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1.
This paper presents a method for solving the linear semi-implicit
immersed boundary equations which avoids the severe time step restriction
presented by explicit-time methods. The Lagrangian variables are
eliminated via a Schur complement to form a purely Eulerian saddle
point system, which is preconditioned by a projection operator and
then solved by a Krylov subspace method. From the viewpoint of
projection methods, we derive an ideal preconditioner for the
saddle point problem and compare the efficiency of a number of simpler
preconditioners that approximate this perfect one.
For low Reynolds number and high stiffness,
one particular projection preconditioner
yields an efficiency improvement of the explicit IB method
by a factor around thirty.
Substantial speed-ups over explicit-time method are
achieved for Reynolds number below 100.
This speedup increases as the Eulerian grid size and/or the Reynolds number are further reduced. 相似文献
2.
Faisal A. Fairag Andrew J. Wathen 《Numerical Methods for Partial Differential Equations》2012,28(3):888-898
Iterative methods of Krylov‐subspace type can be very effective solvers for matrix systems resulting from partial differential equations if appropriate preconditioning is employed. We describe and test block preconditioners based on a Schur complement approximation which uses a multigrid method for finite element approximations of the linearized incompressible Navier‐Stokes equations in streamfunction and vorticity formulation. By using a Picard iteration, we use this technology to solve fully nonlinear Navier‐Stokes problems. The solvers which result scale very well with problem parameters. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011 相似文献
3.
Mohammed Seaïd 《Applied Numerical Mathematics》2009,59(3-4):754-768
We present an Eulerian–Lagrangian method for the numerical solution of coupled parabolic-hyperbolic equations. The method combines advantages of the modified method of characteristics to accurately solve the hyperbolic equations with an Eulerian method to discretize the parabolic equations. The Runge–Kutta Chebyshev scheme is used for the time integration. The implementation of the proposed method differs from its Eulerian counterpart in the fact that it is applied during each time step, along the characteristic curves rather than in the time direction. The focus is on constructing explicit schemes with a large stability region to solve coupled radiation hydrodynamics models. Numerical results are presented for two test examples in coupled convection-radiation and conduction–radiation problems. 相似文献
4.
Gérard Gallice 《Numerische Mathematik》2003,94(4):673-713
Summary. Using the concept of simple Riemann solvers, we present entropic and positive Godunov-type schemes preserving contact discontinuities for both Lagrangian and Eulerian systems of gas dynamics and magnetohydrodynamics (MHD). On the one hand, for the Lagrangian form, we develop positive and entropic Riemann solvers which can be considered as a natural extension of Roe's solvers in which the sound speed is relaxed. On the other hand, for the Eulerian form, we are able to construct by two ways Godunov-type schemes based on Lagrangian simple Riemann solvers. The first method establishes a relation between the jump of the intermediate states and the second one between the intermediate states themselves. 相似文献
5.
Mohamed Al‐Lawatia 《Numerical Methods for Partial Differential Equations》2012,28(5):1481-1496
We develop a mass conservative Eulerian‐Lagrangian control volume scheme (ELCVS) for the solution of the transient advection‐diffusion equations in two space dimensions. This method uses finite volume test functions over the space‐time domain defined by the characteristics within the framework of the class of Eulerian‐Lagrangian localized adjoint characteristic methods (ELLAM). It, therefore, maintains the advantages of characteristic methods in general, and of this class in particular, which include global mass conservation as well as a natural treatment of all types of boundary conditions. However, it differs from other methods in that class in the treatment of the mass storage integrals at the previous time step defined on deformed Lagrangian regions. This treatment is especially attractive for orthogonal rectangular Eulerian grids composed of block elements. In the algorithm, each deformed region is approximated by an eight‐node region with sides drawn parallel to the Eulerian grid, which significantly simplifies the integration compared with the approach used in finite volume ELLAM methods, based on backward tracking, while retaining local mass conservation at no additional expenses in terms of accuracy or CPU consumption. This is verified by numerical tests which show that ELCVS performs as well as standard finite volume ELLAM methods, which have previously been shown to outperform many other well‐received classes of numerical methods for the equations considered. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2012 相似文献
6.
