共查询到20条相似文献,搜索用时 31 毫秒
1.
A Geometry-Preserving Finite Volume Method for Compressible Fluids on Schwarzschild Spacetime
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We consider the relativistic Euler equations governing spherically symmetric, perfect fluid flows on the outer domain of communication of Schwarzschild spacetime, and we introduce a version of the finite volume method which is formulated
from the geometric formulation (and thus takes the geometry into account at the discretization level) and is well-balanced, in the sense that it preserves steady solutions to
the Euler equations on the curved geometry under consideration. In order to formulate our method, we first derive a closed formula describing all steady and spherically
symmetric solutions to the Euler equations posed on Schwarzschild spacetime. Second, we describe a geometry-preserving, finite volume method which is based on the family of steady solutions to the Euler system. Our scheme is second-order accurate and, as required, preserves the family of steady solutions at the discrete level.
Numerical experiments are presented which demonstrate the efficiency and robustness of the proposed method even for solutions containing shock waves and nonlinear
interacting wave patterns. As an application, we investigate the late-time asymptotics
of perturbed steady solutions and demonstrate its convergence for late time toward
another steady solution, taking the overall effect of the perturbation into account. 相似文献
2.
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
3.
A recent work of Li et al. [Numer. Math. Theor. Meth. Appl., 1(2008), pp.
92-112] proposed a finite volume solver to solve 2D steady Euler equations. Although
the Venkatakrishnan limiter is used to prevent the non-physical oscillations nearby the
shock region, the overshoot or undershoot phenomenon can still be observed. Moreover,
the numerical accuracy is degraded by using Venkatakrishnan limiter. To fix the
problems, in this paper the WENO type reconstruction is employed to gain both the
accurate approximations in smooth region and non-oscillatory sharp profiles near the
shock discontinuity. The numerical experiments will demonstrate the efficiency and
robustness of the proposed numerical strategy. 相似文献
4.
Jonas Zeifang Jochen Schü tz Klaus Kaiser rea Beck & Sebastian Noelle 《Communications In Computational Physics》2020,27(1):292-320
In this paper, we introduce an extension of a splitting method for singularly
perturbed equations, the so-called RS-IMEX splitting [Kaiser et al., Journal of Scientific
Computing, 70(3), 1390–1407], to deal with the fully compressible Euler equations. The
straightforward application of the splitting yields sub-equations that are, due to the
occurrence of complex eigenvalues, not hyperbolic. A modification, slightly changing
the convective flux, is introduced that overcomes this issue. It is shown that the splitting gives rise to a discretization that respects the low-Mach number limit of the Euler
equations; numerical results using finite volume and discontinuous Galerkin schemes
show the potential of the discretization. 相似文献
5.
Shuang Tan Wenjun Sun Kun Xu Junxia Wei & Guoxi Ni 《Communications In Computational Physics》2020,28(3):1189-1218
In this paper, a time implicit unified gas kinetic scheme (IUGKS) for 3D
multi-group neutron transport equation with delayed neutron is developed. The
explicit scheme, implicit 1st-order backward Euler scheme, and 2nd-order Crank-Nicholson scheme, become the subsets of the current IUGKS. In neutron transport,
the microscopic angular flux and the macroscopic scalar flux are fully coupled in an
implicit way with the combination of dual-time step technique for the convergence acceleration of unsteady evolution. In IUGKS, the computational time step is no longer
limited by the Courant-Friedrichs-Lewy (CFL) condition, which improves the computational efficiency in both steady and unsteady simulations with a large time step.
Mathematically, the current scheme has the asymptotic preserving (AP) property in
recovering automatically the diffusion solution in the continuum regime. Since the
explicit scanning along neutron traveling direction within the computational domain
is not needed in IUGKS, the scheme can be easily extended to multi-dimensional and
parallel computations. The numerical tests demonstrate that the IUGKS has high computational efficiency, high accuracy, and strong robustness when compared with other
schemes, such as the explicit UGKS, the commonly used finite difference, and finite
volume methods. This study shows that the IUGKS can be used faithfully to study
neutron transport in practical engineering applications. 相似文献
6.
A Stabilized Finite Element Method for Modified Poisson-Nernst-Planck Equations to Determine Ion Flow Through a Nanopore
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Jehanzeb Hameed Chaudhry Jeffrey Comer Aleksei Aksimentiev & Luke N. Olson 《Communications In Computational Physics》2014,15(1):93-125
The conventional Poisson-Nernst-Planck equations do not account for the
finite size of ions explicitly. This leads to solutions featuring unrealistically high ionic
concentrations in the regions subject to external potentials, in particular, near highly
charged surfaces. A modified form of the Poisson-Nernst-Planck equations accounts
for steric effects and results in solutions with finite ion concentrations. Here, we evaluate numerical methods for solving the modified Poisson-Nernst-Planck equations by
modeling electric field-driven transport of ions through a nanopore. We describe a
novel, robust finite element solver that combines the applications of the Newton's
method to the nonlinear Galerkin form of the equations, augmented with stabilization
terms to appropriately handle the drift-diffusion processes.To make direct comparison with particle-based simulations possible, our method is
specifically designed to produce solutions under periodic boundary conditions and
to conserve the number of ions in the solution domain. We test our finite element
solver on a set of challenging numerical experiments that include calculations of the
ion distribution in a volume confined between two charged plates, calculations of the
ionic current though a nanopore subject to an external electric field, and modeling the
effect of a DNA molecule on the ion concentration and nanopore current. 相似文献
7.
