共查询到20条相似文献,搜索用时 15 毫秒
1.
A Well-Balanced Positivity-Preserving Quasi-Lagrange Moving Mesh DG Method for the Shallow Water Equations 下载免费PDF全文
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
2.
High-Order Positivity-Preserving Well-Balanced Discontinuous Galerkin Methods for Euler Equations with Gravitation on Unstructured Meshes 下载免费PDF全文
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
3.
A Well-Balanced Gas Kinetic Scheme for Navier-Stokes Equations with Gravitational Potential 下载免费PDF全文
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
4.
One-Dimensional Blood Flow with Discontinuous Properties and Transport: Mathematical Analysis and Numerical Schemes 下载免费PDF全文
Alessandra Spilimbergo Eleuterio F. Toro & Lucas O. Mü ller 《Communications In Computational Physics》2021,29(3):649-697
In this paper we consider the one-dimensional blood flow model with discontinuous mechanical and geometrical properties, as well as passive scalar transport,
proposed in [E.F. Toro and A. Siviglia. Flow in collapsible tubes with discontinuous
mechanical properties: mathematical model and exact solutions. Communications in
Computational Physics. 13(2), 361-385, 2013], completing the mathematical analysis by
providing new propositions and new proofs of relations valid across different waves.
Next we consider a first order DOT Riemann solver, proposing an integration path that
incorporates the passive scalar and proving the well-balanced properties of the resulting numerical scheme for stationary solutions. Finally we describe a novel and simple
well-balanced, second order, non-linear numerical scheme to solve the equations under study; by using suitable test problems for which exact solutions are available, we
assess the well-balanced properties of the scheme, its capacity to provide accurate solutions in challenging flow conditions and its accuracy. 相似文献
5.
A Computational Model for Simulation of Shallow Water Waves by Elastic Deformations in the Topography 下载免费PDF全文
Alia Al-Ghosoun Ashraf S. Osman & Mohammed Seaid 《Communications In Computational Physics》2021,29(4):1095-1124
We propose a coupled model to simulate shallow water waves induced by
elastic deformations in the bed topography. The governing equations consist of the
depth-averaged shallow water equations including friction terms for the water free-surface and the well-known second-order elastostatics formulation for the bed deformation. The perturbation on the free-surface is assumed to be caused by a sudden
change in the bottom beds. At the interface between the water flow and the bed topography, transfer conditions are implemented. Here, the hydrostatic pressure and
friction forces are considered for the elastostatic equations whereas bathymetric forces
are accounted for in the shallow water equations. The focus in the present study is on
the development of a simple and accurate representation of the interaction between
water waves and bed deformations in order to simulate practical shallow water flows
without relying on complex partial differential equations with free boundary conditions. The effects of location and magnitude of the deformation on the flow fields and
free-surface waves are investigated in details. Numerical simulations are carried out
for several test examples on shallow water waves induced by sudden changes in the
bed. The proposed computational model has been found to be feasible and satisfactory. 相似文献
6.
We present a new Finite Volume Evolution Galerkin (FVEG) scheme for the
solution of the shallow water equations (SWE) with the bottom topography as a source
term. Our new scheme will be based on the FVEG methods presented in (Noelle and
Kraft, J. Comp. Phys., 221 (2007)), but adds the possibility to handle dry boundaries.
The most important aspect is to preserve the positivity of the water height. We present
a general approach to ensure this for arbitrary finite volume schemes. The main idea is
to limit the outgoing fluxes of a cell whenever they would create negative water height.
Physically, this corresponds to the absence of fluxes in the presence of vacuum. Well-balancing
is then re-established by splitting gravitational and gravity driven parts of
the flux. Moreover, a new entropy fix is introduced that improves the reproduction of
sonic rarefaction waves. 相似文献
7.
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
8.
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
9.
Localized Exponential Time Differencing Method for Shallow Water Equations: Algorithms and Numerical Study 下载免费PDF全文
Xucheng Meng Thi-Thao-Phuong Hoang Zhu Wang & Lili Ju 《Communications In Computational Physics》2021,29(1):80-110
In this paper, we investigate the performance of the exponential time differencing (ETD) method applied to the rotating shallow water equations. Comparing
with explicit time stepping of the same order accuracy in time, the ETD algorithms
could reduce the computational time in many cases by allowing the use of large time
step sizes while still maintaining numerical stability. To accelerate the ETD simulations, we propose a localized approach that synthesizes the ETD method and overlapping domain decomposition. By dividing the original problem into many subdomain
problems of smaller sizes and solving them locally, the proposed approach could speed
up the calculation of matrix exponential vector products. Several standard test cases
for shallow water equations of one or multiple layers are considered. The results show
great potential of the localized ETD method for high-performance computing because
each subdomain problem can be naturally solved in parallel at every time step. 相似文献
10.
