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1.
An Efficient Positivity-Preserving Finite Volume Scheme for the Nonequilibrium Three-Temperature Radiation Diffusion Equations on Polygonal Meshes 下载免费PDF全文
This paper develops an efficient positivity-preserving finite volume scheme
for the two-dimensional nonequilibrium three-temperature radiation diffusion equations on general polygonal meshes. The scheme is formed as a predictor-corrector algorithm. The corrector phase obtains the cell-centered solutions on the primary mesh,
while the predictor phase determines the cell-vertex solutions on the dual mesh independently. Moreover, the flux on the primary edge is approximated with a fixed
stencil and the nonnegative cell-vertex solutions are not reconstructed. Theoretically,
our scheme does not require any nonlinear iteration for the linear problems, and can
call the fast nonlinear solver (e.g. Newton method) for the nonlinear problems. The
positivity, existence and uniqueness of the cell-centered solutions obtained on the corrector phase are analyzed, and the scheme on quasi-uniform meshes is proved to be $L^2$- and $H^1$-stable under some assumptions. Numerical experiments demonstrate the
accuracy, efficiency and positivity of the scheme on various distorted meshes. 相似文献
2.
A Conservative Parallel Iteration Scheme for Nonlinear Diffusion Equations on Unstructured Meshes 下载免费PDF全文
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
3.
This paper extends the adaptive moving mesh method developed by Tang
and Tang [36] to two-dimensional (2D) relativistic hydrodynamic (RHD) equations.
The algorithm consists of two "independent" parts: the time evolution of the RHD
equations and the (static) mesh iteration redistribution. In the first part, the RHD
equations are discretized by using a high resolution finite volume scheme on the fixed
but nonuniform meshes without the full characteristic decomposition of the governing equations. The second part is an iterative procedure. In each iteration, the mesh
points are first redistributed, and then the cell averages of the conservative variables
are remapped onto the new mesh in a conservative way. Several numerical examples
are given to demonstrate the accuracy and effectiveness of the proposed method. 相似文献
4.
Mesh Sensitivity for Numerical Solutions of Phase-Field Equations Using r-Adaptive Finite Element Methods 下载免费PDF全文
There have been several recent papers on developing moving mesh methods
for solving phase-field equations. However, it is observed that some of these moving
mesh solutions are essentially different from the solutions on very fine fixed meshes.
One of the purposes of this paper is to understand the reason for the differences. We
carried out numerical sensitivity studies systematically in this paper and it can be concluded that for the phase-field equations, the numerical solutions are very sensitive to
the starting mesh and the monitor function. As a separate issue, an efficient alternating Crank-Nicolson time discretization scheme is developed for solving the nonlinear
system resulting from a finite element approximation to the phase-field equations. 相似文献
5.
Two-Grid Method for Miscible Displacement Problem by Mixed Finite Element Methods and Mixed Finite Element Method of Characteristics 下载免费PDF全文
The miscible displacement of one incompressible fluid by another in a porous
medium is governed by a system of two equations. One is elliptic form equation for
the pressure and the other is parabolic form equation for the concentration of one of
the fluids. Since only the velocity and not the pressure appears explicitly in the concentration
equation, we use a mixed finite element method for the approximation of
the pressure equation and mixed finite element method with characteristics for the
concentration equation. To linearize the mixed-method equations, we use a two-grid
algorithm based on the Newton iteration method for this full discrete scheme problems.
First, we solve the original nonlinear equations on the coarse grid, then, we
solve the linearized problem on the fine grid used Newton iteration once. It is shown
that the coarse grid can be much coarser than the fine grid and achieve asymptotically
optimal approximation as long as the mesh sizes satisfy $h=H^2$ in this paper. Finally,
numerical experiment indicates that two-grid algorithm is very effective. 相似文献
6.
