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1.
A Fourth-Order Upwinding Embedded Boundary Method (UEBM) for Maxwell's Equations in Media with Material Interfaces: Part I
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In this paper, we present a new fourth-order upwinding embedded boundary method (UEBM) over Cartesian grids, originally proposed in the Journal of Computational Physics [190 (2003), pp. 159-183.] as a second-order method for treating material interfaces for Maxwell's equations. In addition to the idea of the UEBM to evolve solutions at interfaces, we utilize the ghost fluid method to construct finite difference approximation of spatial derivatives at Cartesian grid points near the material interfaces. As a result, Runge-Kutta type time discretization can be used for the semidiscretized system to yield an overall fourth-order method, in contrast to the original second-order UEBM based on a Lax-Wendroff type difference. The final scheme allows time step sizes independent of the interface locations. Numerical examples are given to demonstrate the fourth-order accuracy as well as the stability of the method. We tested the scheme for several wave problems with various material interface locations, including electromagnetic scattering of a plane wave incident on a planar boundary and a two-dimensional electromagnetic application with an interface parallel to the y-axis. 相似文献
2.
High-Order Runge-Kutta Discontinuous Galerkin Methods with a New Type of Multi-Resolution WENO Limiters on Tetrahedral Meshes
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In this paper, the second-order and third-order Runge-Kutta discontinuous
Galerkin (RKDG) methods with multi-resolution weighted essentially non-oscillatory
(WENO) limiters are proposed on tetrahedral meshes. The multi-resolution WENO
limiter is an extension of a finite volume multi-resolution WENO scheme developed
in [81], which serves as a limiter for RKDG methods on tetrahedral meshes. This new
WENO limiter uses information of the DG solution essentially only within the troubled cell itself which is identified by a new modified version of the original KXRCF
indicator [42], to build a sequence of hierarchical $L^2$ projection polynomials from zeroth degree to the second or third degree of the DG solution. The second-order and
third-order RKDG methods with the associated multi-resolution WENO limiters are
developed as examples for general high-order RKDG methods, which could maintain
the original order of accuracy in smooth regions and keep essentially non-oscillatory
property near strong discontinuities by gradually degrading from the optimal order
to the first order. The linear weights inside the procedure of the new multi-resolution
WENO limiters can be set as any positive numbers on the condition that they sum
to one. A series of polynomials of different degrees within the troubled cell itself
are applied in a WENO fashion to modify the DG solutions in the troubled cell on
tetrahedral meshes. These new WENO limiters are very simple to construct, and can
be easily implemented to arbitrary high-order accuracy on tetrahedral meshes. Such
spatial reconstruction methodology improves the robustness in the simulation on the
same compact spatial stencil of the original DG methods on tetrahedral meshes. Extensive one-dimensional (run as three-dimensional problems on tetrahedral meshes)
and three-dimensional tests are performed to demonstrate the good performance of
the RKDG methods with new multi-resolution WENO limiters. 相似文献
3.
Implementation of Finite Difference Weighted Compact Nonlinear Schemes with the Two-Stage Fourth-Order Accurate Temporal Discretization
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Zhiwei He Fujie Gao Baolin Tian & Jiequan Li 《Communications In Computational Physics》2020,27(5):1470-1484
In this paper, we present a new two-stage fourth-order finite difference
weighted compact nonlinear scheme (WCNS) for hyperbolic conservation laws with
special application to compressible Euler equations. To construct this algorithm, apart
from the traditional WCNS for the spatial derivative, it was necessary to first construct
a linear compact/explicit scheme utilizing time derivative of flux at midpoints, which,
in turn, was solved by a generalized Riemann solver. Combining these two schemes,
the fourth-order time accuracy was achieved using only the two-stage time-stepping
technique. The final algorithm was numerically tested for various one-dimensional
and two-dimensional cases. The results demonstrated that the proposed algorithm
had an essentially similar performance as that based on the fourth-order Runge-Kutta
method, while it required 25 percent less computational cost for one-dimensional
cases, which is expected to decline further for multidimensional cases. 相似文献
4.
