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1.
Extrapolation Cascadic Multigrid Method for Cell-Centered FV Discretization of Diffusion Equations with Strongly Discontinuous and Anisotropic Coefficients
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Kejia Pan Xiaoxin Wu Yunlong Yu Zhiqiang Sheng & Guangwei Yuan 《Communications In Computational Physics》2022,31(5):1561-1584
Extrapolation cascadic multigrid (EXCMG) method with conjugate gradient
smoother is very efficient for solving the elliptic boundary value problems with linear
finite element discretization. However, it is not trivial to generalize the vertex-centred
EXCMG method to cell-centered finite volume (FV) methods for diffusion equations
with strongly discontinuous and anisotropic coefficients, since a non-nested hierarchy
of grid nodes are used in the cell-centered discretization. For cell-centered FV schemes,
the vertex values (auxiliary unknowns) need to be approximated by cell-centered ones
(primary unknowns). One of the novelties is to propose a new gradient transfer (GT)
method of interpolating vertex unknowns with cell-centered ones, which is easy to implement and applicable to general diffusion tensors. The main novelty of this paper is
to design a multigrid prolongation operator based on the GT method and splitting extrapolation method, and then propose a cell-centered EXCMG method with BiCGStab
smoother for solving the large linear system resulting from linear FV discretization
of diffusion equations with strongly discontinuous and anisotropic coefficients. Numerical experiments are presented to demonstrate the high efficiency of the proposed
method. 相似文献
2.
Weighted Interior Penalty Method with Semi-Implicit Integration Factor Method for Non-Equilibrium Radiation Diffusion Equation
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Rongpei Zhang Xijun Yu Jiang Zhu Abimael F. D. Loula & Xia Cui 《Communications In Computational Physics》2013,14(5):1287-1303
Weighted interior penalty discontinuous Galerkin method is developed to solve the two-dimensional non-equilibrium radiation diffusion equation on unstructured mesh. There are three weights including the arithmetic, the harmonic, and the geometric weight in the weighted discontinuous Galerkin scheme. For the time discretization, we treat the nonlinear diffusion coefficients explicitly, and apply the semi-implicit integration factor method to the nonlinear ordinary differential equations arising from discontinuous Galerkin spatial discretization. The semi-implicit integration factor method can not only avoid severe time step limits, but also take advantage of the local property of DG methods by which small sized nonlinear algebraic systems are solved element by element with the exact Newton iteration method. Numerical results are presented to demonstrate the validity of discontinuous Galerkin method for high nonlinear and tightly coupled radiation diffusion equation. 相似文献
3.
Fast Numerical Simulation of Two-Phase Transport Model in the Cathode of a Polymer Electrolyte Fuel Cell
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Pengtao Sun Guangri Xue Chaoyang Wang & Jinchao Xu 《Communications In Computational Physics》2009,6(1):49-71
In this paper, we apply streamline-diffusion and Galerkin-least-squares finite element methods for 2D steady-state two-phase model in the cathode of polymer
electrolyte fuel cell (PEFC) that contains a gas channel and a gas diffusion layer (GDL).
This two-phase PEFC model is typically modeled by a modified Navier-Stokes equation
for the mass and momentum, with Darcy's drag as an additional source term in
momentum for flows through GDL, and a discontinuous and degenerate convection-diffusion
equation for water concentration. Based on the mixed finite element method
for the modified Navier-Stokes equation and standard finite element method for water
equation, we design streamline-diffusion and Galerkin-least-squares to overcome
the dominant convection arising from the gas channel. Meanwhile, we employ Kirchhoff
transformation to deal with the discontinuous and degenerate diffusivity in water
concentration. Numerical experiments demonstrate that our finite element methods,
together with these numerical techniques, are able to get accurate physical solutions
with fast convergence. 相似文献
4.
