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1.
S. Guisset S. Brull B. Dubroca S. Karpov & I. Potapenko 《Communications In Computational Physics》2016,19(2):301-328
This work deals with the numerical resolution of the M1-Maxwell system
in the quasi-neutral regime. In this regime the stiffness of the stability constraints of
classical schemes causes huge calculation times. That is why we introduce a new stable
numerical scheme consistent with the transitional and limit models. Such schemes
are called Asymptotic-Preserving (AP) schemes in literature. This new scheme is able
to handle the quasi-neutrality limit regime without any restrictions on time and space
steps. This approach can be easily applied to angular moment models by using a moments
extraction. Finally, two physically relevant numerical test cases are presented
for the Asymptotic-Preserving scheme in different regimes. The first one corresponds
to a regime where electromagnetic effects are predominant. The second one on the contrary
shows the efficiency of the Asymptotic-Preserving scheme in the quasi-neutral
regime. In the latter case the illustrative simulations are compared with kinetic and
hydrodynamic numerical results. 相似文献
2.
Fifth-Order A-WENO Path-Conservative Central-Upwind Scheme for Behavioral Non-Equilibrium Traffic Models
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Shaoshuai Chu Alexander Kurganov Saeed Mohammadian & Zuduo Zheng 《Communications In Computational Physics》2023,33(3):692-732
Non-equilibrium hyperbolic traffic models can be derived as continuum
approximations of car-following models and in many cases the resulting continuum
models are non-conservative. This leads to numerical difficulties, which seem to have
discouraged further development of complex behavioral continuum models, which is
a significant research need.In this paper, we develop a robust numerical scheme that solves hyperbolic traffic
flow models based on their non-conservative form. We develop a fifth-order alternative weighted essentially non-oscillatory (A-WENO) finite-difference scheme based
on the path-conservative central-upwind (PCCU) method for several non-equilibrium
traffic flow models. In order to treat the non-conservative product terms, we use a
path-conservative technique. To this end, we first apply the recently proposed second-order finite-volume PCCU scheme to the traffic flow models, and then extend this
scheme to the fifth-order of accuracy via the finite-difference A-WENO framework.
The designed schemes are applied to three different traffic flow models and tested on
a number of challenging numerical examples. Both schemes produce quite accurate results though the resolution achieved by the fifth-order A-WENO scheme is higher. The
proposed scheme in this paper sets the stage for developing more robust and complex
continuum traffic flow models with respect to human psychological factors. 相似文献
3.
Wenbin Chen Jianyu Jing Cheng Wang Xiaoming Wang & Steven M. Wise 《Communications In Computational Physics》2022,31(1):60-93
In this paper we propose and analyze a second order accurate numerical
scheme for the Cahn-Hilliard equation with logarithmic Flory Huggins energy potential. A modified Crank-Nicolson approximation is applied to the logarithmic nonlinear term, while the expansive term is updated by an explicit second order Adams-Bashforth extrapolation, and an alternate temporal stencil is used for the surface diffusion term. A nonlinear artificial regularization term is added in the numerical scheme,
which ensures the positivity-preserving property, i.e., the numerical value of the phase
variable is always between -1 and 1 at a point-wise level. Furthermore, an unconditional energy stability of the numerical scheme is derived, leveraging the special form
of the logarithmic approximation term. In addition, an optimal rate convergence estimate is provided for the proposed numerical scheme, with the help of linearized
stability analysis. A few numerical results, including both the constant-mobility and
solution-dependent mobility flows, are presented to validate the robustness of the proposed numerical scheme. 相似文献
4.
Xiaoqin Shen Qian Yang Lin Bai & Kaitai Li 《Communications In Computational Physics》2021,29(1):186-210
In this study, the dynamics of elliptic membrane shell model has been proposed and discussed numerically for the first time. Firstly, we show that the solution of
this model exists and is unique. Secondly, we consider spatial and time discretizations
of the time-dependent elliptic membrane shell by finite element method and Newmark scheme, respectively. Then, the corresponding existence, uniqueness, stability,
convergence and a priori error estimate are given. Finally, we present numerical results involving a portion of an ellipsoidal shell and a portion of a spherical shell to
verify the efficiency and convergence of the numerical scheme. 相似文献
5.
A Positivity-Preserving Second-Order BDF Scheme for the Cahn-Hilliard Equation with Variable Interfacial Parameters
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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
6.
Rajesh Gandham David Medina & Timothy Warburton 《Communications In Computational Physics》2015,18(1):37-64
We discuss the development, verification, and performance of a GPU accelerated
discontinuous Galerkin method for the solutions of two dimensional nonlinear
shallow water equations. The shallow water equations are hyperbolic partial differential
equations and are widely used in the simulation of tsunami wave propagations.