A Weighted Runge-Kutta Discontinuous Galerkin Method for 3D Acoustic and Elastic Wave-Field Modeling
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Numerically solving 3D seismic wave equations is a key requirement for
forward modeling and inversion. Here, we propose a weighted Runge-Kutta discontinuous Galerkin (WRKDG) method for 3D acoustic and elastic wave-field modeling. For this method, the second-order seismic wave equations in 3D heterogeneous anisotropic media are transformed into a first-order hyperbolic system, and
then we use a discontinuous Galerkin (DG) solver based on numerical-flux formulations for spatial discretization. The time discretization is based on an implicit diagonal Runge-Kutta (RK) method and an explicit iterative technique, which avoids
solving a large-scale system of linear equations. In the iterative process, we introduce
a weighting factor. We investigate the numerical stability criteria of the 3D method in
detail for linear and quadratic spatial basis functions. We also present a 3D analysis of
numerical dispersion for the full discrete approximation of acoustic equation, which
demonstrates that the WRKDG method can efficiently suppress numerical dispersion
on coarse grids. Numerical results for several different 3D models including homogeneous and heterogeneous media with isotropic and anisotropic cases show that the 3D
WRKDG method can effectively suppress numerical dispersion and provide accurate
wave-field information on coarse mesh. 相似文献
5.
A Strong Stability-Preserving Predictor-Corrector Method for the Simulation of Elastic Wave Propagation in Anisotropic Media
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In this paper, we propose a strong stability-preserving predictor-corrector
(SSPC) method based on an implicit Runge-Kutta method to solve the acoustic- and
elastic-wave equations. We first transform the wave equations into a system of ordinary differential equations (ODEs) and apply the local extrapolation method to discretize the spatial high-order derivatives, resulting in a system of semi-discrete ODEs.
Then we use the SSPC method based on an implicit Runge-Kutta method to solve
the semi-discrete ODEs and introduce a weighting parameter into the SSPC method.
On top of such a structure, we develop a robust numerical algorithm to effectively
suppress the numerical dispersion, which is usually caused by the discretization of
wave equations when coarse grids are used or geological models have large velocity
contrasts between adjacent layers. Meanwhile, we investigate the performance of the
SSPC method including numerical errors and convergence rate, numerical dispersion,
and stability criteria with different choices of the weighting parameter to solve 1-D
and 2-D acoustic- and elastic-wave equations. When the SSPC is applied to seismic
simulations, the computational efficiency is also investigated by comparing the SSPC,
the fourth-order Lax-Wendroff correction (LWC) method, and the staggered-grid (SG)
finite difference method. Comparisons of synthetic waveforms computed by the SSPC
and analytic solutions for acoustic and elastic models are given to illustrate the accuracy and the validity of the SSPC method. Furthermore, several numerical experiments
are conducted for the geological models including a 2-D homogeneous transversely
isotropic (TI) medium, a two-layer elastic model, and the 2-D SEG/EAGE salt model.
The results show that the SSPC can be used as a practical tool for large-scale seismic
simulation because of its effectiveness in suppressing numerical dispersion even in the
situations such as coarse grids, strong interfaces, or high frequencies. 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.
In this paper, we investigate the ideal magnetohydrodynamic (MHD) equations with random inputs based on generalized polynomial chaos (gPC) stochastic
Galerkin approximation. A special treatment with symmetrization is carried out for
the gPC stochastic Galerkin method so that the resulting deterministic gPC Galerkin
system is provably symmetric hyperbolic in the spatially one-dimensional case. We
discretize the hyperbolic gPC Galerkin system with a high-order path-conservative finite volume weighted essentially non-oscillatory scheme in space and a third-order total variation diminishing Runge-Kutta method in time. The method is also extended to
two spatial dimensions via the operator splitting technique. Several numerical examples are provided to illustrate the accuracy and effectiveness of the numerical scheme. 相似文献
8.
Jun Zhu & Jianxian Qiu 《Communications In Computational Physics》2020,27(3):897-920
In this paper, a new type of third-order and fourth-order weighted essentially non-oscillatory (WENO) schemes is designed for simulating the Hamilton-Jacobi equations on triangular meshes. We design such schemes with the use of the nodal information defined on five unequal-sized spatial stencils, the application of monotone Hamiltonians as a building block, the artificial set of positive linear weights to make up high-order approximations in smooth regions simultaneously avoiding spurious oscillations nearby discontinuities of the derivatives of the solutions. The spatial reconstructions are convex combinations of the derivatives of a modified cubic/quartic polynomial defined on a big spatial stencil and four quadratic polynomials defined on small spatial stencils, and a third-order TVD Runge-Kutta method is used for the time discretization. The main advantages of these WENO schemes are their efficiency, simplicity, and can be easily implemented to higher dimensional unstructured meshes. Extensive numerical tests are performed to illustrate the good performance of such new WENO schemes. 相似文献
9.