Manuel Jesú s Castro Dí az Yuanzhen Cheng Alina Chertock & Alexander Kurganov 《Communications In Computational Physics》2014,16(5):1323-1354
In this paper, we develop and study numerical methods for the two-mode
shallow water equations recently proposed in [S. STECHMANN, A. MAJDA, and
B. KHOUIDER, Theor. Comput. Fluid Dynamics, 22 (2008), pp. 407–432]. Designing
a reliable numerical method for this system is a challenging task due to its conditional
hyperbolicity and the presence of nonconservative terms. We present several numerical approaches–two operator splitting methods (based on either Roe-type upwind
or central-upwind scheme), a central-upwind scheme and a path-conservative central-upwind scheme– and test their performance in a number of numerical experiments.
The obtained results demonstrate that a careful numerical treatment of nonconservative terms is crucial for designing a robust and highly accurate numerical method. 相似文献
11.
A New Approach of High Order Well-Balanced Finite Volume WENO Schemes and Discontinuous Galerkin Methods for a Class of Hyperbolic Systems with Source Terms 下载免费PDF全文
Hyperbolic balance laws have steady state solutions in which the flux gradients are
nonzero but are exactly balanced by the source terms. In our earlier work [31–33], we designed
high order well-balanced schemes to a class of hyperbolic systems with separable source terms.
In this paper, we present a different approach to the same purpose: designing high order
well-balanced finite volume weighted essentially non-oscillatory (WENO) schemes and RungeKutta
discontinuous Galerkin (RKDG) finite element methods. We make the observation that
the traditional RKDG methods are capable of maintaining certain steady states exactly, if a
small modification on either the initial condition or the flux is provided. The computational
cost to obtain such a well balanced RKDG method is basically the same as the traditional
RKDG method. The same idea can be applied to the finite volume WENO schemes. We
will first describe the algorithms and prove the well balanced property for the shallow water
equations, and then show that the result can be generalized to a class of other balance laws.
We perform extensive one and two dimensional simulations to verify the properties of these
schemes such as the exact preservation of the balance laws for certain steady state solutions,
the non-oscillatory property for general solutions with discontinuities, and the genuine high
order accuracy in smooth regions. 相似文献
12.
A Novel Strong-Coupling Pseudo-Spectral Method for Numerical Studies of Two-Layer Turbulent Channel Flows 下载免费PDF全文
Tong Wu Zixuan Yang Shizhao Wang & Guowei He 《Communications In Computational Physics》2020,28(3):1133-1146
A new method is proposed to simulate a coupled air-water two-layer turbulent channel flow. A numerically effective dynamic viscosity is implemented to calculate the viscous momentum flux at the interface, leading to a strong-coupling scheme
for the evolution of air and water motions. The direct numerical simulation results
are compared with those in the literature obtained from a weak-coupling scheme. It
is discovered that while the turbulence statistics of the air phase based on the strong- and weak-coupling schemes are close to each other, the results on the water side are
influenced by the coupling approach, especially near the water surface. 相似文献
13.
High Order Finite Difference Hermite WENO Fixed-Point Fast Sweeping Method for Static Hamilton-Jacobi Equations 下载免费PDF全文
In this paper, we combine the nonlinear HWENO reconstruction in [43] and
the fixed-point iteration with Gauss-Seidel fast sweeping strategy, to solve the static
Hamilton-Jacobi equations in a novel HWENO framework recently developed in [22].
The proposed HWENO frameworks enjoys several advantages. First, compared with
the traditional HWENO framework, the proposed methods do not need to introduce
additional auxiliary equations to update the derivatives of the unknown function $\phi$.