Our algorithms are tailored to take advantage of the single instruction multiple data
(SIMD) architecture of graphic processing units. The time integration is accelerated by
local time stepping based on a multi-rate Adams-Bashforth scheme. A total variational
bounded limiter is adopted for nonlinear stability of the numerical scheme. This limiter
is coupled with a mass and momentum conserving positivity preserving limiter
for the special treatment of a dry or partially wet element in the triangulation. Accuracy,
robustness and performance are demonstrated with the aid of test cases. Furthermore,
we developed a unified multi-threading model OCCA. The kernels expressed
in OCCA model can be cross-compiled with multi-threading models OpenCL, CUDA,
and OpenMP. We compare the performance of the OCCA kernels when cross-compiled
with these models. 相似文献
7.
In this paper we propose and analyze a (temporally) third order accurate
backward differentiation formula (BDF) numerical scheme for the no-slope-selection
(NSS) equation of the epitaxial thin film growth model, with Fourier pseudo-spectral
discretization in space. The surface diffusion term is treated implicitly, while the nonlinear chemical potential is approximated by a third order explicit extrapolation formula for the sake of solvability. In addition, a third order accurate Douglas-Dupont
regularization term, in the form of $−A∆t^2∆^2_N
(u^{n+1}−u^n)$, is added in the numerical
scheme. A careful energy stability estimate, combined with Fourier eigenvalue analysis, results in the energy stability in a modified version, and a theoretical justification of the coefficient $A$ becomes available. As a result of this energy stability analysis, a uniform in time bound of the numerical energy is obtained. And also, the
optimal rate convergence analysis and error estimate are derived in details, in the $ℓ^∞(0,T;ℓ^2)∩ℓ^2(0,T;H^2_h)$ norm, with the help of a linearized estimate for the nonlinear error terms. Some numerical simulation results are presented to demonstrate the
efficiency of the numerical scheme and the third order convergence. The long time
simulation results for $ε = 0.02$ (up to $T = 3×10^5$) have indicated a logarithm law for
the energy decay, as well as the power laws for growth of the surface roughness and
the mound width. In particular, the power index for the surface roughness and the
mound width growth, created by the third order numerical scheme, is more accurate
than those produced by certain second order energy stable schemes in the existing
literature. 相似文献
8.
A Two-Phase Flow Simulation of Discrete-Fractured Media Using Mimetic Finite Difference Method
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Various conceptual models exist for numerical simulation of fluid flow in
fractured porous media, such as dual-porosity model and equivalent continuum model.
As a promising model, the discrete-fracture model has been received more attention
in the past decade. It can be used both as a stand-alone tool as well as for the evaluation of effective parameters for the continuum models. Various numerical methods
have been applied to the discrete-fracture model, including control volume finite difference, Galerkin and mixed finite element methods. All these methods have inherent
limitations in accuracy and applicabilities. In this work, we developed a new numerical scheme for the discrete-fracture model by using mimetic finite difference method.
The proposed numerical model is applicable in arbitrary unstructured grid cells with
full-tensor permeabilities. The matrix-fracture and fracture-fracture fluxes are calculated based on powerful features of the mimetic finite difference method, while the
upstream finite volume scheme is used for the approximation of the saturation equation. Several numerical tests in 2D and 3D are carried out to demonstrate the efficiency
and robustness of the proposed numerical model. 相似文献
9.
Deep Ray Praveen Chandrashekar Ulrik S. Fjordholm & Siddhartha Mishra 《Communications In Computational Physics》2016,19(5):1111-1140
We propose an entropy stable high-resolution finite volume scheme to approximate
systems of two-dimensional symmetrizable conservation laws on unstructured
grids. In particular we consider Euler equations governing compressible flows.
The scheme is constructed using a combination of entropy conservative fluxes and
entropy-stable numerical dissipation operators. High resolution is achieved based on
a linear reconstruction procedure satisfying a suitable sign property that helps to maintain
entropy stability. The proposed scheme is demonstrated to robustly approximate
complex flow features by a series of benchmark numerical experiments. 相似文献
10.
Numerical Solutions of Coupled Nonlinear Schrödinger Equations by Orthogonal Spline Collocation Method
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Qing-Jiang Meng Li-Ping Yin Xiao-Qing Jin & Fang-Li Qiao 《Communications In Computational Physics》2012,12(5):1392-1416
In this paper, we present the use of the orthogonal spline collocation
method for the semi-discretization scheme of the one-dimensional coupled nonlinear Schrödinger equations. This method uses the Hermite basis functions, by which
physical quantities are approximated with their values and derivatives associated with
Gaussian points. The convergence rate with order O(h4+τ2) and the stability of the
scheme are proved. Conservation properties are shown in both theory and practice.