High-Order Positivity-Preserving Well-Balanced Discontinuous Galerkin Methods for Euler Equations with Gravitation on Unstructured Meshes
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Weijie Zhang Yulong Xing Yinhua Xia & Yan Xu 《Communications In Computational Physics》2022,31(3):771-815
In this paper, we propose a high-order accurate discontinuous Galerkin
(DG) method for the compressible Euler equations under gravitational fields on unstructured meshes. The scheme preserves a general hydrostatic equilibrium state and
provably guarantees the positivity of density and pressure at the same time. Comparing with the work on the well-balanced scheme for Euler equations with gravitation
on rectangular meshes, the extension to triangular meshes is conceptually plausible
but highly nontrivial. We first introduce a special way to recover the equilibrium state
and then design a group of novel variables at the interface of two adjacent cells, which
plays an important role in the well-balanced and positivity-preserving properties. One
main challenge is that the well-balanced schemes may not have the weak positivity
property. In order to achieve the well-balanced and positivity-preserving properties
simultaneously while maintaining high-order accuracy, we carefully design DG spatial discretization with well-balanced numerical fluxes and suitable source term approximation. For the ideal gas, we prove that the resulting well-balanced scheme, coupled with strong stability preserving time discretizations, satisfies a weak positivity
property. A simple existing limiter can be applied to enforce the positivity-preserving
property, without losing high-order accuracy and conservation. Extensive one- and
two-dimensional numerical examples demonstrate the desired properties of the proposed scheme, as well as its high resolution and robustness. 相似文献
10.
Bhuvana Srinivasan Ammar Hakim & Uri Shumlak 《Communications In Computational Physics》2011,10(1):183-215
The finite volume wave propagation method and the finite element RungeKutta
discontinuous Galerkin (RKDG) method are studied for applications to balance
laws describing plasma fluids. The plasma fluid equations explored are dispersive and
not dissipative. The physical dispersion introduced through the source terms leads to
the wide variety of plasma waves. The dispersive nature of the plasma fluid equations
explored separates the work in this paper from previous publications. The linearized
Euler equations with dispersive source terms are used as a model equation system to
compare the wave propagation and RKDG methods. The numerical methods are then
studied for applications of the full two-fluid plasma equations. The two-fluid equations
describe the self-consistent evolution of electron and ion fluids in the presence
of electromagnetic fields. It is found that the wave propagation method, when run
at a CFL number of 1, is more accurate for equation systems that do not have disparate
characteristic speeds. However, if the oscillation frequency is large compared
to the frequency of information propagation, source splitting in the wave propagation
method may cause phase errors. The Runge-Kutta discontinuous Galerkin method
provides more accurate results for problems near steady-state as well as problems with
disparate characteristic speeds when using higher spatial orders. 相似文献
11.
A RKDG Method for 2D Lagrangian Ideal Magnetohydrodynamics Equations with Exactly Divergence-Free Magnetic Field
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Shijun Zou Xiaolong Zhao Xijun Yu & Zihuan Dai 《Communications In Computational Physics》2022,32(2):547-582
In this paper, we present a Runge-Kutta Discontinuous Galerkin (RKDG)
method for solving the two-dimensional ideal compressible magnetohydrodynamics
(MHD) equations under the Lagrangian framework. The fluid part of the ideal MHD
equations along with $z$-component of the magnetic induction equation are discretized
using a DG method based on linear Taylor expansions. By using the magnetic flux-freezing principle which is the integral form of the magnetic induction equation of
the ideal MHD, an exactly divergence-free numerical magnetic field can be obtained.
The nodal velocities and the corresponding numerical fluxes are explicitly calculated
by solving multidirectional approximate Riemann problems. Two kinds of limiter are
proposed to inhibit the non-physical oscillation around the shock wave, and the second limiter can eliminate the phenomenon of mesh tangling in the simulations of the
rotor problems. This Lagrangian RKDG method conserves mass, momentum, and
total energy. Several numerical tests are presented to demonstrate the accuracy and
robustness of the proposed scheme. 相似文献
12.
Dominik Dierkes Florian Kummer & Dominik Plü macher 《Communications In Computational Physics》2021,30(1):288-320
We present a high-order discontinuous Galerkin (DG) scheme to solve the
system of helically symmetric Navier-Stokes equations which are discussed in [28].
