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1.
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
2.
High Order Conservative Semi-Lagrangian Scheme for the BGK Model of the Boltzmann Equation
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Sebastiano Boscarino Seung-Yeon Cho Giovanni Russo & Seok-Bae Yun 《Communications In Computational Physics》2021,29(1):1-33
In this paper, we present a conservative semi-Lagrangian finite-difference
scheme for the BGK model. Classical semi-Lagrangian finite difference schemes, coupled with an L-stable treatment of the collision term, allow large time steps, for all the
range of Knudsen number [17, 27, 30]. Unfortunately, however, such schemes are not
conservative. Lack of conservation is analyzed in detail, and two main sources are
identified as its cause. Firstly, when using classical continuous Maxwellian, conservation error is negligible only if velocity space is resolved with sufficiently large number
of grid points. However, for a small number of grid points in velocity space such error
is not negligible, because the parameters of the Maxwellian do not coincide with the
discrete moments. Secondly, the non-linear reconstruction used to prevent oscillations
destroys the translation invariance which is at the basis of the conservation properties
of the scheme. As a consequence, the schemes show a wrong shock speed in the limit
of small Knudsen number. To treat the first problem and ensure machine precision
conservation of mass, momentum and energy with a relatively small number of velocity grid points, we replace the continuous Maxwellian with the discrete Maxwellian
introduced in [22]. The second problem is treated by implementing a conservative correction procedure based on the flux difference form as in [26]. In this way we can construct conservative semi-Lagrangian schemes which are Asymptotic Preserving (AP)
for the underlying Euler limit, as the Knudsen number vanishes. The effectiveness of
the proposed scheme is demonstrated by extensive numerical tests. 相似文献
3.
Kinetic Energy Preserving and Entropy Stable Finite Volume Schemes for Compressible Euler and Navier-Stokes Equations
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Praveen Chandrashekar 《Communications In Computational Physics》2013,14(5):1252-1286
Centered numerical fluxes can be constructed for compressible Euler equations
which preserve kinetic energy in the semi-discrete finite volume scheme. The essential
feature is that the momentum flux should be of the form where are any consistent approximations to the
pressure and the mass flux. This scheme thus leaves most terms in the numerical
flux unspecified and various authors have used simple averaging. Here we enforce
approximate or exact entropy consistency which leads to a unique choice of all the
terms in the numerical fluxes. As a consequence, a novel entropy conservative flux that
also preserves kinetic energy for the semi-discrete finite volume scheme has been proposed.
These fluxes are centered and some dissipation has to be added if shocks are
present or if the mesh is coarse. We construct scalar artificial dissipation terms which
are kinetic energy stable and satisfy approximate/exact entropy condition. Secondly,
we use entropy-variable based matrix dissipation flux which leads to kinetic energy
and entropy stable schemes. These schemes are shown to be free of entropy violating
solutions unlike the original Roe scheme. For hypersonic flows a blended scheme is
proposed which gives carbuncle free solutions for blunt body flows. Numerical results
for Euler and Navier-Stokes equations are presented to demonstrate the performance
of the different schemes. 相似文献
4.
In this paper, we introduce a new type of troubled-cell indicator to improve
hybrid weighted essentially non-oscillatory (WENO) schemes for solving the hyperbolic conservation laws. The hybrid WENO schemes selectively adopt the high-order
linear upwind scheme or the WENO scheme to avoid the local characteristic decompositions and calculations of the nonlinear weights in smooth regions. Therefore,
they can reduce computational cost while maintaining non-oscillatory properties in
non-smooth regions. Reliable troubled-cell indicators are essential for efficient hybrid
WENO methods. Most of troubled-cell indicators require proper parameters to detect
discontinuities precisely, but it is very difficult to determine the parameters automatically. We develop a new troubled-cell indicator derived from the mean value theorem
that does not require any variable parameters. Additionally, we investigate the characteristics of indicator variable; one of the conserved properties or the entropy is considered as indicator variable. Detailed numerical tests for 1D and 2D Euler equations are
conducted to demonstrate the performance of the proposed indicator. The results with
the proposed troubled-cell indicator are in good agreement with pure WENO schemes.
Also the new indicator has advantages in the computational cost compared with the
other indicators. 相似文献
5.