Two Nonlinear Positivity-Preserving Finite Volume Schemes for Three-Dimensional Heat Conduction Equations on General Polyhedral Meshes
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Menghuan Liu Shi Shu Guangwei Yuan & Xiaoqiang Yue 《Communications In Computational Physics》2021,30(4):1185-1215
In this article we present two types of nonlinear positivity-preserving finite
volume (PPFV) schemes for a class of three-dimensional heat conduction equations on
general polyhedral meshes. First, we present a new parameter selection strategy on the
one-sided flux and establish a nonlinear PPFV scheme based on a two-point flux with
higher efficiency. By comparing with the scheme proposed in [H. Xie, X. Xu, C. Zhai,
H. Yong, Commun. Comput. Phys. 24 (2018) 1375–1408], our scheme avoids the assumption that the values of auxiliary unknowns are nonnegative, which makes our
interpolation formulae suitable to be constructed by existing approaches with high
accuracy and well robustness (e.g., the finite element method), thus enhancing the
adaptability to distorted meshes with large deformations. Then we derive a linear
multi-point flux involving combination coefficients and, via the Patankar trick, obtain
another nonlinear PPFV scheme that is concise and easy to implement. The selection
strategy of combination coefficients is also provided to improve the convergence behavior of the Picard procedure. Furthermore, the existence and positivity-preserving
properties of these two nonlinear PPFV solutions are proved. Numerical experiments
with the discontinuous diffusion scalar as well as discontinuous and anisotropic diffusion tensors are given to confirm our theoretical findings and demonstrate that our
schemes both can achieve ideal-order accuracy even on severely distorted meshes. 相似文献
5.
Propagation of short pulses of light through biological tissues can be studied by numerically solving the diffusion equation.
The boundary integral method was used to convert the differential equation to integral form and the result was solved using
the boundary element method. The effects of different optical parameters of the tissue, i.e. scattering, absorption coefficients
and anisotropic factor, on temporal evolution of the diffusely reflected pulse were studied. The results were compared with
those obtained using the finite difference time domain method and the boundary integral method was found to be more precise
and faster than the last method. The method can be used to investigate reflected pulses in the study of cell morphology and
tumours in different types of tissue. 相似文献
6.
A fully discrete discontinuous Galerkin method is introduced for solving
time-dependent Maxwell's equations. Distinguished from the Runge-Kutta discontinuous Galerkin method (RKDG) and the finite element time domain method (FETD), in
our scheme, discontinuous Galerkin methods are used to discretize not only the spatial
domain but also the temporal domain. The proposed numerical scheme is proved to be
unconditionally stable, and a convergent rate $\mathcal{O}((∆t)^{r+1}+h^{k+1/2})$ is established under the $L^2$ -norm when polynomials of degree at most $r$ and $k$ are used for temporal and
spatial approximation, respectively. Numerical results in both 2-D and 3-D are provided to validate the theoretical prediction. An ultra-convergence of order $(∆t)^{2r+1}$ in
time step is observed numerically for the numerical fluxes w.r.t. temporal variable at
the grid points. 相似文献
7.
A New Interpolation for Auxiliary Unknowns of the Monotone Finite Volume Scheme for 3D Diffusion Equations
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Fei Zhao Xiang Lai Guangwei Yuan & Zhiqiang Sheng 《Communications In Computational Physics》2020,27(4):1201-1233
A monotone cell-centered finite volume scheme for diffusion equations on
tetrahedral meshes is established in this paper, which deals with tensor diffusion coefficients and strong discontinuous diffusion coefficients. The first novelty here is to
propose a new method of interpolating vertex unknowns (auxiliary unknowns) with
cell-centered unknowns (primary unknowns), in which a sufficient condition is given
to guarantee the non-negativity of vertex unknowns. The second novelty of this paper
is to devise a modified Anderson acceleration, which is based on an iterative combination of vertex unknowns and will be denoted as AA-Vertex algorithm, in order to solve
the nonlinear scheme efficiently. Numerical testes indicate that our new method can
obtain almost second order accuracy and is more accurate than some existing methods.
Furthermore, with the same accuracy, the modified Anderson acceleration is much
more efficient than the usual one. 相似文献
8.