In particular, we discretize the helically reduced Navier-Stokes equations emerging
from a reduction of the independent variables such that the remaining variables are: $t$, $r$, $ξ$ with $ξ=az+bϕ$, where $r$, $ϕ$, $z$ are common cylindrical coordinates and $t$ the
time. Beside this, all three velocity components are kept non-zero. A new non-singular
coordinate $η$ is introduced which ensures that a mapping of helical solutions into the
three-dimensional space is well defined. Using that, periodicity conditions for the
helical frame as well as uniqueness conditions at the centerline axis $r=0$ are derived. In
the sector near the axis of the computational domain a change of the polynomial basis
is implemented such that all physical quantities are uniquely defined at the centerline.For the temporal integration, we present a semi-explicit scheme of third order
where the full spatial operator is split into a Stokes operator which is discretized
implicitly and an operator for the nonlinear terms which is treated explicitly. Computations are conducted for a cylindrical shell, excluding the centerline axis, and for
the full cylindrical domain, where the centerline is included. In all cases we obtain the
convergence rates of order $\mathcal{O}(h^{k+1})$ that are expected from DG theory.In addition to the first DG discretization of the system of helically invariant Navier-Stokes equations, the treatment of the central axis, the resulting reduction of the DG
space, and the simultaneous use of a semi-explicit time stepper are of particular novelty. 相似文献
13.
A Simple Solver for the Two-Fluid Plasma Model Based on PseudoSpectral Time-Domain Algorithm
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Benoit Morel Remo Giust Kazem Ardaneh & Francois Courvoisier 《Communications In Computational Physics》2021,29(3):955-978
We present a solver of 3D two-fluid plasma model for the simulation of
short-pulse laser interactions with plasma. This solver resolves the equations of the
two-fluid plasma model with ideal gas closure. We also include the Bhatnagar-Gross-Krook collision model. Our solver is based on PseudoSpectral Time-Domain (PSTD)
method to solve Maxwell's curl equations. We use a Strang splitting to integrate Euler
equations with source term: while Euler equations are solved with a composite scheme
mixing Lax-Friedrichs and Lax-Wendroff schemes, the source term is integrated with
a fourth-order Runge-Kutta scheme. This two-fluid plasma model solver is simple to
implement because it only relies on finite difference schemes and Fast Fourier Transforms. It does not require spatially staggered grids. The solver was tested against
several well-known problems of plasma physics. Numerical simulations gave results
in excellent agreement with analytical solutions or with previous results from the literature. 相似文献
14.
Directional $\mathcal{H}^2$ Compression Algorithm: Optimisations and Application to a Discontinuous Galerkin BEM for the Helmholtz Equation
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Nadir-Alexandre Messaï Sebastien Pernet & Abdesselam Bouguerra 《Communications In Computational Physics》2022,31(5):1585-1635
This study aimed to specialise a directional $\mathcal{H}^2
(\mathcal{D}\mathcal{H}^2)$ compression to matrices arising from the discontinuous Galerkin (DG) discretisation of the hypersingular
equation in acoustics. The significant finding is an algorithm that takes a DG stiffness matrix and finds a near-optimal $\mathcal{D}\mathcal{H}^2$ approximation for low and high-frequency
problems. We introduced the necessary special optimisations to make this algorithm
more efficient in the case of a DG stiffness matrix. Moreover, an automatic parameter
tuning strategy makes it easy to use and versatile. Numerical comparisons with a classical Boundary Element Method (BEM) show that a DG scheme combined with a $\mathcal{D}\mathcal{H}^2$ gives better computational efficiency than a classical BEM in the case of high-order finite elements and $hp$ heterogeneous meshes. The results indicate that DG is suitable
for an auto-adaptive context in integral equations. 相似文献
15.