Numerical Simulation of Compressible Vortical Flows Using a Conservative Unstructured-Grid Adaptive Scheme
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Giuseppe Forestieri Alberto Guardone Dario Isola Filippo Marulli & Giuseppe Quaranta 《Communications In Computational Physics》2012,12(3):866-884
A two-dimensional numerical scheme for the compressible Euler equations
is presented and applied here to the simulation of exemplary compressible vortical
flows. The proposed approach allows to perform computations on unstructured moving grids with adaptation, which is required to capture complex features of the flow-field. Grid adaptation is driven by suitable error indicators based on the Mach number
and by element-quality constraints as well. At the new time level, the computational
grid is obtained by a suitable combination of grid smoothing, edge-swapping, grid
refinement and de-refinement. The grid modifications—including topology modification due to edge-swapping or the insertion/deletion of a new grid node—are interpreted at the flow solver level as continuous (in time) deformations of suitably-defined
node-centered finite volumes. The solution over the new grid is obtained without explicitly resorting to interpolation techniques, since the definition of suitable interface
velocities allows one to determine the new solution by simple integration of the Arbitrary Lagrangian-Eulerian formulation of the flow equations. Numerical simulations
of the steady oblique-shock problem, of the steady transonic flow and of the start-up
unsteady flow around the NACA 0012 airfoil are presented to assess the scheme capabilities to describe these flows accurately. 相似文献
8.
9.
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
10.
Bhuvana Srinivasan Ammar Hakim & Uri Shumlak 《Communications In Computational Physics》2011,10(1):183-215
The finite volume wave propagation method and the finite element RungeKutta
discontinuous Galerkin (RKDG) method are studied for applications to balance
laws describing plasma fluids. The plasma fluid equations explored are dispersive and
not dissipative. The physical dispersion introduced through the source terms leads to
the wide variety of plasma waves. The dispersive nature of the plasma fluid equations
explored separates the work in this paper from previous publications. The linearized
Euler equations with dispersive source terms are used as a model equation system to
compare the wave propagation and RKDG methods. The numerical methods are then
studied for applications of the full two-fluid plasma equations. The two-fluid equations
describe the self-consistent evolution of electron and ion fluids in the presence
of electromagnetic fields. It is found that the wave propagation method, when run
at a CFL number of 1, is more accurate for equation systems that do not have disparate
characteristic speeds. However, if the oscillation frequency is large compared
to the frequency of information propagation, source splitting in the wave propagation
method may cause phase errors. The Runge-Kutta discontinuous Galerkin method
provides more accurate results for problems near steady-state as well as problems with
disparate characteristic speeds when using higher spatial orders. 相似文献
11.
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
12.
Remapping-Free Adaptive GRP Method for Multi-Fluid Flows I: One Dimensional Euler Equations
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In this paper, a remapping-free adaptive GRP method for one dimensional
(1-D) compressible flows is developed. Based on the framework of finite volume
method, the 1-D Euler equations are discretized on moving volumes and the resulting
numerical fluxes are computed directly by the GRP method. Thus the remapping
process in the earlier adaptive GRP algorithm [17,18] is omitted. By adopting a flexible
moving mesh strategy, this method could be applied for multi-fluid problems. The interface
of two fluids will be kept at the node of computational grids and the GRP solver
is extended at the material interfaces of multi-fluid flows accordingly. Some typical numerical
tests show competitive performances of the new method, especially for contact
discontinuities of one fluid cases and the material interface tracking of multi-fluid
cases. 相似文献
13.
A Tailored Finite Point Method for Solving Steady MHD Duct Flow Problems with Boundary Layers
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Po-Wen Hsieh Yintzer Shih & Suh-Yuh Yang 《Communications In Computational Physics》2011,10(1):161-182
In this paper we propose a development of the finite difference method,
called the tailored finite point method, for solving steady magnetohydrodynamic
(MHD) duct flow problems with a high Hartmann number. When the Hartmann number
is large, the MHD duct flow is convection-dominated and thus its solution may exhibit
localized phenomena such as the boundary layer. Most conventional numerical
methods can not efficiently solve the layer problem because they are lacking in either
stability or accuracy. However, the proposed tailored finite point method is capable
of resolving high gradients near the layer regions without refining the mesh. Firstly,
we devise the tailored finite point method for the scalar inhomogeneous convection-diffusion
problem, and then extend it to the MHD duct flow which consists of a coupled
system of convection-diffusion equations. For each interior grid point of a given
rectangular mesh, we construct a finite-point difference operator at that point with
some nearby grid points, where the coefficients of the difference operator are tailored
to some particular properties of the problem. Numerical examples are provided to
show the high performance of the proposed method. 相似文献
14.
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
15.
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
16.
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
17.
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
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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
18.
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
19.
Up to now, several numerical methods have been presented to solve finite horizon fractional optimal control problems by researchers, while solving fractional optimal control problems on infinite domain is a challenging work. Hence, in this article, a numerical method is proposed to solve fractional infinite horizon optimal control problems. At the first stage, a domain transformation technique is used to map the infinite domain to a finite horizon. Also, fractional derivative defined on an unbounded domain is converted into an equivalent derivative on a finite domain. Then, a new shifted Legendre pseudospectral method is utilized to solve the obtained finite problem and a nonlinear programming problem is suggested to approximate the optimal solutions. Finally, some numerical examples are given to show the efficiency of the method. 相似文献
20.
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