Leopold Grinberg & George Em Karniadakis 《Communications In Computational Physics》2008,4(5):1151-1169
Ultra-parallel flow simulations on hundreds of thousands of processors require new multi-level domain decomposition methods. Here we present such a new
two-level method that has features both of discontinuous and continuous Galerkin
formulations. Specifically, at the coarse level the domain is subdivided into several
big patches and within each patch a spectral element discretization (fine level) is employed. New interface conditions for the Navier-Stokes equations are developed to
connect the patches, relaxing the C0continuity and minimizing data transfer at the
patch interface. We perform several 3D flow simulations of a benchmark problem and
of arterial flows to evaluate the performance of the new method and investigate its
accuracy. 相似文献
9.
A Parallel Domain Decomposition Algorithm for Simulating Blood Flow with Incompressible Navier-Stokes Equations with Resistive Boundary Condition
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We introduce and study a parallel domain decomposition algorithm for
the simulation of blood flow in compliant arteries using a fully-coupled system of
nonlinear partial differential equations consisting of a linear elasticity equation and
the incompressible Navier-Stokes equations with a resistive outflow boundary condition. The system is discretized with a finite element method on unstructured moving
meshes and solved by a Newton-Krylov algorithm preconditioned with an overlapping restricted additive Schwarz method. The resistive outflow boundary condition
plays an interesting role in the accuracy of the blood flow simulation and we provide a
numerical comparison of its accuracy with the standard pressure type boundary condition. We also discuss the parallel performance of the implicit domain decomposition
method for solving the fully coupled nonlinear system on a supercomputer with a few
hundred processors. 相似文献
10.
A Bilinear Immersed Finite Volume Element Method for the Diffusion Equation with Discontinuous Coefficient
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This paper is to present a finite volume element (FVE) method based on the
bilinear immersed finite element (IFE) for solving the boundary value problems of the
diffusion equation with a discontinuous coefficient (interface problem). This method
possesses the usual FVE method's local conservation property and can use a structured
mesh or even the Cartesian mesh to solve a boundary value problem whose coefficient
has discontinuity along piecewise smooth nontrivial curves. Numerical examples are
provided to demonstrate features of this method. In particular, this method can produce
a numerical solution to an interface problem with the usual O(h2) (in L2 norm)
and O(h) (in H1 norm) convergence rates. 相似文献
11.
A Conservative Parallel Iteration Scheme for Nonlinear Diffusion Equations on Unstructured Meshes
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Yunlong Yu Yanzhong Yao Guangwei Yuan & Xingding Chen 《Communications In Computational Physics》2016,20(5):1405-1423
In this paper, a conservative parallel iteration scheme is constructed to solve
nonlinear diffusion equations on unstructured polygonal meshes. The design is based
on two main ingredients: the first is that the parallelized domain decomposition is
embedded into the nonlinear iteration; the second is that prediction and correction
steps are applied at subdomain interfaces in the parallelized domain decomposition
method. A new prediction approach is proposed to obtain an efficient conservative
parallel finite volume scheme. The numerical experiments show that our parallel
scheme is second-order accurate, unconditionally stable, conservative and has linear
parallel speed-up. 相似文献
12.
We present a parallel Schwarz type domain decomposition preconditioned
recycling Krylov subspace method for the numerical solution of stochastic indefinite
elliptic equations with two random coefficients. Karhunen-Loève expansions are used
to represent the stochastic variables and the stochastic Galerkin method with double
orthogonal polynomials is used to derive a sequence of uncoupled deterministic
equations. We show numerically that the Schwarz preconditioned recycling GMRES
method is an effective technique for solving the entire family of linear systems and, in
particular, the use of recycled Krylov subspaces is the key element of this successful
approach. 相似文献
13.
In this paper, we investigate the dynamic process of liquid bridge formation between two parallel hydrophobic plates with hydrophilic patches, previously
studied in [1]. We propose a dynamic Hele-Shaw model to take advantage of the
small aspect ratio between the gap width and the plate size. A constrained level set
method is applied to solve the model equations numerically, where a global constraint
is imposed in the evolution [2] stage together with local constraints in the reinitialization [3] stage of level set function in order to limit numerical mass loss. In contrast
to the finite element method used in [2], we use a finite difference method with a
5th order HJWENO scheme for spatial discretization. To illustrate the effectiveness
of the constrained method, we have compared the results obtained by the standard
level set method with those from the constrained version. Our results show that the
constrained level set method produces physically reasonable results while that of the
standard method is less reliable. Our numerical results also show that the dynamic
nature of the flow plays an important role in the process of liquid bridge formation
and criteria based on static energy minimization approach has limited applicability. 相似文献
14.