Jun Zhu Xinghui Zhong Chi-Wang Shu & Jianxian Qiu 《Communications In Computational Physics》2016,19(4):944-969
In this paper, we propose a new type of weighted essentially non-oscillatory
(WENO) limiter, which belongs to the class of Hermite WENO (HWENO) limiters, for
the Runge-Kutta discontinuous Galerkin (RKDG) methods solving hyperbolic conservation
laws. This new HWENO limiter is a modification of the simple WENO limiter
proposed recently by Zhong and Shu [29]. Both limiters use information of the DG
solutions only from the target cell and its immediate neighboring cells, thus maintaining
the original compactness of the DG scheme. The goal of both limiters is to obtain
high order accuracy and non-oscillatory properties simultaneously. The main novelty
of the new HWENO limiter in this paper is to reconstruct the polynomial on the target
cell in a least square fashion [8] while the simple WENO limiter [29] is to use the entire
polynomial of the original DG solutions in the neighboring cells with an addition of
a constant for conservation. The modification in this paper improves the robustness
in the computation of problems with strong shocks or contact discontinuities, without
changing the compact stencil of the DG scheme. Numerical results for both one and
two dimensional equations including Euler equations of compressible gas dynamics
are provided to illustrate the viability of this modified limiter. 相似文献
16.
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
17.
Ziqing Xie Jiangxing Wang Bo Wang & Chuanmiao Chen 《Communications In Computational Physics》2016,19(5):1242-1264
In this paper, an approach combining the DG method in space with CG
method in time (CG-DG method) is developed to solve time-dependent Maxwell's
equations when meta-materials are involved. Both the unconditional $L^2$-stability and
error estimate of order $\mathcal{O}$($τ^ {r+1}$+$h^{k+\frac{1}{2}}$) are obtained when polynomials of degree at
most r is used for the temporal discretization and at most k for the spatial discretization.
Numerical results in 3D are given to validate the theoretical results. 相似文献
18.
A Sparse Grid Discrete Ordinate Discontinuous Galerkin Method for the Radiative Transfer Equation
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Jianguo Huang & Yue Yu 《Communications In Computational Physics》2021,30(4):1009-1036
The radiative transfer equation is a fundamental equation in transport theory and applications, which is a 5-dimensional PDE in the stationary one-velocity case,
leading to great difficulties in numerical simulation. To tackle this bottleneck, we first
use the discrete ordinate technique to discretize the scattering term, an integral with respect to the angular variables, resulting in a semi-discrete hyperbolic system. Then, we
make the spatial discretization by means of the discontinuous Galerkin (DG) method
combined with the sparse grid method. The final linear system is solved by the block
Gauss-Seidal iteration method. The computational complexity and error analysis are
developed in detail, which show the new method is more efficient than the original
discrete ordinate DG method. A series of numerical results are performed to validate
the convergence behavior and effectiveness of the proposed method. 相似文献
19.
A Well-Balanced Positivity-Preserving Quasi-Lagrange Moving Mesh DG Method for the Shallow Water Equations
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A high-order, well-balanced, positivity-preserving quasi-Lagrange moving
mesh DG method is presented for the shallow water equations with non-flat bottom
topography. The well-balance property is crucial to the ability of a scheme to simulate perturbation waves over the lake-at-rest steady state such as waves on a lake or
tsunami waves in the deep ocean. The method combines a quasi-Lagrange moving
mesh DG method, a hydrostatic reconstruction technique, and a change of unknown
variables. The strategies in the use of slope limiting, positivity-preservation limiting,
and change of variables to ensure the well-balance and positivity-preserving properties are discussed. Compared to rezoning-type methods, the current method treats
mesh movement continuously in time and has the advantages that it does not need to
interpolate flow variables from the old mesh to the new one and places no constraint
for the choice of a update scheme for the bottom topography on the new mesh. A selection of one- and two-dimensional examples are presented to demonstrate the well-balance property, positivity preservation, and high-order accuracy of the method and
its ability to adapt the mesh according to features in the flow and bottom topography. 相似文献
20.
Hendrik Ranocha Dimitrios Mitsotakis & David I. Ketcheson 《Communications In Computational Physics》2021,29(4):979-1029
We develop a general framework for designing conservative numerical
methods based on summation by parts operators and split forms in space, combined
with relaxation Runge-Kutta methods in time. We apply this framework to create
new classes of fully-discrete conservative methods for several nonlinear dispersive
wave equations: Benjamin-Bona-Mahony (BBM), Fornberg-Whitham, Camassa-Holm,
Degasperis-Procesi, Holm-Hone, and the BBM-BBM system. These full discretizations
conserve all linear invariants and one nonlinear invariant for each system. The spatial
semidiscretizations include finite difference, spectral collocation, and both discontinuous and continuous finite element methods. The time discretization is essentially
explicit, using relaxation Runge-Kutta methods. We implement some specific schemes
from among the derived classes, and demonstrate their favorable properties through
numerical tests. 相似文献