共查询到20条相似文献,搜索用时 31 毫秒
1.
Development of a Combined Compact Difference Scheme to Simulate Soliton Collision in a Shallow Water Equation 下载免费PDF全文
In this paper a three-step scheme is applied to solve the Camassa-Holm
(CH) shallow water equation. The differential order of the CH equation has been
reduced in order to facilitate development of numerical scheme in a comparatively
smaller grid stencil. Here a three-point seventh-order spatially accurate upwinding
combined compact difference (CCD) scheme is proposed to approximate the first-order
derivative term. We conduct modified equation analysis on the CCD scheme and
eliminate the leading discretization error terms for accurately predicting unidirectional
wave propagation. The Fourier analysis is carried out as well on the proposed numerical
scheme to minimize the dispersive error. For preserving Hamiltonians in Camassa-Holm
equation, a symplecticity conserving time integrator has been employed. The
other main emphasis of the present study is the use of u−P−α formulation to get nondissipative
CH solution for peakon-antipeakon and soliton-anticuspon head-on wave
collision problems. 相似文献
2.
Development of a High-Resolution Scheme for Solving the PNP-NS Equations in Curved Channels 下载免费PDF全文
Tony W. H. Sheu Yogesh G. Bhumkar S. T. Yuan & S. C. Syue 《Communications In Computational Physics》2016,19(2):496-533
A high-order finite difference scheme has been developed to approximate
the spatial derivative terms present in the unsteady Poisson-Nernst-Planck (PNP) equations
and incompressible Navier-Stokes (NS) equations. Near the wall the sharp solution
profiles are resolved by using the combined compact difference (CCD) scheme
developed in five-point stencil. This CCD scheme has a sixth-order accuracy for the
second-order derivative terms while a seventh-order accuracy for the first-order derivative
terms. PNP-NS equations have been also transformed to the curvilinear coordinate
system to study the effects of channel shapes on the development of electroosmotic
flow. In this study, the developed scheme has been analyzed rigorously through
the modified equation analysis. In addition, the developed method has been computationally
verified through four problems which are amenable to their own exact solutions.
The electroosmotic flow details in planar and wavy channels have been explored
with the emphasis on the formation of Coulomb force. Significance of different forces
resulting from the pressure gradient, diffusion and Coulomb origins on the convective
electroosmotic flow motion is also investigated in detail. 相似文献
3.
Simulation of Incompressible Free Surface Flow Using the Volume Preserving Level Set Method 下载免费PDF全文
This study aims to develop a numerical scheme in collocated Cartesian grids
to solve the level set equation together with the incompressible two-phase flow equations.
A seventh-order accurate upwinding combined compact difference (UCCD7)
scheme has been developed for the approximation of the first-order spatial derivative
terms shown in the level set equation. Developed scheme has a higher accuracy with a
three-point grid stencil to minimize phase error. To preserve the mass of each phase all
the time, the temporal derivative term in the level set equation is approximated by the
sixth-order accurate symplectic Runge-Kutta (SRK6) scheme. All the simulated results
for the dam-break, Rayleigh-Taylor instability, bubble rising, two-bubble merging, and
milkcrown problems in two and three dimensions agree well with the available numerical
or experimental results. 相似文献
4.
Numerical Simulation of Time-Harmonic Waves in Inhomogeneous Media Using Compact High Order Schemes 下载免费PDF全文
Steven Britt Semyon Tsynkov & Eli Turkel 《Communications In Computational Physics》2011,9(3):520-541
In many problems, one wishes to solve the Helmholtz equation with variable
coefficients within the Laplacian-like term and use a high order accurate method
(e.g., fourth order accurate) to alleviate the points-per-wavelength constraint by reducing
the dispersion errors. The variation of coefficients in the equation may be due
to an inhomogeneous medium and/or non-Cartesian coordinates. This renders existing
fourth order finite difference methods inapplicable. We develop a new compact
scheme that is provably fourth order accurate even for these problems. We present
numerical results that corroborate the fourth order convergence rate for several model
problems. 相似文献
5.
