section: Navier-stokes Equations On Irregular Domains With Open and Tmentioning

confidence: 99%

Effective matrix-free preconditioning for the augmented immersed interface method

Xia

2015

Journal of Computational Physics

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Please cite this article in press as: J. Xia et al., Effective matrix-free preconditioning for the augmented immersed interface method, J. Comput. Phys. (2015), http://dx. AbstractWe present effective and efficient matrix-free preconditioning techniques for the augmented immersed interface method (AIIM). AIIM has been developed recently and is shown to be very effective for interface problems and problems on irregular domains. GMRES is often used to solve for the augmented variable(s) associated with a Schur complement A in AIIM that is defined along the interface or the irregular boundary. The efficiency of AIIM relies on how quickly the system for A can be solved. For some applications, there are substantial difficulties involved, such as the slow convergence of GMRES (particularly for free boundary and moving interface problems), and the inconvenience in finding a preconditioner (due to the situation that only the products of A and vectors are available). Here, we propose matrix-free structured preconditioning techniques for AIIM via adaptive randomized sampling, using only the products of A and vectors to construct a hierarchically semiseparable matrix approximation to A. Several improvements over existing schemes are shown so as to enhance the efficiency and also avoid potential instability. The significance of the preconditioners includes: (1) they do not require the entries of A or the multiplication of A T with vectors; (2) constructing the preconditioners needs only O(log N) matrix-vector products and O(N) storage, where N is the size of A; (3) applying the preconditioners needs only O(N) flops;(4) they are very flexible and do not require any a priori knowledge of the structure of A. The preconditioners are observed to significantly accelerate the convergence of GMRES, with heuristical justifications of the effectiveness. Comprehensive tests on several important applications are provided, such as Navier-Stokes equations on irregular domains with traction boundary conditions, interface problems in incompressible flows, mixed boundary problems, and free boundary problems. The preconditioning techniques are also useful for several other problems and methods.

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Section: Navier-stokes Equations On Irregular Domains With Open and Tmentioning

confidence: 99%

Effective matrix-free preconditioning for the augmented immersed interface method

Xia

2015

Journal of Computational Physics

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“…Our main results are as follows:

We discretize the Stokes equation using a ghost fluid finite difference scheme. We show that our scheme is second-order accurate;

We refine the method proposed in [14, 27] to design artificial boundary conditions on the boundary of computational domain, allowing the change of solute volume;We couple the level-set method with our Stokes solver to track the motion of solute-solvent interface. Our method captures both dry and wet states for some simple model systems.…”

Section: Introductionmentioning

confidence: 99%

“…In several recent works [13, 14, 26, 27], the authors have initiated the development of a fluid mechanics approach to treat the solvent fluid in molecular systems. The key features of such a new approach include: (1) the aqueous solvent (i.e., water or salted water) is treated as an incompressible fluid and its motion is by the Stokes or Navier-Stokes equation; (2) the solute pressure is simply described by the ideal-gas law; (3) the electrostatic interactions are modeled by the Poisson or Poisson–Boltzmann equation; and (4) all viscous force, electrostatic force, and vdW force are balanced on the solute-solvent interface that moves with solvent velocity.…”

Section: Introductionmentioning

confidence: 99%

“…With the same deterministic model, Li et al [13] study the linear stability of a cylindrical solute-solvent interface, and conclude that the viscosity can modify the critical wavelength of such stabilities. Luo et al [14, 27] make a connection of solvent fluid mechanics model with statistical mechanics theory. They also develop numerical methods to solve the solvent fluid equations.…”

Section: Introductionmentioning

confidence: 99%

“…We refine the method proposed in [14, 27] to design artificial boundary conditions on the boundary of computational domain, allowing the change of solute volume;…”

Section: Introductionmentioning

confidence: 99%

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Numerical Treatment of Stokes Solvent Flow and Solute–Solvent Interfacial Dynamics for Nonpolar Molecules

et al. 2015

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We design and implement numerical methods for the incompressible Stokes solvent flow and solute-solvent interface motion for nonpolar molecules in aqueous solvent. The balance of viscous force, surface tension, and van der Waals type dispersive force leads to a traction boundary condition on the solute-solvent interface. To allow the change of solute volume, we design special numerical boundary conditions on the boundary of a computational domain through a consistency condition. We use a finite difference ghost fluid scheme to discretize the Stokes equation with such boundary conditions. The method is tested to have a second-order accuracy. We combine this ghost fluid method with the level-set method to simulate the motion of the solute-solvent interface that is governed by the solvent fluid velocity. Numerical examples show that our method can predict accurately the blow up time for a test example of curvature flow and reproduce the polymodal (e.g., dry and wet) states of hydration of some simple model molecular systems.

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An efficient second‐order poisson–boltzmann method

Wei

Luo

2019

J Comput Chem

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Immersed interface method (IIM) is a promising high-accuracy numerical scheme for the Poisson-Boltzmann model that has been widely used to study electrostatic interactions in biomolecules. However, the IIM suffers from instability and slow convergence for typical applications. In this study, we introduced both analytical interface and surface regulation into IIM to address these issues. The analytical interface setup leads to better accuracy and its convergence closely follows a quadratic manner as predicted by theory. The surface regulation further speeds up the convergence for nontrivial biomolecules. In addition, uncertainties of the numerical energies for tested systems are also reduced by about half. More interestingly, the analytical setup significantly improves the linear solver efficiency and stability by generating more precise and better-conditioned linear systems. Finally, we implemented the bottleneck linear system solver on GPUs to further improve the efficiency of the method, so it can be widely used for practical biomolecular applications.

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A semi-implicit augmented IIM for Navier–Stokes equations with open, traction, or free boundary conditions

Cited by 10 publications

References 47 publications

Effective matrix-free preconditioning for the augmented immersed interface method

Effective matrix-free preconditioning for the augmented immersed interface method

Numerical Treatment of Stokes Solvent Flow and Solute–Solvent Interfacial Dynamics for Nonpolar Molecules

An efficient second‐order poisson–boltzmann method

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