Extensive numerical experiments are presented to validate the numerical study under
consideration. 相似文献
11.
Numerical Stability Analysis for a Stationary and Translating Droplet at Extremely Low Viscosity Values Using the Lattice Boltzmann Method Color-Gradient Multi-Component Model with Central Moments Formulation
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Karun P. N. Datadien Gianluca Di Staso & Federico Toschi 《Communications In Computational Physics》2023,33(1):330-348
Multicomponent models based on the Lattice Boltzmann Method (LBM)
have clear advantages with respect to other approaches, such as good parallel performances and scalability and the automatic resolution of breakup and coalescence
events. Multicomponent flow simulations are useful for a wide range of applications,
yet many multicomponent models for LBM are limited in their numerical stability and
therefore do not allow exploration of physically relevant low viscosity regimes. Here
we perform a quantitative study and validations, varying parameters such as viscosity,
droplet radius, domain size and acceleration for stationary and translating droplet simulations for the color-gradient method with central moments (CG-CM) formulation, as
this method promises increased numerical stability with respect to the non-CM formulation. We focus on numerical stability and on the effect of decreasing grid-spacing,
i.e. increasing resolution, in the extremely low viscosity regime for stationary droplet
simulations. The effects of small- and large-scale anisotropy, due to grid-spacing and
domain-size, respectively, are investigated for a stationary droplet. The effects on numerical stability of applying a uniform acceleration in one direction on the domain,
i.e. on both the droplet and the ambient, is explored into the low viscosity regime, to
probe the numerical stability of the method under dynamical conditions. 相似文献
12.
Relaxation Schemes for the $M_1$ Model with Space-Dependent Flux: Application to Radiotherapy Dose Calculation
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Teddy Pichard Denise Aregba-Driollet Sté phane Brull Bruno Dubroca & Martin Frank 《Communications In Computational Physics》2016,19(1):168-191
Because of stability constraints, most numerical schemes applied to hyperbolic
systems of equations turn out to be costly when the flux term is multiplied by
some very large scalar. This problem emerges with the $M_1$ system of equations in
the field of radiotherapy when considering heterogeneous media with very disparate
densities. Additionally, the flux term of the $M_1$ system is non-linear, and in order for
the model to be well-posed the numerical solution needs to fulfill conditions called
realizability. In this paper, we propose a numerical method that overcomes the stability
constraint and preserves the realizability property. For this purpose, we relax the
$M_1$ system to obtain a linear flux term. Then we extend the stencil of the difference
quotient to obtain stability. The scheme is applied to a radiotherapy dose calculation
example. 相似文献
13.
A Well-Balanced Gas Kinetic Scheme for Navier-Stokes Equations with Gravitational Potential
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The hydrostatic equilibrium state is the consequence of the exact balance between hydrostatic pressure and external force. Standard finite volume cannot keep this
balance exactly due to their unbalanced truncation errors. In this study, we introduce
an auxiliary variable which becomes constant at isothermal hydrostatic equilibria and
propose a well-balanced gas kinetic scheme for the Navier-Stokes equations. Through
reformulating the convection term and the force term via the auxiliary variable, zero
numerical flux and zero numerical source term are enforced at the hydrostatic equilibrium state instead of the balance between hydrostatic pressure and external force.
Several problems are tested to demonstrate the accuracy and the stability of the new
scheme. The results confirm that, the new scheme can preserve the exact hydrostatic
solution. The small perturbation riding on hydrostatic equilibria can be calculated accurately. More importantly, the new scheme is capable of simulating the process of
converging towards hydrostatic equilibria from a highly unbalanced initial condition.
The ultimate state of zero velocity and constant temperature is achieved up to machine
accuracy. As demonstrated by the numerical experiments, the current scheme is very
suitable for small amplitude perturbation and long time running under gravitational
potential. 相似文献
14.