They are now computed from the current value of $\phi$ and the previous spatial derivatives of $\phi$. This approach saves the computational storage and CPU time, which greatly
improves the computational efficiency of the traditional HWENO scheme. In addition,
compared with the traditional WENO method, reconstruction stencil of the HWENO
methods becomes more compact, their boundary treatment is simpler, and the numerical errors are smaller on the same mesh. Second, the fixed-point fast sweeping method
is used to update the numerical approximation. It is an explicit method and does
not involve the inverse operation of nonlinear Hamiltonian, therefore any Hamilton-Jacobi equations with complex Hamiltonian can be solved easily. It also resolves some
known issues, including that the iterative number is very sensitive to the parameter $ε$ used in the nonlinear weights, as observed in previous studies. Finally, to further
reduce the computational cost, a hybrid strategy is also presented. Extensive numerical experiments are performed on two-dimensional problems, which demonstrate the
good performance of the proposed fixed-point fast sweeping HWENO methods. 相似文献
14.
Simulation of Three-Dimensional Free-Surface Flows Using Two-Dimensional Multilayer Shallow Water Equations 下载免费PDF全文
Saida Sari Thomas Rowan Mohammed Seaid & Fayssal Benkhaldoun 《Communications In Computational Physics》2020,27(5):1413-1442
We present an efficient and conservative Eulerian-Lagrangian method for
solving two-dimensional hydrostatic multilayer shallow water flows with mass exchange between the vertical layers. The method consists of a projection finite volume
method for the Eulerian stage and a method of characteristics to approximate the numerical fluxes for the Lagrangian stage. The proposed method is simple to implement,
satisfies the conservation property and it can be used for multilayer shallow water
equations on non-flat bathymetry including eddy viscosity and Coriolis forces. It offers a novel method of calculating stratified vertical velocities without the use of the
Navier-Stokes equations. Numerical results are presented for several examples and
the obtained results for a free-surface flow problem are in close agreement with the
analytical solutions. We also test the performance of the proposed method for a test
example of wind-driven flows with recirculation 相似文献
15.
This paper is concerned with a new version of the Osher-Solomon Riemann
solver and is based on a numerical integration of the path-dependent dissipation matrix.
The resulting scheme is much simpler than the original one and is applicable to
general hyperbolic conservation laws, while retaining the attractive features of the original
solver: the method is entropy-satisfying, differentiable and complete in the sense
that it attributes a different numerical viscosity to each characteristic field, in particular
to the intermediate ones, since the full eigenstructure of the underlying hyperbolic system
is used. To illustrate the potential of the proposed scheme we show applications
to the following hyperbolic conservation laws: Euler equations of compressible gasdynamics
with ideal gas and real gas equation of state, classical and relativistic MHD
equations as well as the equations of nonlinear elasticity. To the knowledge of the authors,
apart from the Euler equations with ideal gas, an Osher-type scheme has never
been devised before for any of these complicated PDE systems. Since our new general
Riemann solver can be directly used as a building block of high order finite volume
and discontinuous Galerkin schemes we also show the extension to higher order of
accuracy and multiple space dimensions in the new framework of PNPM schemes on
unstructured meshes recently proposed in [9]. 相似文献
16.
A One-Dimensional Second-Order Cell-Centered Lagrangian Scheme Satisfying the Entropy Condition 下载免费PDF全文
The numerical solutions of gas dynamics equations have to be consistent
with the second law of thermodynamics, which is termed entropy condition. However, most cell-centered Lagrangian (CL) schemes do not satisfy the entropy condition.
Until 2020, for one-dimensional gas dynamics equations, the first-order CL scheme
with the hybridized flux developed by combining the acoustic approximate (AA) flux
and the entropy conservative (EC) flux developed by Maire et al. was used. This hybridized CL scheme satisfies the entropy condition; however, it is under-entropic in
the part zones of rarefaction waves. Moreover, the EC flux may result in nonphysical
numerical oscillations in simulating strong rarefaction waves. Another disadvantage
of this scheme is that it is of only first-order accuracy. In this paper, we firstly construct
a modified entropy conservative (MEC) flux which can damp effectively numerical oscillations in simulating strong rarefaction waves. Then we design a new hybridized
CL scheme satisfying the entropy condition for one-dimensional complex flows. This
new hybridized CL scheme is a combination of the AA flux and the MEC flux.In order to prevent the specific entropy of the hybridized CL scheme from being
under-entropic, we propose using the third-order TVD-type Runge-Kutta time discretization method. Based on the new hybridized flux, we develop the second-order
CL scheme that satisfies the entropy condition.Finally, the characteristics of our new CL scheme using the improved hybridized
flux are demonstrated through several numerical examples. 相似文献
17.