Arbitrary-Lagrangian-Eulerian One-Step WENO Finite Volume Schemes on Unstructured Triangular Meshes 下载免费PDF全文
In this article we present a new class of high order accurate ArbitraryEulerian-Lagrangian (ALE) one-step WENO finite volume schemes for solving nonlinear hyperbolic systems of conservation laws on moving two dimensional unstructured triangular meshes. A WENO reconstruction algorithm is used to achieve high
order accuracy in space and a high order one-step time discretization is achieved by
using the local space-time Galerkin predictor proposed in [25]. For that purpose, a
new element-local weak formulation of the governing PDE is adopted on moving
space-time elements. The space-time basis and test functions are obtained considering Lagrange interpolation polynomials passing through a predefined set of nodes.
Moreover, a polynomial mapping defined by the same local space-time basis functions
as the weak solution of the PDE is used to map the moving physical space-time element onto a space-time reference element. To maintain algorithmic simplicity, the
final ALE one-step finite volume scheme uses moving triangular meshes with straight
edges. This is possible in the ALE framework, which allows a local mesh velocity that
is different from the local fluid velocity. We present numerical convergence rates for
the schemes presented in this paper up to sixth order of accuracy in space and time and
show some classical numerical test problems for the two-dimensional Euler equations
of compressible gas dynamics. 相似文献
7.
Convergent Adaptive Finite Element Method Based on Centroidal Voronoi Tessellations and Superconvergence 下载免费PDF全文
Yunqing Huang Hengfeng Qin Desheng Wang & Qiang Du 《Communications In Computational Physics》2011,10(2):339-370
We present a novel adaptive finite element method (AFEM) for elliptic equations
which is based upon the Centroidal Voronoi Tessellation (CVT) and superconvergent
gradient recovery. The constructions of CVT and its dual Centroidal Voronoi
Delaunay Triangulation (CVDT) are facilitated by a localized Lloyd iteration to produce
almost equilateral two dimensional meshes. Working with finite element solutions
on such high quality triangulations, superconvergent recovery methods become
particularly effective so that asymptotically exact a posteriori error estimations can be
obtained. Through a seamless integration of these techniques, a convergent adaptive
procedure is developed. As demonstrated by the numerical examples, the new AFEM
is capable of solving a variety of model problems and has great potential in practical
applications. 相似文献
8.
A Nodal Finite Element Method for a Thermally Coupled Eddy-Current Problem with Moving Conductors 下载免费PDF全文
Zezhong Wang & Qiya Hu 《Communications In Computational Physics》2021,29(3):767-801
This paper aims to design and analyze a solution method for a time-dependent, nonlinear and thermally coupled eddy-current problem with a moving
conductor on hyper-velocity. We transform the problem into an equivalent coupled
system and use the nodal finite element discretization (in space) and the implicit Euler method (in time) for the coupled system. The resulting discrete coupled system is
decoupled and implicitly solved by a time step-length iteration method and the Picard
iteration. We numerically and theoretically prove that the finite element approximations have the optimal error estimates and both the two iteration methods possess the
linear convergence. For the proposed method, numerical stability and accuracy of the
approximations can be held even for coarser mesh partitions and larger time steps.
We also construct a preconditioner for the discrete operator defined by the linearized
bilinear form and show that this preconditioner is uniformly effective. Numerical experiments are done to confirm the theoretical results and illustrate that the proposed
method is well behaved in large-scale numerical simulations. 相似文献
9.
An Implicit Algorithm for High-Order DG/FV Schemes for Compressible Flows on 2D Arbitrary Grids 下载免费PDF全文
Laiping Zhang Ming Li Wei Liu & Xin He 《Communications In Computational Physics》2015,17(1):287-316
A Newton/LU-SGS (lower-upper symmetric Gauss-Seidel) iteration implicit
method was developed to solve two-dimensional Euler and Navier-Stokes equations
by the DG/FV hybrid schemes on arbitrary grids. The Newton iteration was employed
to solve the nonlinear system, while the linear system was solved with LU-SGS iteration.