Hong Wang Mohamed Al‐Lawatia 《Numerical Methods for Partial Differential Equations》2001,17(6):565-583
We develop an Eulerian‐Lagrangian substructuring domain decomposition method for the solution of unsteady‐state advection‐diffusion transport equations. This method reduces to an Eulerian‐Lagrangian scheme within each subdomain and to a type of Dirichlet‐Neumann algorithm at subdomain interfaces. The method generates accurate and stable solutions that are free of artifacts even if large time‐steps are used in the simulation. Numerical experiments are presented to show the strong potential of the method. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17:565–583, 2001 相似文献
7.
Mixed discrete least squares meshfree (MDLSM) method has been developed as a truly meshfree method and successfully used to solve single-phase flow problems. In the MDLSM, a residual functional is minimized in terms of the nodal unknown parameters leading to a set of positive-definite system of algebraic equations. The functional is defined using a least square summation of the residual of the governing partial differential equations and its boundary conditions at all nodal points discretizing the computational domain. Unlike the discrete least squares meshfree (DLSM) which uses an irreducible form of the governing equations, the MDLSM uses a mixed form of the original governing equations allowing for direct calculation of the gradients leading to more accurate computational results. In this study, an Eulerian–Lagrangian MDLSM method is proposed to solve incompressible multiphase flow problems. In the Eulerian step, the MDLSM method is used to solve the governing phase averaged Navier–Stokes equations discretized at fixed nodal points to get the velocity and pressure fields. A Lagrangian based approach is then used to track different flow phases indexed by a set of marker points. The velocities of marker points are calculated by interpolating the velocity of fixed nodal points using a kernel approximation, which are then used to move the marker points as Lagrangian particles to track phases. To avoid unphysical clustering and dispersing of the marker points, as a common drawback of Lagrangian point tracking methods, a new approach is proposed to smooth the distribution of marker points. The hybrid Eulerian and Lagrangian characteristics of the approach used here provides clear advantages for the proposed method. Since the nodal points are static on the Eulerian step, the time-consuming moving least squares (MLS) approximation is implemented only once making the proposed method more efficient than corresponding fully Lagrangian methods. Furthermore, phases can be simply tracked using the Lagrangian phase tracking procedure. Efficiency of the proposed MDLSM multiphase method is evaluated using several benchmark problems and the results are presented and discussed. The results verify the efficiency and accuracy of the proposed method for solving multiphase flow problems. 相似文献
8.
Hong Wang 《Numerical Methods for Partial Differential Equations》1998,14(6):739-780
A family of ELLAM (Eulerian–Lagrangian localized adjoint method) schemes is developed and analyzed for linear advection-diffusion-reaction transport partial differential equations with any combination of inflow and outflow Dirichlet, Neumann, or flux boundary conditions. The formulation uses space-time finite elements, with edges oriented along Lagrangian flow paths, in a time–stepping procedure, where space-time test functions are chosen to satisfy a local adjoint condition. This allows Eulerian–Lagrangian concepts to be applied in a systematic mass-conservative manner, yielding numerical schemes defined at each discrete time level. Optimal-order error estimates and superconvergence results are derived. Numerical experiments are performed to verify the theoretical estimates. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 739–780, 1998 相似文献
9.
Kaixin Wang Hong Wang Mohamed Al‐Lawatia 《Numerical Methods for Partial Differential Equations》2007,23(6):1343-1367
We develop an Eulerian‐Lagrangian discontinuous Galerkin method for time‐dependent advection‐diffusion equations. The derived scheme has combined advantages of Eulerian‐Lagrangian methods and discontinuous Galerkin methods. The scheme does not contain any undetermined problem‐dependent parameter. An optimal‐order error estimate and superconvergence estimate is derived. Numerical experiments are presented, which verify the theoretical estimates.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2007 相似文献
10.
