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1.
A Conservative Numerical Method for the Cahn–Hilliard Equation with Generalized Mobilities on Curved Surfaces in Three-Dimensional Space
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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
2.
Extended Physics-Informed Neural Networks (XPINNs): A Generalized Space-Time Domain Decomposition Based Deep Learning Framework for Nonlinear Partial Differential Equations
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Ameya D. Jagtap & George Em Karniadakis 《Communications In Computational Physics》2020,28(5):2002-2041
We propose a generalized space-time domain decomposition approach for
the physics-informed neural networks (PINNs) to solve nonlinear partial differential
equations (PDEs) on arbitrary complex-geometry domains. The proposed framework,
named eXtended PINNs ($XPINNs$), further pushes the boundaries of both PINNs as
well as conservative PINNs (cPINNs), which is a recently proposed domain decomposition approach in the PINN framework tailored to conservation laws. Compared to
PINN, the XPINN method has large representation and parallelization capacity due to
the inherent property of deployment of multiple neural networks in the smaller subdomains. Unlike cPINN, XPINN can be extended to any type of PDEs. Moreover, the
domain can be decomposed in any arbitrary way (in space and time), which is not
possible in cPINN. Thus, XPINN offers both space and time parallelization, thereby
reducing the training cost more effectively. In each subdomain, a separate neural network is employed with optimally selected hyperparameters, e.g., depth/width of the
network, number and location of residual points, activation function, optimization
method, etc. A deep network can be employed in a subdomain with complex solution,
whereas a shallow neural network can be used in a subdomain with relatively simple
and smooth solutions. We demonstrate the versatility of XPINN by solving both forward and inverse PDE problems, ranging from one-dimensional to three-dimensional
problems, from time-dependent to time-independent problems, and from continuous
to discontinuous problems, which clearly shows that the XPINN method is promising in many practical problems. The proposed XPINN method is the generalization of
PINN and cPINN methods, both in terms of applicability as well as domain decomposition approach, which efficiently lends itself to parallelized computation. The XPINN
code is available on $https://github.com/AmeyaJagtap/XPINNs$. 相似文献
3.
A Constrained Finite Element Method Based on Domain Decomposition Satisfying the Discrete Maximum Principle for Diffusion Problems
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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
4.
In the paper, we develop and analyze a new mass-preserving splitting domain
decomposition method over multiple sub-domains for solving parabolic equations.
The domain is divided into non-overlapping multi-bock sub-domains. On the
interfaces of sub-domains, the interface fluxes are computed by the semi-implicit (explicit)
flux scheme. The solutions and fluxes in the interiors of sub-domains are computed
by the splitting one-dimensional implicit solution-flux coupled scheme. The
important feature is that the proposed scheme is mass conservative over multiple non-overlapping
sub-domains. Analyzing the mass-preserving S-DDM scheme is difficult
over non-overlapping multi-block sub-domains due to the combination of the splitting
technique and the domain decomposition at each time step. We prove theoretically
that our scheme satisfies conservation of mass over multi-block non-overlapping sub-domains
and it is unconditionally stable. We further prove the convergence and obtain
the error estimate in $L^2$-norm. Numerical experiments confirm theoretical results. 相似文献
5.
We present a parallel Cartesian method to solve elliptic problems with complex immersed interfaces. This method is based on a finite-difference scheme and is second-order accurate in the whole domain. The originality of the method lies in the use of additional unknowns located on the interface, allowing to express straightforwardly the interface transmission conditions. We describe the method and the details of its parallelization performed with the PETSc library. Then we present numerical validations in two dimensions, assorted with comparisons to other related methods, and a numerical study of the parallelized method. 相似文献
6.
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
7.
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
8.
Implementation of 2D Domain Decomposition in the UCAN Gyrokinetic Particle-in-Cell Code and Resulting Performance of UCAN2
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Jean-Noel G. Leboeuf Viktor K. Decyk David E. Newman & Raul Sanchez 《Communications In Computational Physics》2016,19(1):205-225
The massively parallel, nonlinear, three-dimensional (3D), toroidal, electrostatic,
gyrokinetic, particle-in-cell (PIC), Cartesian geometry UCAN code, with particle
ions and adiabatic electrons, has been successfully exercised to identify non-diffusive
transport characteristics in present day tokamak discharges. The limitation in applying
UCAN to larger scale discharges is the 1D domain decomposition in the toroidal (or
z-) direction for massively parallel implementation using MPI which has restricted the
calculations to a few hundred ion Larmor radii or gyroradii per plasma minor radius.