The effect of several parameters in the implicit scheme, such as the CFL number,
the Newton sub-iteration steps, and the update frequency of Jacobian matrix, was investigated
to evaluate the performance of convergence history. Several typical test
cases were simulated, and compared with the traditional explicit Runge-Kutta (RK)
scheme. Firstly the Couette flow was tested to validate the order of accuracy of the
present DG/FV hybrid schemes. Then a subsonic inviscid flow over a bump in a channel
was simulated and the effect of parameters was alsoinvestigated. Finally, the implicit
algorithm was applied to simulate a subsonic inviscid flow over a circular cylinder
and the viscous flow in a square cavity. The numerical results demonstrated that
the present implicit scheme can accelerate the convergence history efficiently. Choosing
proper parameters would improve the efficiency of the implicit scheme. Moreover,
in the same framework, the DG/FV hybrid schemes are more efficient than the same
order DG schemes. 相似文献
10.
A New Family of High Order Unstructured MOOD and ADER Finite Volume Schemes for Multidimensional Systems of Hyperbolic Conservation Laws 下载免费PDF全文
Rapha& euml l Loub& egrave re Michael Dumbser & Steven Diot 《Communications In Computational Physics》2014,16(3):718-763
In this paper, we investigate the coupling of the Multi-dimensional Optimal
Order Detection (MOOD) method and the Arbitrary high order DERivatives (ADER)
approach in order to design a new high order accurate, robust and computationally
efficient Finite Volume (FV) scheme dedicated to solving nonlinear systems of hyperbolic conservation laws on unstructured triangular and tetrahedral meshes in two and
three space dimensions, respectively. The Multi-dimensional Optimal Order Detection
(MOOD) method for 2D and 3D geometries has been introduced in a recent series of
papers for mixed unstructured meshes. It is an arbitrary high-order accurate Finite
Volume scheme in space, using polynomial reconstructions with a posteriori detection
and polynomial degree decrementing processes to deal with shock waves and other
discontinuities. In the following work, the time discretization is performed with an
elegant and efficient one-step ADER procedure. Doing so, we retain the good properties of the MOOD scheme, that is to say, the optimal high-order of accuracy is reached
on smooth solutions, while spurious oscillations near singularities are prevented. The
ADER technique not only reduces the cost of the overall scheme as shown
on a set of numerical tests in 2D and 3D, but also increases the stability of the overall scheme. A systematic comparison between classical unstructured ADER-WENO
schemes and the new ADER-MOOD approach has been carried out for high-order
schemes in space and time in terms of cost, robustness, accuracy and efficiency. The
main finding of this paper is that the combination of ADER with MOOD generally
outperforms the one of ADER and WENO either because at given accuracy MOOD isless expensive (memory and/or CPU time), or because it is more accurate for a given
grid resolution. A large suite of classical numerical test problems has been solved
on unstructured meshes for three challenging multi-dimensional systems of conservation laws: the Euler equations of compressible gas dynamics, the classical equations
of ideal magneto-Hydrodynamics (MHD) and finally the relativistic MHD equations
(RMHD), which constitutes a particularly challenging nonlinear system of hyperbolic
partial differential equation. All tests are run on genuinely unstructured grids composed of simplex elements. 相似文献
11.
A Conservative Numerical Method for the Cahn–Hilliard Equation with Generalized Mobilities on Curved Surfaces in Three-Dimensional Space 下载免费PDF全文
Darae Jeong Yibao Li Chaeyoung Lee Junxiang Yang & Junseok Kim 《Communications In Computational Physics》2020,27(2):412-430
In this paper, we develop a conservative numerical method for the Cahn–
Hilliard equation with generalized mobilities on curved surfaces in three-dimensional
space. We use an unconditionally gradient stable nonlinear splitting numerical scheme
and solve the resulting system of implicit discrete equations on a discrete narrow band
domain by using a Jacobi-type iteration. For the domain boundary cells, we use the
trilinear interpolation using the closest point method. The proposing numerical algorithm is computationally efficient because we can use the standard finite difference
Laplacian scheme on three-dimensional Cartesian narrow band mesh instead of discrete Laplace–Beltrami operator on triangulated curved surfaces. In particular, we employ a mass conserving correction scheme, which enforces conservation of total mass.