Construction and Analysis of an Adapted Spectral Finite Element Method to Convective Acoustic Equations 下载免费PDF全文
Andreas Hü ppe Gary Cohen Sé bastien Imperiale & Manfred Kaltenbacher 《Communications In Computational Physics》2016,20(1):1-22
The paper addresses the construction of a non spurious mixed spectral finite
element (FE) method to problems in the field of computational aeroacoustics. Based
on a computational scheme for the conservation equations of linear acoustics, the extension
towards convected wave propagation is investigated. In aeroacoustic applications,
the mean flow effects can have a significant impact on the generated sound
field even for smaller Mach numbers. For those convective terms, the initial spectral
FE discretization leads to non-physical, spurious solutions. Therefore, a regularization
procedure is proposed and qualitatively investigated by means of discrete eigenvalues
analysis of the discrete operator in space. A study of convergence and an application
of the proposed scheme to simulate the flow induced sound generation in the process
of human phonation underlines stability and validity. 相似文献
6.
Implementation of Finite Difference Weighted Compact Nonlinear Schemes with the Two-Stage Fourth-Order Accurate Temporal Discretization 下载免费PDF全文
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
7.
A fourth-order finite difference method is proposed and studied for the
primitive equations (PEs) of large-scale atmospheric and oceanic flow based on mean
vorticity formulation. Since the vertical average of the horizontal velocity field is
divergence-free, we can introduce mean vorticity and mean stream function which are
connected by a 2-D Poisson equation. As a result, the PEs can be reformulated such that
the prognostic equation for the horizontal velocity is replaced by evolutionary equations for the mean vorticity field and the vertical derivative of the horizontal velocity.
The mean vorticity equation is approximated by a compact difference scheme due to
the difficulty of the mean vorticity boundary condition, while fourth-order long-stencil
approximations are utilized to deal with transport type equations for computational
convenience. The numerical values for the total velocity field (both horizontal and
vertical) are statically determined by a discrete realization of a differential equation at
each fixed horizontal point. The method is highly efficient and is capable of producing highly resolved solutions at a reasonable computational cost. The full fourth-order
accuracy is checked by an example of the reformulated PEs with force terms. Additionally, numerical results of a large-scale oceanic circulation are presented. 相似文献
8.
We apply in this study an area preserving level set method to simulate
gas/water interface flow. For the sake of accuracy, the spatial derivative terms in the
equations of motion for an incompressible fluid flow are approximated by the fifth-order accurate upwinding combined compact difference (UCCD) scheme. This scheme
development employs two coupled equations to calculate the first- and second-order
derivative terms in the momentum equations. For accurately predicting the level set
value, the interface tracking scheme is also developed to minimize phase error of the
first-order derivative term shown in the pure advection equation. For the purpose of
retaining the long-term accurate Hamiltonian in the advection equation for the level
set function, the time derivative term is discretized by the sixth-order accurate symplectic Runge-Kutta scheme. Also, to keep as a distance function for ensuring the front
having a finite thickness for all time, the re-initialization equation is used. For the verification of the optimized UCCD scheme for the pure advection equation, two benchmark problems have been chosen to investigate in this study. The level set method
with excellent area conservation property proposed for capturing the interface in incompressible fluid flows is also verified by solving the dam-break, Rayleigh-Taylor
instability, two-bubble rising in water, and droplet falling problems. 相似文献
9.
Development of an Explicit Symplectic Scheme that Optimizes the Dispersion-Relation Equation of the Maxwell's Equations 下载免费PDF全文
Tony W. H. Sheu L. Y. Liang & J. H. Li 《Communications In Computational Physics》2013,13(4):1107-1133
In this paper an explicit finite-difference time-domain scheme for solving
the Maxwell's equations in non-staggered grids is presented. The proposed scheme
for solving the Faraday's and Ampère's equations in a theoretical manner is aimed to
preserve discrete zero-divergence for the electric and magnetic fields. The inherent local conservation laws in Maxwell's equations are also preserved discretely all the time
using the explicit second-order accurate symplectic partitioned Runge-Kutta scheme.
The remaining spatial derivative terms in the semi-discretized Faraday's and Ampère's
equations are then discretized to provide an accurate mathematical dispersion relation
equation that governs the numerical angular frequency and the wavenumbers in two
space dimensions. To achieve the goal of getting the best dispersive characteristics, we
propose a fourth-order accurate space centered scheme which minimizes the difference
between the exact and numerical dispersion relation equations. Through the computational exercises, the proposed dual-preserving solver is computationally demonstrated
to be efficient for use to predict the long-term accurate Maxwell's solutions. 相似文献
10.