To exceed these sizes, we have implemented 2D domain decomposition in UCAN with
the addition of the y-direction to the processor mix. This has been facilitated by use
of relevant components in the P2LIB library of field and particle management routines
developed for UCLA's UPIC Framework of conventional PIC codes. The gyro-averaging
specific to gyrokinetic codes is simplified by the use of replicated arrays
for efficient charge accumulation and force deposition. The 2D domain-decomposed
UCAN2 code reproduces the original 1D domain nonlinear results within round-off.
Benchmarks of UCAN2 on the Cray XC30 Edison at NERSC demonstrate ideal scaling
when problem size is increased along with processor number up to the largest power
of 2 available, namely 131,072 processors. These particle weak scaling benchmarks
also indicate that the 1 nanosecond per particle per time step and 1 TFlops barriers are
easily broken by UCAN2 with 1 billion particles or more and 2000 or more processors. 相似文献
9.
Jinjing Xu Fei Zhao Zhiqiang Sheng & Guangwei Yuan 《Communications In Computational Physics》2021,29(3):747-766
In this paper we propose a new nonlinear cell-centered finite volume scheme
on general polygonal meshes for two dimensional anisotropic diffusion problems,
which preserves discrete maximum principle (DMP). The scheme is based on the so-called diamond scheme with a nonlinear treatment on its tangential flux to obtain a
local maximum principle (LMP) structure. It is well-known that existing DMP preserving diffusion schemes suffer from the fact that auxiliary unknowns should be presented as a convex combination of primary unknowns. In this paper, to get rid of
this constraint a nonlinearization strategy is introduced and it requires only a second-order accurate approximation for auxiliary unknowns. Numerical results show that
this scheme has second-order accuracy, preserves maximum and minimum for solutions and is conservative. 相似文献
10.
An Efficient Positivity-Preserving Finite Volume Scheme for the Nonequilibrium Three-Temperature Radiation Diffusion Equations on Polygonal Meshes
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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
11.
Explicit Computation of Robin Parameters in Optimized Schwarz Waveform Relaxation Methods for Schrödinger Equations Based on Pseudodifferential Operators
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Xavier Antoine & Emmanuel Lorin 《Communications In Computational Physics》2020,27(4):1032-1052
The Optimized Schwarz Waveform Relaxation algorithm, a domain decomposition method based on Robin transmission condition, is becoming a popular computational method for solving evolution partial differential equations in parallel. Along
with well-posedness, it offers a good balance between convergence rate, efficient computational complexity and simplicity of the implementation. The fundamental question is the selection of the Robin parameter to optimize the convergence of the algorithm. In this paper, we propose an approach to explicitly estimate the Robin parameter which is based on the approximation of the transmission operators at the subdomain interfaces, for the linear/nonlinear Schrödinger equation. Some illustrating
numerical experiments are proposed for the one- and two-dimensional problems. 相似文献
12.
A Decoupled and Positivity-Preserving DDFVS Scheme for Diffusion Problems on Polyhedral Meshes
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We propose a decoupled and positivity-preserving discrete duality finite
volume (DDFV) scheme for anisotropic diffusion problems on polyhedral meshes with
star-shaped cells and planar faces. Under the generalized DDFV framework, two sets
of finite volume (FV) equations are respectively constructed on the dual and primary
meshes, where the ones on the dual mesh are derived from the ingenious combination
of a geometric relationship with the construction of the cell matrix. The resulting system on the dual mesh is symmetric and positive definite, while the one on the primary
mesh possesses an M-matrix structure. To guarantee the positivity of the two categories of unknowns, a cutoff technique is introduced. As for the local conservation, it
is conditionally maintained on the dual mesh while strictly preserved on the primary
mesh. More interesting is that the FV equations on the dual mesh can be solved independently, so that the two sets of FV equations are decoupled. As a result, no nonlinear
iteration is required for linear problems and a general nonlinear solver could be used
for nonlinear problems. In addition, we analyze the well-posedness of numerical solutions for linear problems. The properties of the presented scheme are examined by
numerical experiments. The efficiency of the Newton method is also demonstrated by
comparison with those of the fixed-point iteration method and its Anderson acceleration. 相似文献
13.