We perform numerical experiments on the various curved surfaces such as sphere,
torus, bunny, cube, and cylinder to demonstrate the performance and effectiveness of
the proposed method. We also present the dynamics of the CH equation with constant
and space-dependent mobilities on the curved surfaces. 相似文献
12.
A Finite Volume Upwind-Biased Centred Scheme for Hyperbolic Systems of Conservation Laws: Application to Shallow Water Equations 下载免费PDF全文
Guglielmo Stecca Annunziato Siviglia & Eleuterio F. Toro 《Communications In Computational Physics》2012,12(4):1183-1214
We construct a new first-order central-upwind numerical method for solving systems of hyperbolic equations in conservative form. It applies in multidimensional structured and unstructured meshes. The proposed method is an extension of
the UFORCE method developed by Stecca, Siviglia and Toro [25], in which the upwind
bias for the modification of the staggered mesh is evaluated taking into account the
smallest and largest wave of the entire Riemann fan. The proposed first-order method
is shown to be identical to the Godunov upwind method in applications to a 2×2 linear
hyperbolic system. The method is then extended to non-linear systems and its performance is assessed by solving the two-dimensional inviscid shallow water equations.
Extension to second-order accuracy is carried out using an ADER-WENO approach in
the finite volume framework on unstructured meshes. Finally, numerical comparison
with current competing numerical methods enables us to identify the salient features
of the proposed method. 相似文献
13.
Two Nonlinear Positivity-Preserving Finite Volume Schemes for Three-Dimensional Heat Conduction Equations on General Polyhedral Meshes 下载免费PDF全文
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
14.
15.
Shuangzhang Tu Gordon W. Skelton & Qing Pang 《Communications In Computational Physics》2011,9(2):441-480
This paper presents a novel high-order space-time method for hyperbolic
conservation laws. Two important concepts, the staggered space-time mesh of the
space-time conservation element/solution element (CE/SE) method and the local discontinuous
basis functions of the space-time discontinuous Galerkin (DG) finite element
method, are the two key ingredients of the new scheme. The staggered space-time
mesh is constructed using the cell-vertex structure of the underlying spatial mesh.
The universal definitions of CEs and SEs are independent of the underlying spatial
mesh and thus suitable for arbitrarily unstructured meshes. The solution within each
physical time step is updated alternately at the cell level and the vertex level. For
this solution updating strategy and the DG ingredient, the new scheme here is termed
as the discontinuous Galerkin cell-vertex scheme (DG-CVS). The high order of accuracy
is achieved by employing high-order Taylor polynomials as the basis functions
inside each SE. The present DG-CVS exhibits many advantageous features such as
Riemann-solver-free, high-order accuracy, point-implicitness, compactness, and ease
of handling boundary conditions. Several numerical tests including the scalar advection
equations and compressible Euler equations will demonstrate the performance of
the new method. 相似文献
16.
A Constrained Finite Element Method Based on Domain Decomposition Satisfying the Discrete Maximum Principle for Diffusion Problems 下载免费PDF全文
In this paper, we are concerned with the constrained finite element method
based on domain decomposition satisfying the discrete maximum principle for diffusion
problems with discontinuous coefficients on distorted meshes. The basic idea of
domain decomposition methods is used to deal with the discontinuous coefficients. To
get the information on the interface, we generalize the traditional Neumann-Neumann
method to the discontinuous diffusion tensors case. Then, the constrained finite element
method is used in each subdomain. Comparing with the method of using the
constrained finite element method on the global domain, the numerical experiments
show that not only the convergence order is improved, but also the nonlinear iteration
time is reduced remarkably in our method. 相似文献
17.