Improvement on Spherical Symmetry in Two-Dimensional Cylindrical Coordinates for a Class of Control Volume Lagrangian Schemes
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In [14], Maire developed a class of cell-centered Lagrangian schemes for
solving Euler equations of compressible gas dynamics in cylindrical coordinates. These
schemes use a node-based discretization of the numerical fluxes. The control volume version has several distinguished properties, including the conservation of mass,
momentum and total energy and compatibility with the geometric conservation law
(GCL). However, it also has a limitation in that it cannot preserve spherical symmetry
for one-dimensional spherical flow. An alternative is also given to use the first order
area-weighted approach which can ensure spherical symmetry, at the price of sacrificing conservation of momentum. In this paper, we apply the methodology proposed in
our recent work [8] to the first order control volume scheme of Maire in [14] to obtain
the spherical symmetry property. The modified scheme can preserve one-dimensional
spherical symmetry in a two-dimensional cylindrical geometry when computed on an
equal-angle-zoned initial grid, and meanwhile it maintains its original good properties
such as conservation and GCL. Several two-dimensional numerical examples in cylindrical coordinates are presented to demonstrate the good performance of the scheme
in terms of symmetry, non-oscillation and robustness properties. 相似文献
6.
Arbitrary-Lagrangian-Eulerian One-Step WENO Finite Volume Schemes on Unstructured Triangular Meshes
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In this article we present a new class of high order accurate ArbitraryEulerian-Lagrangian (ALE) one-step WENO finite volume schemes for solving nonlinear hyperbolic systems of conservation laws on moving two dimensional unstructured triangular meshes. A WENO reconstruction algorithm is used to achieve high
order accuracy in space and a high order one-step time discretization is achieved by
using the local space-time Galerkin predictor proposed in [25]. For that purpose, a
new element-local weak formulation of the governing PDE is adopted on moving
space-time elements. The space-time basis and test functions are obtained considering Lagrange interpolation polynomials passing through a predefined set of nodes.
Moreover, a polynomial mapping defined by the same local space-time basis functions
as the weak solution of the PDE is used to map the moving physical space-time element onto a space-time reference element. To maintain algorithmic simplicity, the
final ALE one-step finite volume scheme uses moving triangular meshes with straight
edges. This is possible in the ALE framework, which allows a local mesh velocity that
is different from the local fluid velocity. We present numerical convergence rates for
the schemes presented in this paper up to sixth order of accuracy in space and time and
show some classical numerical test problems for the two-dimensional Euler equations
of compressible gas dynamics. 相似文献
7.
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
8.
A High Order Sharp-Interface Method with Local Time Stepping for Compressible Multiphase Flows
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Angela Ferrari Claus-Dieter Munz & Bernhard Weigand 《Communications In Computational Physics》2011,9(1):205-230
In this paper, a new sharp-interface approach to simulate compressible
multiphase flows is proposed. The new scheme consists of a high order WENO finite volume scheme for solving the Euler equations coupled with a high order path-conservative
discontinuous Galerkin finite element scheme to evolve an indicator function
that tracks the material interface. At the interface our method applies ghost cells
to compute the numerical flux, as the ghost fluid method. However, unlike the original
ghost fluid scheme of Fedkiw et al. [15], the state of the ghost fluid is derived
from an approximate-state Riemann solver, similar to the approach proposed in [25],
but based on a much simpler formulation. Our formulation leads only to one single
scalar nonlinear algebraic equation that has to be solved at the interface, instead of
the system used in [25]. Away from the interface, we use the new general Osher-type
flux recently proposed by Dumbser and Toro [13], which is a simple but complete Riemann
solver, applicable to general hyperbolic conservation laws. The time integration
is performed using a fully-discrete one-step scheme, based on the approaches recently
proposed in [5, 7]. This allows us to evolve the system also with time-accurate local
time stepping. Due to the sub-cell resolution and the subsequent more restrictive
time-step constraint of the DG scheme, a local evolution for the indicator function is
applied, which is matched with the finite volume scheme for the solution of the Euler
equations that runs with a larger time step. The use of a locally optimal time step
avoids the introduction of excessive numerical diffusion in the finite volume scheme.
Two different fluids have been used, namely an ideal gas and a weakly compressible
fluid modeled by the Tait equation. Several tests have been computed to assess the
accuracy and the performance of the new high order scheme. A verification of our
algorithm has been carefully carried out using exact solutions as well as a comparison
with other numerical reference solutions. The material interface is resolved sharply
and accurately without spurious oscillations in the pressure field. 相似文献
9.