Arthur Bousquet Madalina Petcu Ming-Cheng Shiue Roger Temam & Joseph Tribbia 《Communications In Computational Physics》2013,14(3):664-702
A new set of boundary conditions has been derived by rigorous methods for the shallow water equations in a limited domain. The aim of this article is to present these boundary conditions and to report on numerical simulations which have been performed using these boundary conditions. The new boundary conditions which are mildly dissipative let the waves move freely inside and outside the domain. The problems considered include a one-dimensional shallow water system with two layers of fluids and a two-dimensional inviscid shallow water system in a rectangle. 相似文献
18.
Consistent Forcing Scheme in the Simplified Lattice Boltzmann Method for Incompressible Flows 下载免费PDF全文
Yuan Gao Liuming Yang Yang Yu Guoxiang Hou & Zhongbao Hou 《Communications In Computational Physics》2021,30(5):1427-1452
Considering the fact that the lattice discrete effects are neglected while introducing a body force into the simplified lattice Boltzmann method (SLBM), we propose
a consistent forcing scheme in SLBM for incompressible flows with external forces. The
lattice discrete effects are considered at the level of distribution functions in the present
forcing scheme. Consequently, it is more accurate compared with the original forcing
scheme used in SLBM. Through Taylor series expansion and Chapman-Enskog (CE)
expansion analysis, the present forcing scheme can be proven to recover the macroscopic Navier-Stokes (N-S) equations. Then, the macroscopic equations are resolved
through a fractional step technique. Furthermore, the material derivative term is discretized by the central difference method. To verify the results of the present scheme,
we simulate with multiple forms of external force interactions including the space- and
time-dependent body forces. Hence, the present forcing scheme overcomes the disadvantages of the original forcing scheme and the body force can be accurately imposed
in the present scheme even when a coarse mesh is applied while the original scheme
fails. Excellent agreements between the analytical solutions and our numerical results
can be observed. 相似文献
19.
Discrete-Velocity Vector-BGK Models Based Numerical Methods for the Incompressible Navier-Stokes Equations 下载免费PDF全文
Jin Zhao 《Communications In Computational Physics》2021,29(2):420-444
In this paper, we propose a class of numerical methods based on discrete-velocity vector-BGK models for the incompressible Navier-Stokes equations. By analyzing a splitting method with Maxwell iteration, we show that the usual lattice Boltzmann discretization of the vector-BGK models provides a good numerical scheme.
Moreover, we establish the stability of the numerical scheme. The stability and second-order accuracy of the scheme are validated through numerical simulations of the two-dimensional Taylor-Green vortex flows. Further numerical tests are conducted to exhibit some potential advantages of the vector-BGK models, which can be regarded as
competitive alternatives of the scalar-BGK models. 相似文献
20.
A Conservative Lagrangian Scheme for Solving Compressible Fluid Flows with Multiple Internal Energy Equations 下载免费PDF全文
Juan Cheng Chi-Wang Shu & Qinghong Zeng 《Communications In Computational Physics》2012,12(5):1307-1328
Lagrangian methods are widely used in many fields for multi-material compressible flow simulations such as in astrophysics and inertial confinement fusion
(ICF), due to their distinguished advantage in capturing material interfaces automatically. In some of these applications, multiple internal energy equations such as those
for electron, ion and radiation are involved. In the past decades, several staggered-grid based Lagrangian schemes have been developed which are designed to solve the
internal energy equation directly. These schemes can be easily extended to solve problems with multiple internal energy equations. However, such schemes are typically
not conservative for the total energy. Recently, significant progress has been made
in developing cell-centered Lagrangian schemes which have several good properties
such as conservation for all the conserved variables and easiness for remapping. However, these schemes are commonly designed to solve the Euler equations in the form
of the total energy, therefore they cannot be directly applied to the solution of either
the single internal energy equation or the multiple internal energy equations without
significant modifications. Such modifications, if not designed carefully, may lead to
the loss of some of the nice properties of the original schemes such as conservation of
the total energy. In this paper, we establish an equivalency relationship between the
cell-centered discretizations of the Euler equations in the forms of the total energy and
of the internal energy. By a carefully designed modification in the implementation,
the cell-centered Lagrangian scheme can be used to solve the compressible fluid flow
with one or multiple internal energy equations and meanwhile it does not lose its total
energy conservation property. An advantage of this approach is that it can be easily
applied to many existing large application codes which are based on the framework
of solving multiple internal energy equations. Several two dimensional numerical examples for both Euler equations and three-temperature hydrodynamic equations in cylindrical coordinates are presented to demonstrate the performance of the scheme in
terms of symmetry preserving, accuracy and non-oscillatory performance. 相似文献