共查询到20条相似文献,搜索用时 31 毫秒
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.
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
3.
We propose an a-posteriori error/smoothness indicator for standard semi-discrete
finite volume schemes for systems of conservation laws, based on the numerical
production of entropy. This idea extends previous work by the first author limited
to central finite volume schemes on staggered grids. We prove that the indicator converges
to zero with the same rate of the error of the underlying numerical scheme on
smooth flows under grid refinement. We construct and test an adaptive scheme for
systems of equations in which the mesh is driven by the entropy indicator. The adaptive
scheme uses a single nonuniform grid with a variable timestep. We show how
to implement a second order scheme on such a space-time non uniform grid, preserving
accuracy and conservation properties. We also give an example of a p-adaptive
strategy. 相似文献
4.
Local Discontinuous Galerkin (LDG) schemes in the sense of [5] are a flexible numerical tool to approximate solutions of nonlinear convection problems with complicated dissipative terms. Such terms frequently appear in evolution equations which describe the dynamics of phase changes in e.g. liquid-vapour mixtures or in elastic solids. We report on results for one-dimensional model problems with dissipative terms including third-order and convolution operators. Cell entropy inequalities and L2-stability results are proved for those model problems. As is common in phase transition theory the solution structure sensitively depends on the coupling parameter between viscosity and capillarity. To avoid spurious solutions due to the counteracting effect of artificial dissipation by the numerical flux and the actual dissipation terms we introduce Tadmors' entropy conservative fluxes. Various numerical experiments underline the reliability of our approach and also illustrate interesting and (partly) new phase transition phenomena. 相似文献
5.
Georgij Bispen K. R. Arun Má ria Luká čová -Medvid'ová & Sebastian Noelle 《Communications In Computational Physics》2014,16(2):307-347
We present new large time step methods for the shallow water flows in the
low Froude number limit. In order to take into account multiscale phenomena that typically
appear in geophysical flows nonlinear fluxes are split into a linear part governing
the gravitational waves and the nonlinear advection. We propose to approximate fast
linear waves implicitly in time and in space by means of a genuinely multidimensional
evolution operator. On the other hand, we approximate nonlinear advection part explicitly
in time and in space by means of the method of characteristics or some standard
numerical flux function. Time integration is realized by the implicit-explicit (IMEX)
method. We apply the IMEX Euler scheme, two step Runge Kutta Cranck Nicolson
scheme, as well as the semi-implicit BDF scheme and prove their asymptotic preserving
property in the low Froude number limit. Numerical experiments demonstrate
stability, accuracy and robustness of these new large time step finite volume schemes
with respect to small Froude number. 相似文献
6.
A One-Dimensional Second-Order Cell-Centered Lagrangian Scheme Satisfying the Entropy Condition
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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
7.
The concept of diffusion regulation (DR) was originally proposed by
Jaisankar for traditional second order finite volume Euler solvers. This was used to
decrease the inherent dissipation associated with using approximate Riemann solvers.
In this paper, the above concept is extended to the high order spectral volume (SV)
method. The DR formulation was used in conjunction with the Rusanov flux to handle the inviscid flux terms. Numerical experiments were conducted to compare and
contrast the original and the DR formulations. These experiments demonstrated (i)
retention of high order accuracy for the new formulation, (ii) higher fidelity of the DR
formulation, when compared to the original scheme for all orders and (iii) straightforward extension to Navier Stokes equations, since the DR does not interfere with
the discretization of the viscous fluxes. In general, the 2D numerical results are very
promising and indicate that the approach has a great potential for 3D flow problems. 相似文献
8.
Andrea Thomann Markus Zenk Gabriella Puppo & Christian Klingenberg 《Communications In Computational Physics》2020,28(2):591-620
We present an implicit-explicit finite volume scheme for the Euler equations.
We start from the non-dimensionalised Euler equations where we split the pressure in
a slow and a fast acoustic part. We use a Suliciu type relaxation model which we split
in an explicit part, solved using a Godunov-type scheme based on an approximate
Riemann solver, and an implicit part where we solve an elliptic equation for the fast
pressure. The relaxation source terms are treated projecting the solution on the equilibrium manifold. The proposed scheme is positivity preserving with respect to the
density and internal energy and asymptotic preserving towards the incompressible
Euler equations. For this first order scheme we give a second order extension which
maintains the positivity property. We perform numerical experiments in 1D and 2D to
show the applicability of the proposed splitting and give convergence results for the
second order extension. 相似文献
9.
J. Vides B. Braconnier E. Audit C. Berthon & B. Nkonga 《Communications In Computational Physics》2014,15(1):46-75
We present a new numerical method to approximate the solutions of an
Euler-Poisson model, which is inherent to astrophysical flows where gravity plays an
important role. We propose a discretization of gravity which ensures adequate coupling of the Poisson and Euler equations, paying particular attention to the gravity
source term involved in the latter equations. In order to approximate this source term,
its discretization is introduced into the approximate Riemann solver used for the Euler equations. A relaxation scheme is involved and its robustness is established. The
method has been implemented in the software HERACLES [29] and several numerical
experiments involving gravitational flows for astrophysics highlight the scheme. 相似文献
10.
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
11.
12.