High-Order Gas-Kinetic Scheme in Curvilinear Coordinates for the Euler and Navier-Stokes Solutions
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Liang Pan & Kun Xu 《Communications In Computational Physics》2020,28(4):1321-1351
The high-order gas-kinetic scheme (HGKS) has achieved success in simulating compressible flows with Cartesian meshes. To study the flow problems in general
geometries, such as the flow over a wing-body, the development of HGKS in general
curvilinear coordinates becomes necessary. In this paper, a two-stage fourth-order gas-kinetic scheme is developed for the Euler and Navier-Stokes solutions in the curvilinear coordinates from one-dimensional to three-dimensional computations. Based on
the coordinate transformation, the kinetic equation is transformed first to the computational space, and the flux function in the gas-kinetic scheme is obtained there and
is transformed back to the physical domain for the update of flow variables inside
each control volume. To achieve the expected order of accuracy, the dimension-by-dimension reconstruction based on the WENO scheme is adopted in the computational domain, where the reconstructed variables are the cell averaged Jacobian and the
Jacobian-weighted conservative variables. In the two-stage fourth-order gas-kinetic
scheme, the point values as well as the spatial derivatives of conservative variables at
Gaussian quadrature points have to be used in the evaluation of the time dependent
flux function. The point-wise conservative variables are obtained by ratio of the above
reconstructed data, and the spatial derivatives are reconstructed through orthogonalization in physical space and chain rule. A variety of numerical examples from the
accuracy tests to the solutions with strong discontinuities are presented to validate the
accuracy and robustness of the current scheme for both inviscid and viscous flows.
The precise satisfaction of the geometrical conservation law in non-orthogonal mesh is
also demonstrated through the numerical example. 相似文献
14.
An Implicit Algorithm for High-Order DG/FV Schemes for Compressible Flows on 2D Arbitrary Grids
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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
15.
John C. Morrison Kyle Steffen Blake Pantoja Asha Nagaiya Jacek Kobus & Thomas Ericsson 《Communications In Computational Physics》2016,19(3):632-647
In order to solve the partial differential equations that arise in the Hartree-Fock
theory for diatomic molecules and in molecular theories that include electron correlation,
one needs efficient methods for solving partial differential equations. In this
article, we present numerical results for a two-variable model problem of the kind that
arises when one solves the Hartree-Fock equations for a diatomic molecule. We compare
results obtained using the spline collocation and domain decomposition methods
with third-order Hermite splines to results obtained using the more-established finite
difference approximation and the successive over-relaxation method. The theory of
domain decomposition presented earlier is extended to treat regions that are divided
into an arbitrary number of subregions by families of lines parallel to the two coordinate
axes. While the domain decomposition method and the finite difference approach
both yield results at the micro-Hartree level, the finite difference approach with a 9-point difference formula produces the same level of accuracy with fewer points. The
domain decomposition method has the strength that it can be applied to problems with
a large number of grid points. The time required to solve a partial differential equation
for a fine grid with a large number of points goes down as the number of partitions
increases. The reason for this is that the length of time necessary for solving a set of
linear equations in each subregion is very much dependent upon the number of equations.
Even though a finer partition of the region has more subregions, the time for
solving the set of linear equations in each subregion is very much smaller. This feature
of the theory may well prove to be a decisive factor for solving the two-electron pair
equation, which – for a diatomic molecule – involves solving partial differential equations
with five independent variables. The domain decomposition theory also makes
it possible to study complex molecules by dividing them into smaller fragments thatare calculated independently. Since the domain decomposition approach makes it possible
to decompose the variable space into separate regions in which the equations are
solved independently, this approach is well-suited to parallel computing. 相似文献
16.
E. Abreu J. Douglas F. Furtado & F. Pereira 《Communications In Computational Physics》2009,6(1):72-84
We describe an operator splitting technique based on physics rather than
on dimension for the numerical solution of a nonlinear system of partial differential
equations which models three-phase flow through heterogeneous porous media. The
model for three-phase flow considered in this work takes into account capillary forces,
general relations for the relative permeability functions and variable porosity and permeability
fields. In our numerical procedure a high resolution, nonoscillatory, second
order, conservative central difference scheme is used for the approximation of the nonlinear
system of hyperbolic conservation laws modeling the convective transport of the
fluid phases. This scheme is combined with locally conservative mixed finite elements
for the numerical solution of the parabolic and elliptic problems associated with the
diffusive transport of fluid phases and the pressure-velocity problem. This numerical
procedure has been used to investigate the existence and stability of nonclassical shock
waves (called transitional or undercompressive shock waves) in two-dimensional heterogeneous
flows, thereby extending previous results for one-dimensional flow problems.