Crouzeix-Raviart MsFEM with Bubble Functions for Diffusion and Advection-Diffusion in Perforated Media
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Pierre Degond Alexei Lozinski Bagus Putra Muljadi & Jacek Narski 《Communications In Computational Physics》2015,17(4):887-907
The adaptation of Crouzeix-Raviart finite element in the context of multiscale
finite element method (MsFEM) is studied and implemented on diffusion and
advection-diffusion problems in perforated media. It is known that the approximation
of boundary condition on coarse element edges when computing the multiscale basis
functions critically influences the eventual accuracy of any MsFEM approaches. The
weakly enforced continuity of Crouzeix-Raviart function space across element edges
leads to a natural boundary condition for the multiscale basis functions which relaxes
the sensitivity of our method to complex patterns of perforations. Another ingredient
to our method is the application of bubble functions which is shown to be instrumental
in maintaining high accuracy amid dense perforations. Additionally, the application
of penalization method makes it possible to avoid complex unstructured domain and
allows extensive use of simpler Cartesian meshes. 相似文献
15.
An Adaptive Modal Discontinuous Galerkin Finite Element Parallel Method Using Unsplit Multi-Axial Perfectly Matched Layer for Seismic Wave Modeling
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Yang Xu Xiaofei Chen Wei Zhang & Xiao Pan 《Communications In Computational Physics》2022,31(4):1083-1113
The discontinuous Galerkin finite element method (DG-FEM) is a high-precision numerical simulation method widely used in various disciplines. In this paper, we derive the auxiliary ordinary differential equation complex frequency-shifted
multi-axial perfectly matched layer (AODE CFS-MPML) in an unsplit format and combine it with any high-order adaptive DG-FEM based on an unstructured mesh to simulate seismic wave propagation. To improve the computational efficiency, we implement
Message Passing Interface (MPI) parallelization for the simulation. Comparisons of
the numerical simulation results with the analytical solutions verify the accuracy and
effectiveness of our method. The results of numerical experiments also confirm the
stability and effectiveness of the AODE CFS-MPML. 相似文献
16.
Linear Scaling Discontinuous Galerkin Density Matrix Minimization Method with Local Orbital Enriched Finite Element Basis: 1-D Lattice Model System
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Tiao Lu Wei Cai Jianguo Xin & Yinglong Guo 《Communications In Computational Physics》2013,14(2):276-300
In the first of a series of papers, we will study a discontinuous Galerkin (DG)
framework for many electron quantum systems. The salient feature of this framework
is the flexibility of using hybrid physics-based local orbitals and accuracy-guaranteed
piecewise polynomial basis in representing the Hamiltonian of the many body system.
Such a flexibility is made possible by using the discontinuous Galerkin method
to approximate the Hamiltonian matrix elements with proper constructions of numerical
DG fluxes at the finite element interfaces. In this paper, we will apply the DG
method to the density matrix minimization formulation, a popular approach in the
density functional theory of many body Schrödinger equations. The density matrix
minimization is to find the minima of the total energy, expressed as a functional of the
density matrix ρ(r,r′), approximated by the proposed enriched basis, together with
two constraints of idempotency and electric neutrality. The idempotency will be handled
with the McWeeny's purification while the neutrality is enforced by imposing the
number of electrons with a penalty method. A conjugate gradient method (a Polak-Ribiere
variant) is used to solve the minimization problem. Finally, the linear-scaling
algorithm and the advantage of using the local orbital enriched finite element basis in
the DG approximations are verified by studying examples of one dimensional lattice
model systems. 相似文献
17.
