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A Multigrid Method for a Model of the implicit Immersed Boundary Equations

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Communications in Computational Physics 2012年第7期12卷 378-400页

作者： Robert D.Guy Bobby Philip Department of Mathematics University of California DavisDavisCA 95616USA Oak Ridge National Laboratory Oak RidgeTN 37831USA

Explicit time stepping schemes for the immersed boundary method require very small time steps in order to maintain *** the equations that arise from an implicit discretization is ***,several different approaches have been proposed,but a complete understanding of this problem is still emerging.A multigrid method is developed and explored for solving the equations in an implicittime discretization of a model of the immersed boundary *** model problem consists of a scalar Poisson equation with conformation-dependent singular forces on an immersed *** model does not include the inertial terms or the incompressibility *** method is more efficient than an explicit method,but the efficiency gain is *** multigrid method alone may not be an effective solver,but when used as a preconditioner for Krylov methods,the speed-up over the explicit-time method is *** example,depending on the constitutive law for the boundary force,with a time step 100 times larger than the explicit method,the implicit method is about 15-100 times more efficient than the explicit method.A very attractive feature of this method is that the efficiency of the multigrid preconditioned Krylov solver is shown to be independent of the number of immersed boundary points.

关键词： Preconditioning implicit methods fluid-structure interaction

Finite Elementθ-Schemes for the Acoustic Wave Equation

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Advances in Applied Mathematics and Mechanics 2011年第2期3卷 181-203页

作者： Samir Karaa Department of Mathematics and Statistics Sultan Qaboos UniversityP.O.Box 36Al-Khod 123MuscatSultanate of Oman

In this paper,we investigate the stability and convergence of a family of implicit finite difference schemes in time and Galerkin finite element methods in space for the numerical solution of the acoustic wave *** schemes cover the classical explicit second-order leapfrog scheme and the fourth-order accurate scheme in time obtained by the modified equation *** derive general stability conditions for the family of implicit schemes covering some well-known CFL *** error estimates are *** sufficiently smooth solutions,we demonstrate that the maximal error in the L^(2)-norm error over a finite time interval converges optimally as O(h^(p+1)+∆t^(s)),where p denotes the polynomial degree,s=2 or 4,h the mesh size,and∆t the time step.

关键词： Finite element methods discontinuous Galerkin methods wave equation implicit methods energy method stability condition optimal error estimates

On the Fully implicit Solution of a Phase-Field Model for Binary Alloy Solidification in Three Dimensions

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Advances in Applied Mathematics and Mechanics 2012年第6期4卷 665-684页

作者： Christopher E.Goodyer Peter K.Jimack Andrew M.Mullis Hongbiao Dong Yu Xie School of Computing University of LeedsWoodhouse LaneLeedsLS29JTUK School of Process Environmental and Materials EngineeringUniversity of LeedsWoodhouse LaneLeedsLS29JTUK Department of Engineering University of LeicesterUniversity RoadLeicesterLE17RHUK

A fully implicit numerical method,based upon a combination of adaptively refined hierarchical meshes and geometric multigrid,is presented for the simulation of binary alloy solidification in three space *** computational techniques are presented for a particular mathematical model,based upon the phase-field approach,however their applicability is of greater generality than for the specific phase-field model used *** particular,an implicit second order time discretization is combined with the use of second order spatial differences to yield a large nonlinear system of algebraic equations as each time *** is demonstrated that these equations may be solved reliably and efficiently through the use of a nonlinear multigrid scheme for locally refined *** effect this paper presents an extension of earlier research in two space dimensions(***.,225(2007),pp.1271–1287)to fully three-dimensional *** extension is validated against earlier two-dimensional results and against some of the limited results available in three dimensions,obtained using an explicit *** efficiency of the implicit approach and the multigrid solver are then demonstrated and some sample computational results for the simulation of the growth of dendrite structures are presented.

关键词： Phase-field simulations binary alloys mesh adaptivity implicit methods nonlinear multigrid

GRID-INDEPENDENT CONSTRUCTION OF MULTISTEP methods

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Journal of Computational Mathematics 2017年第5期35卷 672-692页

作者： Carmen Arevalo Gustaf Soderlind Numerical Analysis Centre for Mathematical Sciences Box 118 SE-22100 Lund Sweden.

A new polynomial formulation of variable step size linear multistep methods is pre- sented, where each k-step method is characterized by a fixed set of k - 1 or k parameters. This construction includes all methods of maximal order （p = k for stiff, and p = k ＋ 1 for nonstiff problems）. Supporting time step adaptivity by construction, the new formulation is not based on extending classical fixed step size methods; instead classical methods are obtained as fixed step size restrictions within a unified framework. The methods are imple- mented in MATLAB, with local error estimation and a wide range of step size controllers. This provides a platform for investigating and comparing different multistep method in realistic operational conditions. Computational experiments show that the new multi- step method construction and implementation compares favorably to existing software, although variable order has not yet been included.

关键词： Linear multistep methods Variable step size Adaptive step size Step sizecontrol Explicit methods implicit methods Nonstiff methods Stiff methods Initial valueproblems Ordinary differential equations Differential-algebraic equations Implementa-tion.

Magnetohydrodynamics with implicit Plasma Simulation

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Communications in Computational Physics 2008年第8期4卷 433-456页

作者： J.U.Brackbill G.Lapenta Los Alamos National Laboratory(retired) Los AlamosNM 87544USA Centrum voor Plasma-Astrofysica DepartementWiskundeKatholieke Universiteit LeuvenCelestijnenlaan 200B3001 LeuvenBelgium

We considerwhether implicit simulation techniques can be extended in time and space scales to magnetohydrodynamics without any change but the addition of *** goal is to couple fluid and kinetic models together for application to multi-scale *** a simulation framework,transition from one model to the other would occur not by a change of algorithm,but by a change of *** would greatly simplify the *** the way,we have found new ways to impose consistent boundary conditions for the field solver that result in charge and energy conservation,and establish that numerically-generated stochastic heating is the problem to *** an MHD-like problem,collisions are clearly necessary to reduce the stochastic *** collisions,the heating rate is *** collisions,the heating rate is significantly reduced.

关键词： Plasma simulation MHD particle-in-cell implicit methods

A Hybrid Backward Euler Control Volume Method to Solve the Concentration-Dependent Solid-State Diffusion Problem in Battery Modeling

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Journal of Applied Mathematics and Physics 2020年第6期8卷 1066-1080页

作者： Kudakwashe Chayambuka Grietus Mulder Dmitri L. Danilov Peter H. L. Notten VITO Mol Belgium EnergyVille Genk Belgium Eindhoven University of Technology Eindhoven Netherlands Forschungszentrum Jülich Fundamental Electrochemistry (IEK-9) Jülich Germany University of Technology Sydney Broadway Sydney NSW Australia

Several efficient analytical methods have been developed to solve the solid-state diffusion problem, for constant diffusion coefficient problems. However, these methods cannot be applied for concentration-dependent diffusion coefficient problems and numerical methods are used instead. Herein, grid-based numerical methods derived from the control volume discretization are presented to resolve the characteristic nonlinear system of partial differential equations. A novel hybrid backward Euler control volume (HBECV) method is presented which requires only one iteration to reach an implicit solution. The HBECV results are shown to be stable and accurate for a moderate number of grid points. The computational speed and accuracy of the HBECV, justify its use in battery simulations, in which the solid-state diffusion coefficient is a strong function of the concentration.

关键词： Solid-State Diffusion implicit methods Backward Euler