Bobby Philip Timothy P. Chartier 《Journal of Computational and Applied Mathematics》2012,236(9):2277-2297
This paper will present a new method of adaptively constructing block iterative methods based on Local Sensitivity Analysis (LSA). The method can be used in the context of geometric and algebraic multigrid methods for constructing smoothers, and in the context of Krylov methods for constructing block preconditioners. It is suitable for both constant and variable coefficient problems. Furthermore, the method can be applied to systems arising from both scalar and coupled system partial differential equations (PDEs), as well as linear systems that do not arise from PDEs. The simplicity of the method will allow it to be easily incorporated into existing multigrid and Krylov solvers while providing a powerful tool for adaptively constructing methods tuned to a problem. 相似文献
11.
The Stochastic Field Method was introduced in the field of combustion and represents a Eulerian Monte-Carlo technique. It was first transfered to the field of multiphase flow in [1] facing a problem in nuclear technology. Within this work it is applied to cavitating flows in the automotive sector. Both phases, the continuous and the dispersed phase, are observed using a Eulerian perspective which fits the architecture of CFD solvers resulting in efficient computations. Instead of solving the behaviour of individual bubbles, as in Lagrangian Particle Methods, variations of volume fractions of the dispersed phase in each computational mesh cell are considered. A stochastic term is introduced into the transport equations for the volume fractions of the dispersed phase via a Wiener process. Thus sampling over several time steps provides the probability density function of volume fraction in each mesh cell. The implementation is carried out by compiling user-functions to a commercial CFD code. A test case representing an injection system in an automotive application is presented. Moreover strategies to implement the effect of coalescence are shown. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
12.
Kaixin Wang Hong Wang Mohamed Al‐Lawatia 《Numerical Methods for Partial Differential Equations》2010,26(3):561-595
We develop a CFL‐free, explicit characteristic interior penalty scheme (CHIPS) for one‐dimensional first‐order advection‐reaction equations by combining a Eulerian‐Lagrangian approach with a discontinuous Galerkin framework. The CHIPS method retains the numerical advantages of the discontinuous Galerkin methods as well as characteristic methods. An optimal‐order error estimate in the L2 norm for the CHIPS method is derived and numerical experiments are presented to confirm the theoretical estimates. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010 相似文献
13.
H. J. van Roessel W. H. Hui 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》1989,40(5):677-710
A new Lagrangian formulation for steady three dimensional inviscid flow over rigid bodies is developed. First, the continuity equation is eliminated by the use of two stream functions. This is followed by a transformation to new independent variables, two of which are these stream functions and the third one is a Lagrangian time distinct from the Eulerian time. This Lagrangian formulation with the use of Lagrangian time requires only three independent variables and allows the free boundary problem of flow with shock wave to be rendered a fixed boundary one thereby making it easier to solve. In the Newtonian limit the governing equations reduce to the geodesic equations of the body surface. For flow past two-dimensional bodies, bodies of revolution, and conical bodies and wings, the problems are solved to within quadrature. All known Newtonian flow solutions are found to be special cases of the present theory. 相似文献
14.
Zheng Wang Mohamed Al‐Lawatia Hong Wang 《Numerical Methods for Partial Differential Equations》2007,23(2):293-329
This article is devoted to the development and application of an Eulerian‐Lagrangian method (ELM) for the solution of the Black‐Scholes partial differential equation for the valuation of European option contracts. This method fully utilizes the transient behavior of the governing equations and generates very accurate option's fair values and their derivatives also known as option Greeks, even if coarse spatial grids and large time steps are used. Numerical experiments on two standard option contracts are presented which show that the ELM method (favorably) compares in terms of accuracy and efficiency to many other well‐perceived methods. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 23: 293–329, 2007 相似文献
15.