Numerical experiments indicate that the operator splitting technique discussed
here leads to computational efficiency and accurate numerical results. 相似文献
17.
Parallelization of an Implicit Algorithm for Multi-Dimensional Particle-in-Cell Simulations
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The implicit 2D3V particle-in-cell (PIC) code developed to study the interaction of ultrashort pulse lasers with matter [G. M. Petrov and J. Davis, Computer Phys.
Comm. 179, 868 (2008); Phys. Plasmas 18, 073102 (2011)] has been parallelized using
MPI (Message Passing Interface). The parallelization strategy is optimized for a small
number of computer cores, up to about 64. Details on the algorithm implementation
are given with emphasis on code optimization by overlapping computations with communications. Performance evaluation for 1D domain decomposition has been made
on a small Linux cluster with 64 computer cores for two typical regimes of PIC operation: "particle dominated", for which the bulk of the computation time is spent on
pushing particles, and "field dominated", for which computing the fields is prevalent.
For a small number of computer cores, less than 32, the MPI implementation offers a
significant numerical speed-up. In the "particle dominated" regime it is close to the
maximum theoretical one, while in the "field dominated" regime it is about 75-80%
of the maximum speed-up. For a number of cores exceeding 32, performance degradation takes place as a result of the adopted 1D domain decomposition. The code
parallelization will allow future implementation of atomic physics and extension to
three dimensions. 相似文献
18.
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
19.
Transition Operator Approach to Seismic Full-Waveform Inversion in Arbitrary Anisotropic Elastic Media
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Morten Jakobsen Ivan P&scaron enčí k Einar Iversen & Bjø rn Ursin 《Communications In Computational Physics》2020,28(1):297-327
We generalize the existing distorted Born iterative T-matrix (DBIT) method
to seismic full-waveform inversion (FWI) based on the scalar wave equation, so that it
can be used for seismic FWI in arbitrary anisotropic elastic media with variable mass
densities and elastic stiffness tensors. The elastodynamic wave equation for an arbitrary anisotropic heterogeneous medium is represented by an integral equation of
the Lippmann-Schwinger type, with a 9-dimensional wave state (displacement-strain)
vector. We solve this higher-dimensional Lippmann-Schwinger equation using a transition operator formalism used in quantum scattering theory. This allows for domain
decomposition and novel variational estimates. The tensorial nonlinear inverse scattering problem is solved iteratively by using an expression for the Fréchet derivatives
of the scattered wavefield with respect to elastic stiffness tensor fields in terms of modified Green's functions and wave state vectors that are updated after each iteration.
Since the generalized DBIT method is consistent with the Gauss-Newton method, it
incorporates approximate Hessian information that is essential for the reduction of
multi-parameter cross-talk effects. The DBIT method is implemented efficiently using
a variant of the Levenberg-Marquard method, with adaptive selection of the regularization parameter after each iteration. In a series of numerical experiments based
on synthetic waveform data for transversely isotropic media with vertical symmetry
axes, we obtained a very good match between the true and inverted models when
using the traditional Voigt parameterization. This suggests that the effects of cross-talk can be sufficiently reduced by the incorporation of Hessian information and the
use of suitable regularization methods. Since the generalized DBIT method for FWI
in anisotropic elastic media is naturally target-oriented, it may be particularly suitable
for applications to seismic reservoir characterization and monitoring. However, the
theory and method presented here is general. 相似文献
20.
Linghua Kong Jialin Hong & Jingjing Zhang 《Communications In Computational Physics》2013,14(1):219-241
The local one-dimensional multisymplectic scheme (LOD-MS) is developed for the three-dimensional (3D) Gross-Pitaevskii (GP) equation in Bose-Einstein condensates. The idea is originated from the advantages of multisymplectic integrators and from the cheap computational cost of the local one-dimensional (LOD) method. The 3D GP equation is split into three linear LOD Schrödinger equations and an exactly solvable nonlinear Hamiltonian ODE. The three linear LOD Schrödinger equations are multisymplectic which can be approximated by multisymplectic integrator (MI). The conservative properties of the proposed scheme are investigated. It is mass-preserving. Surprisingly, the scheme preserves the discrete local energy conservation laws and global energy conservation law if the wave function is variable separable. This is impossible for conventional MIs in nonlinear Hamiltonian context. The numerical results show that the LOD-MS can simulate the original problems very well. They are consistent with the numerical analysis. 相似文献