A Space-Time Conservative Method for Hyperbolic Systems with Stiff and Non-Stiff Source Terms 下载免费PDF全文
In this article we propose a higher-order space-time conservative method
for hyperbolic systems with stiff and non-stiff source terms as well as relaxation systems. We call the scheme a slope propagation (SP) method. It is an extension of our
scheme derived for homogeneous hyperbolic systems [1]. In the present inhomogeneous
systems the relaxation time may vary from order of one to a very small value. These
small values make the relaxation term stronger and highly stiff. In such situations
underresolved numerical schemes may produce spurious numerical results. However,
our present scheme has the capability to correctly capture the behavior of the physical
phenomena with high order accuracy even if the initial layer and the small relaxation
time are not numerically resolved. The scheme treats the space and time in a unified
manner. The flow variables and their slopes are the basic unknowns in the scheme. The
source term is treated by its volumetric integration over the space-time control volume
and is a direct part of the overall space-time flux balance. We use two approaches
for the slope calculations of the flow variables, the first one results directly from the
flux balance over the control volumes, while in the second one we use a finite difference approach. The main features of the scheme are its simplicity, its Jacobian-free
and Riemann solver-free recipe, as well as its efficiency and high order accuracy. In
particular we show that the scheme has a discrete analog of the continuous asymptotic limit. We have implemented our scheme for various test models available in the
literature such as the Broadwell model, the extended thermodynamics equations, the
shallow water equations, traffic flow and the Euler equations with heat transfer. The
numerical results validate the accuracy, versatility and robustness of the present scheme. 相似文献
11.
On the Stability and CPU Time of the Implicit Runge-Kutta Schemes for Steady State Simulations 下载免费PDF全文
Implicit time integration schemes are popular because their relaxed stability
constraints can result in better computational efficiency. For time-accurate unsteady
simulations, it has been well recognized that the inherent dispersion and dissipation
errors of implicit Runge-Kutta schemes will reduce the computational accuracy for
large time steps. Yet for steady state simulations using the time-dependent governing
equations, these errors are often overlooked because the intermediate solutions are of
less interest. Based on the model equation dy/dt = (µ+iλ)y of scalar convection diffusion
systems, this study examines the stability limits, dispersion and dissipation errors
of four diagonally implicit Runge-Kutta-type schemes on the complex (µ+iλ)∆t
plane. Through numerical experiments, it is shown that, as the time steps increase,
the A-stable implicit schemes may not always have reduced CPU time and the computations
may not always remain stable, due to the inherent dispersion and dissipation
errors of the implicit Runge-Kutta schemes. The dissipation errors may decelerate the
convergence rate, and the dispersion errors may cause large oscillations of the numerical
solutions. These errors, especially those of high wavenumber components, grow
at large time steps. They lead to difficulty in the convergence of the numerical computations,
and result in increasing CPU time or even unstable computations as the time
step increases. It is concluded that an optimal implicit time integration scheme for
steady state simulations should have high dissipation and low dispersion. 相似文献
12.
In this paper, a new five-point targeted essentially non-oscillatory (TENO)
scheme with adaptive dissipation is proposed. With the standard TENO weighting
strategy, the cut-off parameter $C_T$ determines the nonlinear numerical dissipation of
the resultant TENO scheme. Moreover, according to the dissipation-adaptive TENO5-A scheme, the choice of the cut-off parameter $C_T$ highly depends on the effective scale
sensor. However, the scale sensor in TENO5-A can only roughly detect the discontinuity locations instead of evaluating the local flow wavenumber as desired. In this
work, a new five-point scale sensor, which can estimate the local flow wavenumber accurately, is proposed to further improve the performance of TENO5-A. In combination
with a hyperbolic tangent function, the new scale sensor is deployed to the TENO5-A
framework for adapting the cut-off parameter $C_T,$ i.e., the local nonlinear dissipation,
according to the local flow wavenumber. Overall, sufficient numerical dissipation is
generated to capture discontinuities, whereas a minimum amount of dissipation is delivered for better resolving the smooth flows. A set of benchmark cases is simulated to
demonstrate the performance of the new TENO5-A scheme. 相似文献
13.