Conservative Semi-Lagrangian Finite Difference WENO Formulations with Applications to the Vlasov Equation
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In this paper, we propose a new conservative semi-Lagrangian (SL) finite
difference (FD) WENO scheme for linear advection equations, which can serve as a
base scheme for the Vlasov equation by Strang splitting [4]. The reconstruction procedure
in the proposed SL FD scheme is the same as the one used in the SL finite volume
(FV) WENO scheme [3]. However, instead of inputting cell averages and approximate
the integral form of the equation in a FV scheme, we input point values and approximate
the differential form of equation in a FD spirit, yet retaining very high order
(fifth order in our experiment) spatial accuracy. The advantage of using point values,
rather than cell averages, is to avoid the second order spatial error, due to the shearing
in velocity (v) and electrical field (E) over a cell when performing the Strang splitting
to the Vlasov equation. As a result, the proposed scheme has very high spatial accuracy,
compared with second order spatial accuracy for Strang split SL FV scheme for
solving the Vlasov-Poisson (VP) system. We perform numerical experiments on linear
advection, rigid body rotation problem; and on the Landau damping and two-stream
instabilities by solving the VP system. For comparison, we also apply (1) the conservative
SL FD WENO scheme, proposed in [22] for incompressible advection problem, (2)
the conservative SL FD WENO scheme proposed in [21] and (3) the non-conservative
version of the SL FD WENO scheme in [3] to the same test problems. The performances
of different schemes are compared by the error table, solution resolution of sharp interface,
and by tracking the conservation of physical norms, energies and entropies,
which should be physically preserved. 相似文献
10.
A conservative modification to the ghost fluid method (GFM) is developed
for compressible multiphase flows. The motivation is to eliminate or reduce the conservation
error of the GFM without affecting its performance. We track the conservative
variables near the material interface and use this information to modify the numerical
solution for an interfacing cell when the interface has passed the cell. The modification
procedure can be used on the GFM with any base schemes. In this paper we use the
fifth order finite difference WENO scheme for the spatial discretization and the third
order TVD Runge-Kutta method for the time discretization. The level set method is
used to capture the interface. Numerical experiments show that the method is at least
mass and momentum conservative and is in general comparable in numerical resolution
with the original GFM. 相似文献
11.
An Adaptive Grid Method for Singularly Perturbed Time-Dependent Convection-Diffusion Problems
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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
12.
A Posteriori Error Estimate and Adaptive Mesh Refinement Algorithm for Atomistic/Continuum Coupling with Finite Range Interactions in Two Dimensions
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In this paper, we develop the residual based a posteriori error estimates
and the corresponding adaptive mesh refinement algorithm for atomistic/continuum
(a/c) coupling with finite range interactions in two dimensions. We have systematically derived a new explicitly computable stress tensor formula for finite range interactions. In particular, we use the geometric reconstruction based consistent atomistic/continuum (GRAC) coupling scheme, which is quasi-optimal if the continuum
model is discretized by P1 finite elements. The numerical results of the adaptive mesh
refinement algorithm is consistent with the quasi-optimal a priori error estimates. 相似文献
13.
On the Order of Accuracy and Numerical Performance of Two Classes of Finite Volume WENO Schemes
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Rui Zhang Mengping Zhang & Chi-Wang Shu 《Communications In Computational Physics》2011,9(3):807-827
In this paper we consider two commonly used classes of finite volume
weighted essentially non-oscillatory (WENO) schemes in two dimensional Cartesian
meshes. We compare them in terms of accuracy, performance for smooth and shocked
solutions, and efficiency in CPU timing. For linear systems both schemes are high
order accurate, however for nonlinear systems, analysis and numerical simulation results
verify that one of them (Class A) is only second order accurate, while the other
(Class B) is high order accurate. The WENO scheme in Class A is easier to implement
and costs less than that in Class B. Numerical experiments indicate that the resolution
for shocked problems is often comparable for schemes in both classes for the same
building blocks and meshes, despite of the difference in their formal order of accuracy.
The results in this paper may give some guidance in the application of high order finite
volume schemes for simulating shocked flows. 相似文献
14.