We developed a nonconventional Eulerian‐Lagrangian single‐node collocation method for transient advection‐diffusion transport partial differential equations in multiple space dimensions. This method greatly reduces the number of unknowns in conventional collocation method, generates accurate numerical solutions, and allows large time steps to be used in numerical simulations. We perform numerical experiments to show the strong potential of the method. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 284–301, 2004 相似文献
16.
We consider numerical methods for the incompressible Reynolds averaged Navier–Stokes equations discretized by finite difference
techniques on non-staggered grids in body-fitted coordinates. A segregated approach is used to solve the pressure–velocity
coupling problem. Several iterative pressure linear solvers including Krylov subspace and multigrid methods and their combination
have been developed to compare the efficiency of each method and to design a robust solver. Three-dimensional numerical experiments
carried out on scalar and vector machines and performed on different fluid flow problems show that a combination of multigrid
and Krylov subspace methods is a robust and efficient pressure solver.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
17.
18.
We analyze the convergence rate of an asynchronous space decomposition method for constrained convex minimization in a reflexive Banach space. This method includes as special cases parallel domain decomposition methods and multigrid methods for solving elliptic partial differential equations. In particular, the method generalizes the additive Schwarz domain decomposition methods to allow for asynchronous updates. It also generalizes the BPX multigrid method to allow for use as solvers instead of as preconditioners, possibly with asynchronous updates, and is applicable to nonlinear problems. Applications to an overlapping domain decomposition for obstacle problems are also studied. The method of this work is also closely related to relaxation methods for nonlinear network flow. Accordingly, we specialize our convergence rate results to the above methods. The asynchronous method is implementable in a multiprocessor system, allowing for communication and computation delays among the processors.
19.
Wurigen Bo & Mikhail Shashkov 《数学研究》2015,48(2):125-167
We present a new R-adaptive Arbitrary Lagrangian Eulerian (ALE) method,
based on the reconnection-based ALE - ReALE methodology [5, 41, 42]. The main elements
in a standard ReALE method are: an explicit Lagrangian phase on an arbitrary
polygonal (in 2D) mesh, followed by a rezoning phase in which a new grid is defined,
and a remapping phase in which the Lagrangian solution is transferred onto the new
grid. The rezoned mesh is smoothed by using one or several steps toward centroidal
Voronoi tessellation, but it is not adapted to the solution in any way. We present a
new R-adaptive ReALE method (R-ReALE, where R stands for Relocation). The new
method is based on the following design principles. First, a monitor function (or error
indicator) based on Hessian of some flow parameter(s), is utilized. Second, the new
algorithm uses the equidistribution principle with respect to the monitor function as
criterion for defining an adaptive mesh. Third, centroidal Voronoi tessellation is used
for the construction of the adaptive mesh. Fourth, we modify the raw monitor function
(scale it to avoid extremely small and large cells and smooth it to create a smooth
mesh), in order to utilize theoretical results related to centroidal Voronoi tessellation.
In the R-ReALE method, the number of mesh cells is chosen at the beginning of the calculation
and does not change with time, but the mesh is adapted according to the modified
monitor function during the rezone stage at each time step. We present all details
required for implementation of the new adaptive R-ReALE method and demonstrate
its performance relative to standard ReALE method on a series of numerical examples. 相似文献
20.
Akira IMAKURA 《数学研究及应用》2021,41(1):87-98
Multigrid methods are widely used and well studied for linear solvers and preconditioners of Krylov subspace methods. The multigrid method is one of the most powerful approaches for solving large scale linear systems;however, it may show low parallel efficiency on coarse grids. There are several kinds of research on this issue. In this paper, we intend to overcome this difficulty by proposing a novel multigrid algorithm that has multiple grids on each layer.Numerical results indicate that the proposed method shows a better convergence rate compared with the existing multigrid method. 相似文献