A multigrid method is proposed to compute the ground state solution of
Bose-Einstein condensations by the finite element method based on the multilevel correction
for eigenvalue problems and the multigrid method for linear boundary value
problems. In this scheme, obtaining the optimal approximation for the ground state
solution of Bose-Einstein condensates includes a sequence of solutions of the linear
boundary value problems by the multigrid method on the multilevel meshes and some
solutions of nonlinear eigenvalue problems some very low dimensional finite element
space. The total computational work of this scheme can reach almost the same optimal
order as solving the corresponding linear boundary value problem. Therefore,
this type of multigrid scheme can improve the overall efficiency for the simulation of
Bose-Einstein condensations. Some numerical experiments are provided to validate
the efficiency of the proposed method. 相似文献
18.
Alina Chertock & Yongle Liu 《Communications In Computational Physics》2020,27(2):480-502
We study the two-component Camassa-Holm (2CH) equations as a model
for the long time water wave propagation. Compared with the classical Saint-Venant
system, it has the advantage of preserving the waves amplitude and shape for a long
time. We present two different numerical methods—finite volume (FV) and hybrid
finite-volume-particle (FVP) ones. In the FV setup, we rewrite the 2CH equations in a
conservative form and numerically solve it by the central-upwind scheme, while in the
FVP method, we apply the central-upwind scheme to the density equation only while
solving the momentum and velocity equations by a deterministic particle method. Numerical examples are shown to verify the accuracy of both FV and FVP methods. The
obtained results demonstrate that the FVP method outperforms the FV method and
achieves a superior resolution thanks to a low-diffusive nature of a particle approximation. 相似文献
19.
A Conservative and Monotone Characteristic Finite Element Solver for Three-Dimensional Transport and Incompressible Navier-Stokes Equations on Unstructured Grids 下载免费PDF全文
Bassou Khouya Mofdi El-Amrani & Mohammed Seaid 《Communications In Computational Physics》2022,31(1):224-256
We propose a mass-conservative and monotonicity-preserving characteristic finite element method for solving three-dimensional transport and incompressible
Navier-Stokes equations on unstructured grids. The main idea in the proposed algorithm consists of combining a mass-conservative and monotonicity-preserving modified method of characteristics for the time integration with a mixed finite element
method for the space discretization. This class of computational solvers benefits from
the geometrical flexibility of the finite elements and the strong stability of the modified method of characteristics to accurately solve convection-dominated flows using
time steps larger than its Eulerian counterparts. In the current study, we implement
three-dimensional limiters to convert the proposed solver to a fully mass-conservative
and essentially monotonicity-preserving method in addition of a low computational
cost. The key idea lies on using quadratic and linear basis functions of the mesh element where the departure point is localized in the interpolation procedures. The
proposed method is applied to well-established problems for transport and incompressible Navier-Stokes equations in three space dimensions. The numerical results
illustrate the performance of the proposed solver and support its ability to yield accurate and efficient numerical solutions for three-dimensional convection-dominated
flow problems on unstructured tetrahedral meshes. 相似文献
20.
Numerical Study of Heat and Moisture Transfer in Textile Materials by a Finite Volume Method 下载免费PDF全文
This paper focuses on the numerical study of heat and moisture transfer in
clothing assemblies, based on a multi-component and multiphase flow model which
includes heat/moisture convection and conduction/diffusion as well as phase change.
A splitting semi-implicit finite volume method is proposed for solving a set of nonlinear convection-diffusion-reaction equations, in which the calculation of liquid water
content absorbed by fiber is decoupled from the rest of the computation. The method
maintains the conservation of air, vapor and heat flux (energy). Four types of clothing
assemblies are investigated and comparison with experimental measurements are also
presented. 相似文献