Cell-centered Finite Difference Scheme Research Articles

Analysis of the CCFD method for MC-based image denoising problems

Image denoising using mean curvature leads to the problem of solving a nonlinear fourth-order integro-differential equation. The nonlinear fourth-order term comes from the mean curvature regularization functional. In this paper, we treat this high-order nonlinearity by reducing the nonlinear fourth-order integro-differential equation to a system of first-order equations. Then a cell-centered finite difference scheme is applied to this system. With a lexicographical ordering of the unknowns, the discretization of the mean curvature functional leads to a block pentadiagonal matrix. Our contributions are fourfold: (i) we give a new method for treating the high-order nonlinearity term; (ii) we express the discretization of this term in terms of simple matrices; (iii) we give an analysis for this new method and establish that the error is of first order; and (iv) we verify this theoretical result by illustrating the convergence rates in numerical experiments.

A parallel domain decomposition-based implicit method for the Cahn–Hilliard–Cook phase-field equation in 3D

We present a numerical algorithm for simulating the spinodal decomposition described by the three dimensional Cahn–Hilliard–Cook (CHC) equation, which is a fourth-order stochastic partial differential equation with a noise term. The equation is discretized in space and time based on a fully implicit, cell-centered finite difference scheme, with an adaptive time-stepping strategy designed to accelerate the progress to equilibrium. At each time step, a parallel Newton–Krylov–Schwarz algorithm is used to solve the nonlinear system. We discuss various numerical and computational challenges associated with the method. The numerical scheme is validated by a comparison with an explicit scheme of high accuracy (and unreasonably high cost). We present steady state solutions of the CHC equation in two and three dimensions. The effect of the thermal fluctuation on the spinodal decomposition process is studied. We show that the existence of the thermal fluctuation accelerates the spinodal decomposition process and that the final steady morphology is sensitive to the stochastic noise. We also show the evolution of the energies and statistical moments. In terms of the parallel performance, it is found that the implicit domain decomposition approach scales well on supercomputers with a large number of processors.

Cell-centered finite difference method for parabolic equation

We develop two cell-centered finite difference schemes for parabolic problems on quadrilateral grids. One scheme is backward Euler scheme with first order accuracy in time increment while the other is Crank–Nicolson scheme with second order accuracy in time increment. The method is based on the lowest order Brezzi–Douglas–Marini (BDM) mixed finite element method. A quadrature rule gives a block-diagonal mass matrix and allows for local velocity elimination, which leads to a symmetric and positive definite cell-centered system for the pressures. Theoretical results indicate first-order convergence in spacial meshsize both for pressures and for subedge flues. Numerical experiments using the schemes show that the convergence rates are in agreement with the theoretical analysis.

A multipoint flux mixed finite element method on distorted quadrilaterals and hexahedra

In this paper, we develop a new mixed finite element method for elliptic problems on general quadrilateral and hexahedral grids that reduces to a cell-centered finite difference scheme. A special non-symmetric quadrature rule is employed that yields a positive definite cell-centered system for the pressure by eliminating local velocities. The method is shown to be accurate on highly distorted rough quadrilateral and hexahedral grids, including hexahedra with non-planar faces. Theoretical and numerical results indicate first-order convergence for the pressure and face fluxes.

Multigrid preconditioned conjugate-gradient solver for mixed finite-element method

The mixed finite-element approximation to a second-order elliptic PDE results in a saddle-point problem and leads to an indefinite linear system of equations. The mixed system of equations can be transformed into coupled symmetric positive-definite matrix equations, or a Schur complement problem, using block Gauss elimination. A preconditioned conjugate-gradient algorithm is used for solving the Schur complement problem. The mixed finite-element method is closely related to the cell-centered finite difference scheme for solving second-order elliptic problems with variable coefficients. For the cell-centered finite difference scheme, a simple multigrid algorithm can be defined and used as a preconditioner. For distorted grids, an additional iteration is needed. Nested iteration with a multigrid preconditioned conjugate gradient inner iteration results in an effective numerical solution technique for the mixed system of linear equations arising from a discretization on distorted grids. Numerical results show that the preconditioned conjugate-gradient inner iteration is robust with respect to grid size and variability in the hydraulic conductivity tensor.

Computational issues of hybrid and multipoint mixed methods for groundwater flow in anisotropic media

In this work, lowest-order Raviart–Thomas and Brezzi–Douglas–Marini mixed methods are considered for groundwater flow simulations. Typically, mixed methods lead to a saddle-point problem, which is expensive to solve. Two approaches are numerically compared here to allow an explicit velocity elimination: (1) the well-known hybrid formulation leading to a symmetric positive definite system where the only unknowns are the Lagrange multipliers and (2) a more recent approach, inspired from the multipoint flux approximation method, reducing low-order mixed methods to cell-centered finite difference schemes. Selected groundwater flow scenarios are used for the comparison between hybrid and multipoint approaches. The simulations are performed in the bidimensional case with a general triangular discretization because of its practical interest for hydrogeologists.

Local flux mimetic finite difference methods

We develop a local flux mimetic finite difference method for second order elliptic equations with full tensor coefficients on polyhedral meshes. To approximate the velocity (vector variable), the method uses two degrees of freedom per element edge in two dimensions and n degrees of freedom per n-gonal mesh face in three dimensions. To approximate the pressure (scalar variable), the method uses one degree of freedom per element. A specially chosen quadrature rule for the L 2-product of vector-functions allows for a local flux elimination and reduction of the method to a cell-centered finite difference scheme for the pressure unknowns. Under certain assumptions, first-order convergence is proved for both variables and second-order convergence is proved for the pressure. The assumptions are verified on simplicial meshes for a particular quadrature rule that leads to a symmetric method. For general polyhedral meshes, non-symmetric methods are constructed based on quadrature rules that are shown to satisfy some of the assumptions. Numerical results confirm the theory.