Yifei Wan & Yinhua Xia 《Communications In Computational Physics》2023,33(5):1270-1331
For steady Euler equations in complex boundary domains, high-order shockcapturing schemes usually suffer not only from the difficulty of steady-state convergence but also from the problem of dealing with physical boundaries on Cartesian
grids to achieve uniform high-order accuracy. In this paper, we utilize a fifth-order
finite difference hybrid WENO scheme to simulate steady Euler equations, and the
same fifth-order WENO extrapolation methods are developed to handle the curved
boundary. The values of the ghost points outside the physical boundary can be obtained by applying WENO extrapolation near the boundary, involving normal derivatives acquired by the simplified inverse Lax-Wendroff procedure. Both equivalent expressions involving curvature and numerical differentiation are utilized to transform
the tangential derivatives along the curved solid wall boundary. This hybrid WENO
scheme is robust for steady-state convergence and maintains high-order accuracy in
the smooth region even with the solid wall boundary condition. Besides, the essentially non-oscillation property is achieved. The numerical spectral analysis also shows
that this hybrid WENO scheme has low dispersion and dissipation errors. Numerical
examples are presented to validate the high-order accuracy and robust performance of
the hybrid scheme for steady Euler equations in curved domains with Cartesian grids. 相似文献
13.
Constraint Preserving Schemes Using Potential-Based Fluxes I Multidimensional Transport Equations
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We consider constraint preserving multidimensional evolution equations.
A prototypical example is provided by the magnetic induction equation of plasma
physics. The constraint of interest is the divergence of the magnetic field. We design
finite volume schemes which approximate these equations in a stable manner and
preserve a discrete version of the constraint. The schemes are based on reformulating
standard edge centered finite volume fluxes in terms of vertex centered potentials.
The potential-based approach provides a general framework for faithful discretizations
of constraint transport and we apply it to both divergence preserving as well as curl
preserving equations. We present benchmark numerical tests which confirm that our
potential-based schemes achieve high resolution, while being constraint preserving. 相似文献
14.
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
15.
A Conservative Lagrangian Scheme for Solving Compressible Fluid Flows with Multiple Internal Energy Equations
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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
16.
Keiichi Kitamura Eiji Shima Keiichiro Fujimoto & Z. J. Wang 《Communications In Computational Physics》2011,10(1):90-119
In low speed flow computations, compressible finite-volume solvers are
known to a) fail to converge in acceptable time and b) reach unphysical solutions.
These problems are known to be cured by A) preconditioning on the time-derivative
term, and B) control of numerical dissipation, respectively. There have been several
methods of A) and B) proposed separately. However, it is unclear which combination
is the most accurate, robust, and efficient for low speed flows. We carried out a
comparative study of several well-known or recently-developed low-dissipation Euler
fluxes coupled with a preconditioned LU-SGS (Lower-Upper Symmetric Gauss-Seidel)
implicit time integration scheme to compute steady flows. Through a series of numerical
experiments, accurate, efficient, and robust methods are suggested for low speed
flow computations. 相似文献
17.
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
18.
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
19.
Arbitrarily High-Order (Weighted) Essentially Non-Oscillatory Finite Difference Schemes for Anelastic Flows on Staggered Meshes
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Siddhartha Mishra Carlos Paré s-Pulido & Kyle G. Pressel 《Communications In Computational Physics》2021,29(5):1299-1335
We propose a WENO finite difference scheme to approximate anelastic flows,
and scalars advected by them, on staggered grids. In contrast to existing WENO
schemes on staggered grids, the proposed scheme is designed to be arbitrarily high-order accurate as it judiciously combines ENO interpolations of velocities with WENO
reconstructions of spatial derivatives. A set of numerical experiments are presented
to demonstrate the increase in accuracy and robustness with the proposed scheme,
when compared to existing WENO schemes and state-of-the-art central finite difference schemes. 相似文献
20.
A Nominally Second-Order Cell-Centered Finite Volume Scheme for Simulating Three-Dimensional Anisotropic Diffusion Equations on Unstructured Grids
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Pascal Jacq Pierre-Henri Maire & R& eacute mi Abgrall 《Communications In Computational Physics》2014,16(4):841-891
We present a finite volume based cell-centered method for solving diffusion
equations on three-dimensional unstructured grids with general tensor conduction.
Our main motivation concerns the numerical simulation of the coupling between fluid
flows and heat transfers. The corresponding numerical scheme is characterized by
cell-centered unknowns and a local stencil. Namely, the scheme results in a global
sparse diffusion matrix, which couples only the cell-centered unknowns. The space
discretization relies on the partition of polyhedral cells into sub-cells and on the partition of cell faces into sub-faces. It is characterized by the introduction of sub-face
normal fluxes and sub-face temperatures, which are auxiliary unknowns. A sub-cell-based variational formulation of the constitutive Fourier law allows to construct an
explicit approximation of the sub-face normal heat fluxes in terms of the cell-centered
temperature and the adjacent sub-face temperatures. The elimination of the sub-face
temperatures with respect to the cell-centered temperatures is achieved locally at each
node by solving a small and sparse linear system. This system is obtained by enforcing
the continuity condition of the normal heat flux across each sub-cell interface impinging at the node under consideration. The parallel implementation of the numerical
algorithm and its efficiency are described and analyzed. The accuracy and the robustness of the proposed finite volume method are assessed by means of various numerical
test cases. 相似文献