Adaptive Order WENO Reconstructions for the Semi-Lagrangian Finite Difference Scheme for Advection Problem
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Jiajie Chen Xiaofeng Cai Jianxian Qiu & Jing-Mei Qiu 《Communications In Computational Physics》2021,30(1):67-96
We present a new conservative semi-Lagrangian finite difference weighted
essentially non-oscillatory scheme with adaptive order. This is an extension of the
conservative semi-Lagrangian (SL) finite difference WENO scheme in [Qiu and Shu,
JCP, 230 (4) (2011), pp. 863-889], in which linear weights in SL WENO framework
were shown not to exist for variable coefficient problems. Hence, the order of accuracy is not optimal from reconstruction stencils. In this paper, we incorporate a recent
WENO adaptive order (AO) technique [Balsara et al., JCP, 326 (2016), pp. 780-804]
to the SL WENO framework. The new scheme can achieve an optimal high order of
accuracy, while maintaining the properties of mass conservation and non-oscillatory
capture of solutions from the original SL WENO. The positivity-preserving limiter is
further applied to ensure the positivity of solutions. Finally, the scheme is applied to
high dimensional problems by a fourth-order dimensional splitting. We demonstrate
the effectiveness of the new scheme by extensive numerical tests on linear advection
equations, the Vlasov-Poisson system, the guiding center Vlasov model as well as the
incompressible Euler equations. 相似文献
15.
A Decoupled Energy Stable Adaptive Finite Element Method for Cahn–Hilliard–Navier–Stokes Equations
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In this paper, we propose, analyze, and numerically validate an adaptive
finite element method for the Cahn–Hilliard–Navier–Stokes equations. The adaptive
method is based on a linear, decoupled scheme introduced by Shen and Yang [30].
An unconditionally energy stable discrete law for the modified energy is shown for
the fully discrete scheme. A superconvergent cluster recovery based a posteriori error
estimations are constructed for both the phase field variable and velocity field function,
respectively. Based on the proposed space and time discretization error estimators, a
time-space adaptive algorithm is designed for numerical approximation of the Cahn–Hilliard–Navier–Stokes equations. Numerical experiments are presented to illustrate
the reliability and efficiency of the proposed error estimators and the corresponding
adaptive algorithm. 相似文献
16.
In this work, two fully discrete grad-div stabilized finite element schemes
for the fluid-fluid interaction model are considered. The first scheme is standard grad-div stabilized scheme, and the other one is modular grad-div stabilized scheme which
adds to Euler backward scheme an update step and does not increase computational
time for increasing stabilized parameters. Moreover, stability and error estimates of
these schemes are given. Finally, computational tests are provided to verify both the
numerical theory and efficiency of the presented schemes. 相似文献
17.
Algorithms for adaptive mesh refinement using a residual error estimator
are proposed for fluid flow problems in a finite volume framework. The residual error
estimator, referred to as the ℜ-parameter is used to derive refinement and coarsening
criteria for the adaptive algorithms. An adaptive strategy based on the ℜ-parameter
is proposed for continuous flows, while a hybrid adaptive algorithm employing a
combination of error indicators and the ℜ-parameter is developed for discontinuous
flows. Numerical experiments for inviscid and viscous flows on different grid topologies demonstrate the effectiveness of the proposed algorithms on arbitrary polygonal
grids. 相似文献
18.
An Improved Second-Order Finite-Volume Algorithm for Detached-Eddy Simulation Based on Hybrid Grids
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Yang Zhang Laiping Zhang Xin He & Xiaogang Deng 《Communications In Computational Physics》2016,20(2):459-485
A hybrid grid based second-order finite volume algorithm has been developed
for Detached-Eddy Simulation (DES) of turbulent flows. To alleviate the
effect caused by the numerical dissipation of the commonly used second order upwind
schemes in implementing DES with unstructured computational fluid dynamics
(CFD) algorithms, an improved second-order hybrid scheme is established through
modifying the dissipation term of the standard Roe's flux-difference splitting scheme
and the numerical dissipation of the scheme can be self-adapted according to the DES
flow field information. By Fourier analysis, the dissipative and dispersive features of
the new scheme are discussed. To validate the numerical method, DES formulations
based on the two most popular background turbulence models, namely, the one equation
Spalart-Allmaras (SA) turbulence model and the two equation k−ω Shear Stress
Transport model (SST), have been calibrated and tested with three typical numerical
examples (decay of isotropic turbulence, NACA0021 airfoil at 60◦incidence and 65◦swept delta wing). Computational results indicate that the issue of numerical dissipation
in implementing DES can be alleviated with the hybrid scheme, the resolution
for turbulence structures is significantly improved and the corresponding solutions
match the experimental data better. The results demonstrate the potentiality of the
present DES solver for complex geometries. 相似文献
19.