A combined mixed finite element ADI scheme for solving Richards' equation with mixed derivatives on irregular grids

We propose and analyze an efficient numerical method for solving semilinear parabolic problems with mixed derivative terms on non-rectangular domains. The spatial semidiscretization process is based on an expanded mixed finite element scheme which, combined with suitable quadrature rules, is converted into a cell-centered finite difference scheme. This choice preserves the asymptotic accuracy and local conservation of mass of the method, while substantially reducing the computational cost of the totally discrete scheme. To obtain it, an alternating direction implicit scheme is used for the integration in time. The resulting numerical algorithm involves sets of uncoupled tridiagonal systems which can be solved in parallel. We set out some theoretical results of unconditional convergence (of second order in space and first order in time) for our method. Finally, a numerical experiment is shown in order to illustrate the theoretical results.

Supraconvergent cell-centered scheme for two dimensional elliptic problems

In this paper we study the convergence properties of a cell-centered finite difference scheme for second order elliptic equations with variable coefficients subject to Dirichlet boundary conditions. We prove that the finite difference scheme on nonuniform meshes although not even being consistent are nevertheless second order convergent. More precisely, second order convergence with respect to a discrete version of L 2 ( Ω ) -norm is shown provided that the exact solution is in H 4 ( Ω ) . Estimates for the difference between the pointwise restriction of the exact solution on the discretization nodes and the finite difference solution are proved. The convergence is studied with the aid of an appropriate negative norm. A numerical example support the convergence result.

Comparison of V‐cycle multigrid method for cell‐centered finite difference on triangular meshes

AbstractWe consider a multigrid algorithm (MG) for the cell centered finite difference scheme (CCFD) on general triangular meshes using a new prolongation operator. This prolongation is designed to solve the diffusion equation with strongly discontinuous coefficient as well as with smooth one. We compare our new prolongation with the natural injection and the weighted operator in Kwak, Kwon, and Lee (Appl Math Comput 21 (1999), 552–564) and the behaviors of these three prolongation are discussed. Numerical experiments show that (i) for smooth problems, the multigrid with our new prolongation is fastest, the next is the weighted prolongation, and the third is the natural injection; and (ii) for nonsmooth problems, our new prolongation is again fastest, the next is the natural injection, and the third is the weighted prolongation. In conclusion, our new prolongation works better than the natural injection and the weighted operator for both smooth and nonsmooth problems. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006

Stability and dispersion analysis for higher order 3-D ADI-FDTD method

In this paper, the stability and dispersion analysis for higher order three-dimensional (3-D) alternating-direction-implicit (ADI) finite-difference time-domain (FDTD) methods is presented. Starting with the stability and dispersion analysis of the fourth order 3-D ADI-FDTD method, which adopts the fourth order cell-centered finite difference scheme for the spatial differential operator, we generalize the analysis to other higher order methods based on the sixth and tenth order cell-centered finite difference schemes. To the best of our knowledge, this is the first time that a comprehensive study of the stability and dispersion characteristics for one series of higher order 3-D ADI-FDTD methods is presented. Our analysis results show that all the higher order ADI-FDTD methods that are based on cell-centered finite difference schemes are unconditionally stable. The generalized form of the dispersion relations for these unconditionally stable ADI-FDTD methods is also presented. Using the relations attained, the effects of the order of schemes, mesh size and time step on the dispersion are illustrated through numerical results. This study will be useful for the selection and evaluation of various higher order ADI methods.

Multigrid algorithm for the cell‐centered finite difference method II: discontinuous coefficient case

AbstractWe consider a multigrid algorithm for the cell centered finite difference scheme with a prolongation operator depending on the diffusion coefficient. This prolongation operator is designed mainly for solving diffusion equations with strong varying or discontinuous coefficient and it reduces to the usual bilinear interpolation for Laplace equation. For simple interface problem, we show that the energy norm of this operator is uniformly bounded by 11/8, no matter how large the jump is, from which one can prove that W‐cycle with one smoothing converges with reduction factor independent of the size of jump using the theory developed by Bramble et al. (Math Comp 56 (1991), 1–34). For general interface problem, we show that the energy norm is bounded by some constant C* (independent of the jumps of the coefficient). In this case, we can conclude W‐cycle converges with sufficiently many smoothings. Numerical experiment shows that even V‐cycle multigrid algorithm with our prolongation works well for various interface problems. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004.

Finite volume methods on Voronoi meshes

Two cell-centered finite difference schemes on Voronoi meshes are derived and investigated. Stability and error estimates in a discrete H1-norm for both symmetric and nonsymmetric problems, including convection dominated, are proven. The theoretical results are illustrated with several numerical experiments. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14:193–212, 1998

Discretization schemes on triangular grids

Physical applications posed on irregular domains have caused difficulties in the use of many, otherwise effective, discretization schemes. In two-dimensional examples, we consider discretization schemes on triangles to approximate irregular domains. Although finite element, mixed finite element, and finite volume schemes for triangles are well understood, efficient solvers are essential for their effective use. In this paper, we consider certain cell-centered finite difference schemes on triangles, both for their theoretical properties on regular grids, and as effective preconditioners for other discretization schemes for more general triangulations. Two cell-centered finite difference schemes (four-point star stencils) are derived from the simplification of the lowest-order Raviart-Thomas mixed finite element discretization on certain triangular grids. Optimal and superconvergence error estimates for both the solution and the gradient of the solution are established. The accuracy of these schemes is also discussed in the case of discontinuous coefficients. Computational experiments verify the error estimates. Finally, the use of these methods as preconditioners is also discussed.