A Hermite WENO Method with Modified Ghost Fluid Method for Compressible Two-Medium Flow Problems 下载免费PDF全文
Zhuang Zhao Yong-Tao Zhang Yibing Chen & Jianxian Qiu 《Communications In Computational Physics》2021,30(3):851-873
In this paper, we develop a novel approach by combining a new robust finite difference Hermite weighted essentially non-oscillatory (HWENO) method [51]
with the modified ghost fluid method (MGFM) [25] to simulate the compressible two-medium flow problems. The main idea is that we first use the technique of the MGFM
to transform a two-medium flow problem to two single-medium cases by defining the
ghost fluids status based on the predicted interface status. Then the efficient and robust
HWENO finite difference method is applied for solving the single-medium flow cases.
By using immediate neighbor information to deal with both the solution and its derivatives, the fifth order finite difference HWENO scheme adopted in this paper is more
compact and has higher resolution than the classical fifth order finite difference WENO
scheme of Jiang and Shu [14]. Furthermore, by combining the HWENO scheme with
the MGFM to simulate the two-medium flow problems, less ghost point information
is needed than that in using the classical WENO scheme in order to obtain the same
numerical accuracy. Various one-dimensional and two-dimensional two-medium flow
problems are solved to illustrate the good performances of the proposed method. 相似文献
14.
An Adaptive Grid Method for Singularly Perturbed Time-Dependent Convection-Diffusion Problems 下载免费PDF全文
In this paper, we study the numerical solution of singularly perturbed time-dependent
convection-diffusion problems. To solve these problems, the backward Euler
method is first applied to discretize the time derivative on a uniform mesh, and the
classical upwind finite difference scheme is used to approximate the spatial derivative
on an arbitrary nonuniform grid. Then, in order to obtain an adaptive grid for all
temporal levels, we construct a positive monitor function, which is similar to the arc-length
monitor function. Furthermore, the ε-uniform convergence of the fully discrete
scheme is derived for the numerical solution. Finally, some numerical results are given
to support our theoretical results. 相似文献
15.
Arnab Ghosh Alessandro Gabbana Herman Wijshoff & Federico Toschi 《Communications In Computational Physics》2023,33(1):349-366
The immersed boundary method has emerged as an efficient approach for
the simulation of finite-sized solid particles in complex fluid flows. However, one of
the well known shortcomings of the method is the limited support for the simulation
of light particles, i.e. particles with a density lower than that of the surrounding fluid,
both in terms of accuracy and numerical stability.Although a broad literature exists, with several authors reporting different approaches
for improving the stability of the method, most of these attempts introduce extra complexities and are very costly from a computational point of view.In this work, we introduce an effective force stabilizing technique, allowing to extend
the stability range of the method by filtering spurious oscillations arising when dealing
with light-particles, pushing down the particle-to-fluid density ratio as low as 0.04.
We thoroughly validate the method comparing with both experimental and numerical
data available in literature. 相似文献
16.
R. K. Mohanty M. K. Jain & B. N. Mishra 《Communications In Computational Physics》2012,12(5):1417-1433
In this article, we present two new novel finite difference approximations of order two and four, respectively, for the three dimensional non-linear triharmonic partial differential equations on a compact stencil where the values of u, ∂2u/∂n2 and ∂4u/∂n4 are prescribed on the boundary. We introduce new ideas to handle the boundary conditions and there is no need to discretize the derivative boundary conditions. We require only 7- and 19-grid points on the compact cell for the second and fourth order approximation, respectively. The Laplacian and the biharmonic of the solution are obtained as by-product of the methods. We require only system of three equations to obtain the solution. Numerical results are provided to illustrate the usefulness of the proposed methods. 相似文献
17.
A Sub-Grid Structure Enhanced Discontinuous Galerkin Method for Multiscale Diffusion and Convection-Diffusion Problems 下载免费PDF全文
In this paper, we present an efficient computational methodology for diffusion and convection-diffusion problems in highly heterogeneous media as well as convection-dominated diffusion problem. It is well known that the numerical computation for these problems requires a significant amount of computer memory and time. Nevertheless, the solutions to these problems typically contain a coarse component, which is usually the quantity of interest and can be represented with a small number of degrees of freedom. There are many methods that aim at the computation of the coarse component without resolving the full details of the solution. Our proposed method falls into the framework of interior penalty discontinuous Galerkin method, which is proved to be an effective and accurate class of methods for numerical solutions of partial differential equations. A distinctive feature of our method is that the solution space contains two components, namely a coarse space that gives a polynomial approximation to the coarse component in the traditional way and a multiscale space which contains sub-grid structures of the solution and is essential to the computation of the coarse component. In addition, stability of the method is proved. The numerical results indicate that the method can accurately capture the coarse behavior of the solution for problems in highly heterogeneous media as well as boundary and internal layers for convection-dominated problems. 相似文献
18.