Effective Two-Level Domain Decomposition Preconditioners for Elastic Crack Problems Modeled by Extended Finite Element Method
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Xingding Chen & Xiao-Chuan Cai 《Communications In Computational Physics》2020,28(4):1561-1584
In this paper, we propose some effective one- and two-level domain decomposition preconditioners for elastic crack problems modeled by extended finite element method. To construct the preconditioners, the physical domain is decomposed
into the "crack tip" subdomain, which contains all the degrees of freedom (dofs) of the
branch enrichment functions, and the "regular" subdomains, which contain the standard dofs and the dofs of the Heaviside enrichment function. In the one-level additive
Schwarz and restricted additive Schwarz preconditioners, the "crack tip" subproblem
is solved directly and the "regular" subproblems are solved by some inexact solvers,
such as ILU. In the two-level domain decomposition preconditioners, traditional interpolations between the coarse and the fine meshes destroy the good convergence property. Therefore, we propose an unconventional approach in which the coarse mesh
is exactly the same as the fine mesh along the crack line, and adopt the technique of
a non-matching grid interpolation between the fine and the coarse meshes. Numerical experiments demonstrate the effectiveness of the two-level domain decomposition
preconditioners applied to elastic crack problems. 相似文献
18.
Performance Analysis of a High-Order Discontinuous Galerkin Method Application to the Reverse Time Migration
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Caroline Baldassari Hé lè ne Barucq Henri Calandra Bertrand Denel & Julien Diaz 《Communications In Computational Physics》2012,11(2):660-673
This work pertains to numerical aspects of a finite element method based
discontinuous functions. Our study focuses on the Interior Penalty Discontinuous
Galerkin method (IPDGM) because of its high-level of flexibility for solving the full
wave equation in heterogeneous media. We assess the performance of IPDGM through
a comparison study with a spectral element method (SEM). We show that IPDGM is
as accurate as SEM. In addition, we illustrate the efficiency of IPDGM when employed
in a seismic imaging process by considering two-dimensional problems involving the
Reverse Time Migration. 相似文献
19.
Enforcing the Discrete Maximum Principle for Linear Finite Element Solutions of Second-Order Elliptic Problems
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The maximum principle is a basic qualitative property of the solution of
second-order elliptic boundary value problems. The preservation of the qualitative
characteristics, such as the maximum principle, in discrete model is one of the key
requirements. It is well known that standard linear finite element solution does not
satisfy maximum principle on general triangular meshes in 2D. In this paper we consider how to enforce discrete maximum principle for linear finite element solutions for
the linear second-order self-adjoint elliptic equation. First approach is based on repair
technique, which is a posteriori correction of the discrete solution. Second method
is based on constrained optimization. Numerical tests that include anisotropic cases
demonstrate how our method works for problems for which the standard finite element methods produce numerical solutions that violate the discrete maximum principle. 相似文献
20.
Electrostatics interactions play a major role in the stabilization of
biomolecules: as such, they remain a major focus of theoretical and computational
studies in biophysics. Electrostatics in solution is strongly dependent on the nature of
the solvent and on the ions it contains. While methods that treat the solvent and ions
explicitly provide an accurate estimate of these interactions, they are usually computationally too demanding to study large macromolecular systems. Implicit solvent
methods provide a viable alternative, especially those based on Poisson theory. The
Poisson-Boltzmann equation (PBE) treats the system in a mean field approximation,
providing reasonable estimates of electrostatics interactions in a solvent treated as continuum. In the first part of this paper, we review the theory behind the PBE, including
recent improvement in which ions size and dipolar features of solvent molecules are
taken into account explicitly. The PBE is a non linear second order differential equation with discontinuous coefficients, for which no analytical solution is available for
large molecular systems. Many numerical solvers have been developed that solve a
discretized version of the PBE on a mesh, either using finite difference, finite element,
or boundary element methods. The accuracy of the solutions provided by these solvers
highly depend on the geometry of their underlying meshes, as well as on the method
used to embed the physical system on the mesh. In the second part of the paper, we
describe a new geometric approach for generating unstructured tetrahedral meshes as
well as simplifications of these meshes that are well fitted for solving the PBE equation
using multigrid approaches. 相似文献