Liang Wang Zhaoli Guo Baochang Shi & Chuguang Zheng 《Communications In Computational Physics》2013,13(4):1151-1172
A comparative study is conducted to evaluate three types of lattice Boltzmann equation (LBE) models for fluid flows with finite-sized particles, including the
lattice Bhatnagar-Gross-Krook (BGK) model, the model proposed by Ladd [Ladd AJC,
J. Fluid Mech., 271, 285-310 (1994); Ladd AJC, J. Fluid Mech., 271, 311-339 (1994)], and
the multiple-relaxation-time (MRT) model. The sedimentation of a circular particle in
a two-dimensional infinite channel under gravity is used as the first test problem. The
numerical results of the three LBE schemes are compared with the theoretical results
and existing data. It is found that all of the three LBE schemes yield reasonable results in general, although the BGK scheme and Ladd's scheme give some deviations
in some cases. Our results also show that the MRT scheme can achieve a better numerical stability than the other two schemes. Regarding the computational efficiency,
it is found that the BGK scheme is the most superior one, while the other two schemes
are nearly identical. We also observe that the MRT scheme can unequivocally reduce
the viscosity dependence of the wall correction factor in the simulations, which reveals
the superior robustness of the MRT scheme. The superiority of the MRT scheme over
the other two schemes is also confirmed by the simulation of the sedimentation of an
elliptical particle. 相似文献
15.
Jiwei Zhang Zhizhong Sun Xiaonan Wu & Desheng Wang 《Communications In Computational Physics》2011,10(3):742-766
The paper is concerned with the numerical solution of Schrödinger equations
on an unbounded spatial domain. High-order absorbing boundary conditions
for one-dimensional domain are derived, and the stability of the reduced initial boundary
value problem in the computational interval is proved by energy estimate. Then a
second order finite difference scheme is proposed, and the convergence of the scheme
is established as well. Finally, numerical examples are reported to confirm our error
estimates of the numerical methods. 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.
An Explicit MUSCL Scheme on Staggered Grids with Kinetic-Like Fluxes for the Barotropic and Full Euler System
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Thierry Goudon Julie Llobell & Sebastian Minjeaud 《Communications In Computational Physics》2020,27(3):672-724
We present a second order scheme for the barotropic and full Euler equations. The scheme works on staggered grids, with numerical unknowns stored at dual locations, while the numerical fluxes are derived in the spirit of kinetic schemes. We identify stability conditions ensuring the positivity of the discrete density and energy. We illustrate the ability of the scheme to capture the structure of complex flows with 1D and 2D simulations on MAC grids. 相似文献
18.
A Space-Time Conservative Method for Hyperbolic Systems with Stiff and Non-Stiff Source Terms
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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
19.
An All-Regime Lagrange-Projection Like Scheme for the Gas Dynamics Equations on Unstructured Meshes
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Christophe Chalons Mathieu Girardin & Samuel Kokh 《Communications In Computational Physics》2016,20(1):188-233
We propose an all regime Lagrange-Projection like numerical scheme for the
gas dynamics equations. By all regime, we mean that the numerical scheme is able to
compute accurate approximate solutions with an under-resolved discretization with
respect to the Mach number M, i.e. such that the ratio between the Mach number M
and the mesh size or the time step is small with respect to 1. The key idea is to decouple
acoustic and transport phenomenon and then alter the numerical flux in the
acoustic approximation to obtain a uniform truncation error in term of M. This modified
scheme is conservative and endowed with good stability properties with respect
to the positivity of the density and the internal energy. A discrete entropy inequality
under a condition on the modification is obtained thanks to a reinterpretation of the
modified scheme in the Harten Lax and van Leer formalism. A natural extension to
multi-dimensional problems discretized over unstructured mesh is proposed. Then
a simple and efficient semi-implicit scheme is also proposed. The resulting scheme
is stable under a CFL condition driven by the (slow) material waves and not by the
(fast) acoustic waves and so verifies the all regime property. Numerical evidences are
proposed and show the ability of the scheme to deal with tests where the flow regime
may vary from low to high Mach values. 相似文献
20.
Two-phase flow and heat transfer, such as boiling and condensing flows, are
complicated physical phenomena that generally prohibit an exact solution and even
pose severe challenges for numerical approaches. If numerical solution time is also an
issue the challenge increases even further. We present here a numerical implementation and novel study of a fully distributed dynamic one-dimensional model of two-phase flow in a tube, including pressure drop, heat transfer, and variations in tube
cross-section. The model is based on a homogeneous formulation of the governing
equations, discretized by a high resolution finite difference scheme due to Kurganov
and Tadmore.
The homogeneous formulation requires a set of thermodynamic relations to cover the
entire range from liquid to gas state. This leads a number of numerical challenges
since these relations introduce discontinuities in the derivative of the variables and are
usually very slow to evaluate. To overcome these challenges, we use an interpolation
scheme with local refinement.
The simulations show that the method handles crossing of the saturation lines for
both liquid to two-phase and two-phase to gas regions. Furthermore, a novel result
obtained in this work, the method is stable towards dynamic transitions of the inlet/outlet boundaries across the saturation lines. Results for these cases are presented
along with a numerical demonstration of conservation of mass under dynamically
varying boundary conditions. Finally we present results for the stability of the code in
a case of a tube with a narrow section. 相似文献