Higher-Order Compact Scheme for the Incompressible Navier-Stokes Equations in Spherical Geometry 下载免费PDF全文
T. V. S. Sekhar B. Hema Sundar Raju & Y. V. S. S. Sanyasiraju 《Communications In Computational Physics》2012,11(1):99-113
A higher-order compact scheme on the nine point 2-D stencil is developed
for the steady stream-function vorticity form of the incompressible Navier-Stokes (NS) equations in spherical polar coordinates, which was used earlier only for the cartesian and cylindrical geometries. The steady, incompressible, viscous and axially symmetric flow past a sphere is used as a model problem. The non-linearity in the N-S
equations is handled in a comprehensive manner avoiding complications in calculations. The scheme is combined with the multigrid method to enhance the convergence
rate. The solutions are obtained over a non-uniform grid generated using the transformation r = eξ while maintaining a uniform grid in the computational plane. The
superiority of the higher order compact scheme is clearly illustrated in comparison
with upwind scheme and defect correction technique at high Reynolds numbers by
taking a large domain. This is a pioneering effort, because for the first time, the fourth
order accurate solutions for the problem of viscous flow past a sphere are presented
here. The drag coefficient and surface pressures are calculated and compared with
available experimental and theoretical results. It is observed that these values simulated over coarser grids using the present scheme are more accurate when compared to
other conventional schemes. It has also been observed that the flow separation initially
occurred at Re=21. 相似文献
19.
Shuang Tan Wenjun Sun Kun Xu Junxia Wei & Guoxi Ni 《Communications In Computational Physics》2020,28(3):1189-1218
In this paper, a time implicit unified gas kinetic scheme (IUGKS) for 3D
multi-group neutron transport equation with delayed neutron is developed. The
explicit scheme, implicit 1st-order backward Euler scheme, and 2nd-order Crank-Nicholson scheme, become the subsets of the current IUGKS. In neutron transport,
the microscopic angular flux and the macroscopic scalar flux are fully coupled in an
implicit way with the combination of dual-time step technique for the convergence acceleration of unsteady evolution. In IUGKS, the computational time step is no longer
limited by the Courant-Friedrichs-Lewy (CFL) condition, which improves the computational efficiency in both steady and unsteady simulations with a large time step.
Mathematically, the current scheme has the asymptotic preserving (AP) property in
recovering automatically the diffusion solution in the continuum regime. Since the
explicit scanning along neutron traveling direction within the computational domain
is not needed in IUGKS, the scheme can be easily extended to multi-dimensional and
parallel computations. The numerical tests demonstrate that the IUGKS has high computational efficiency, high accuracy, and strong robustness when compared with other
schemes, such as the explicit UGKS, the commonly used finite difference, and finite
volume methods. This study shows that the IUGKS can be used faithfully to study
neutron transport in practical engineering applications. 相似文献
20.
A Positivity-Preserving Second-Order BDF Scheme for the Cahn-Hilliard Equation with Variable Interfacial Parameters 下载免费PDF全文
Lixiu Dong Cheng Wang Hui Zhang & Zhengru Zhang 《Communications In Computational Physics》2020,28(3):967-998
We present and analyze a new second-order finite difference scheme for
the Macromolecular Microsphere Composite hydrogel, Time-Dependent Ginzburg-Landau (MMC-TDGL) equation, a Cahn-Hilliard equation with Flory-Huggins-deGennes energy potential. This numerical scheme with unconditional energy stability is based on the Backward Differentiation Formula (BDF) method in time derivation
combining with Douglas-Dupont regularization term. In addition, we present a pointwise bound of the numerical solution for the proposed scheme in the theoretical level.
For the convergent analysis, we treat three nonlinear logarithmic terms as a whole and
deal with all logarithmic terms directly by using the property that the nonlinear error
inner product is always non-negative. Moreover, we present the detailed convergent
analysis in $ℓ^∞$(0,$T$;$H_h^{-1}$)∩$ℓ^2$(0,$T$;$H_h^1$) norm. At last, we use the local Newton approximation and multigrid method to solve the nonlinear numerical scheme, and various
numerical results are presented, including the numerical convergence test, positivity-preserving property test, spinodal decomposition, energy dissipation and mass conservation properties. 相似文献