Wenhua Ma Dongmi Luo Wenjun Ying Guoxi Ni Min Xiao & Yibing Chen 《Communications In Computational Physics》2023,33(3):849-883
For compressible reactive flows with stiff source terms, a new block-based
adaptive multi-resolution method coupled with the adaptive multi-resolution representation model for ZND detonation and a conservative front capturing method based
on a level-set technique is presented. When simulating stiff reactive flows, underresolution in space and time can lead to incorrect propagation speeds of discontinuities, and numerical dissipation makes it impossible for traditional shock-capturing
methods to locate the detonation front. To solve these challenges, the proposed method
leverages an adaptive multi-resolution representation model to separate the scales of
the reaction from those of fluid dynamics, achieving both high-resolution solutions
and high efficiency. A level set technique is used to capture the detonation front
sharply and reduce errors due to the inaccurate prediction of detonation speed. In
order to ensure conservation, a conservative modified finite volume scheme is implemented, and the front transition fluxes are calculated by considering a Riemann problem. A series of numerical examples of stiff detonation simulations are performed to
illustrate that the present method can acquire the correct propagation speed and accurately capture the sharp detonation front. Comparative numerical results also validate
the approach’s benefits and excellent performance. 相似文献
20.
Comparison of Fifth-Order WENO Scheme and Finite Volume WENO-Gas-Kinetic Scheme for Inviscid and Viscous Flow Simulation
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The development of high-order schemes has been mostly concentrated on
the limiters and high-order reconstruction techniques. In this paper, the effect of the
flux functions on the performance of high-order schemes will be studied. Based on the
same WENO reconstruction, two schemes with different flux functions, i.e., the fifth-order WENO method and the WENO-Gas-Kinetic scheme (WENO-GKS), will be compared. The fifth-order finite difference WENO-SW scheme is a characteristic variable
reconstruction based method which uses the Steger-Warming flux splitting for inviscid terms, the sixth-order central difference for viscous terms, and three stages Runge-Kutta time stepping for the time integration. On the other hand, the finite volume
WENO-GKS is a conservative variable reconstruction based method with the same
WENO reconstruction. But it evaluates a time dependent gas distribution function
along a cell interface, and updates the flow variables inside each control volume by
integrating the flux function along the boundary of the control volume in both space
and time. In order to validate the robustness and accuracy of the schemes, both methods are tested under a wide range of flow conditions: vortex propagation, Mach 3
step problem, and the cavity flow at Reynolds number 3200. Our study shows that
both WENO-SW and WENO-GKS yield quantitatively similar results and agree with
each other very well provided a sufficient grid resolution is used. With the reduction of mesh points, the WENO-GKS behaves to have less numerical dissipation and
present more accurate solutions than those from the WENO-SW in all test cases. For
the Navier-Stokes equations, since the WENO-GKS couples inviscid and viscous terms
in a single flux evaluation, and the WENO-SW uses an operator splitting technique, it
appears that the WENO-SW is more sensitive to the WENO reconstruction and boundary treatment. In terms of efficiency, the finite volume WENO-GKS is about 4 times
slower than the finite difference WENO-SW in two dimensional simulations. The current study clearly shows that besides high-order reconstruction, an accurate gas evolution model or flux function in a high-order scheme is also important in the capturing of physical solutions. In a physical flow, the transport, stress deformation, heat conduction, and viscous heating are all coupled in a single gas evolution process. Therefore,
it is preferred to develop such a scheme with multi-dimensionality, and unified treatment of inviscid and dissipative terms. A high-order scheme does prefer a high-order
gas evolution model. Even with the rapid advances of high-order reconstruction techniques, the first-order dynamics of the Riemann solution becomes the bottleneck for
the further development of high-order schemes. In order to avoid the weakness of the
low order flux function, the development of high-order schemes relies heavily on the
weak solution of the original governing equations for the update of additional degree
of freedom, such as the non-conservative gradients of flow variables, which cannot be
physically valid in discontinuous regions. 相似文献