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1.
Summary. A third-order accurate Godunov-type scheme for the approximate solution of hyperbolic systems of conservation laws is presented.
Its two main ingredients include: 1. A non-oscillatory piecewise-quadratic reconstruction of pointvalues from their given
cell averages; and 2. A central differencing based on staggered evolution of the reconstructed cell averages. This results in a third-order central scheme, an extension along the lines
of the second-order central scheme of Nessyahu and Tadmor \cite{NT}. The scalar scheme is non-oscillatory (and hence – convergent),
in the sense that it does not increase the number of initial extrema (– as does the exact entropy solution operator). Extension to systems is carried out by componentwise application of the scalar framework. In particular, we have the advantage that, unlike upwind schemes, no (approximate) Riemann
solvers, field-by-field characteristic decompositions, etc., are required. Numerical experiments confirm the high-resolution
content of the proposed scheme. Thus, a considerable amount of simplicity and robustness is gained while retaining the expected
third-order resolution.
Received April 10, 1996 / Revised version received January 20, 1997 相似文献
2.
给出了一种真正多维的HLL Riemann解法器.采用TV(Toro-Vázquez)分裂将通量分裂成对流通量和压力通量,其中对流通量的计算采用类似于AUSM格式的迎风方法,压力通量的计算采用波速基于压力系统特征值的HLL格式,并将HLL格式耗散项中的密度差用压力差代替,来克服传统的HLL格式不能分辨接触间断的缺点.为了实现数值格式真正多维的特性,分别计算网格界面中点和角点上的数值通量,并且采用Simpson公式加权中点和角点上的数值通量来得到网格界面上的数值通量.采用基于SDWLS(solution dependent weighted least squares)梯度的线性重构来获得空间的二阶精度,时间离散采用二阶Runge-Kutta格式.数值实验表明,相比于传统的一维HLL格式,该文的真正多维HLL格式具有能够分辨接触间断,消除慢行激波波后振荡以及更大的时间步长等优点.并且,与其他能够分辨接触间断的格式(例如HLLC格式)不同的是,真正多维的HLL格式在计算二维问题时不会出现数值激波不稳定现象. 相似文献
3.
The Generalized Riemann Problem (GRP) for a nonlinear hyperbolic system of m balance laws (or alternatively “quasi-conservative” laws) in one space dimension is now well-known and can be formulated
as follows: Given initial-data which are analytic on two sides of a discontinuity, determine the time evolution of the solution
at the discontinuity. In particular, the GRP numerical scheme (second-order high resolution) is based on an analytical evaluation
of the first time derivative. It turns out that this derivative depends only on the first-order spatial derivatives, hence
the initial data can be taken as piecewise linear. The analytical solution is readily obtained for a single equation (m = 1) and, more generally, if the system is endowed with a complete (coordinate) set of Riemann invariants. In this case it
can be “diagonalized” and reduced to the scalar case. However, most systems with m > 2 do not admit such a set of Riemann invariants. This paper introduces a generalization of this concept: weakly coupled
systems (WCS). Such systems have only “partial set” of Riemann invariants, but these sets are weakly coupled in a way which
enables a “diagonalized” treatment of the GRP. An important example of a WCS is the Euler system of compressible, nonisentropic
fluid flow (m = 3). The solution of the GRP discussed here is based on a careful analysis of rarefaction waves. A “propagation of singularities”
argument is applied to appropriate Riemann invariants across the rarefaction fan. It serves to “rotate” initial spatial slopes
into “time derivative”. In particular, the case of a “sonic point” is incorporated easily into the general treatment. A GRP
scheme based on this solution is derived, and several numerical examples are presented. Special attention is given to the
“acoustic approximation” of the analytical solution. It can be viewed as a proper linearization (different from the approach
of Roe) of the nonlinear system. The resulting numerical scheme is the simplest (second-order, high-resolution) generalization
of the Godunov scheme. 相似文献
4.
Summary. We first analyse a semi-discrete operator splitting method for nonlinear, possibly strongly degenerate, convection-diffusion
equations. Due to strong degeneracy, solutions can be discontinuous and are in general not uniquely determined by their data.
Hence weak solutions satisfying an entropy condition are sought. We then propose and analyse a fully discrete splitting method
which employs a front tracking scheme for the convection step and a finite difference scheme for the diffusion step. Numerical
examples are presented which demonstrate that our method can be used to compute physically correct solutions to mixed hyperbolic-parabolic
convection-diffusion equations.
Received November 4, 1997 / Revised version received June 22, 1998 相似文献
5.
Wolfgang Dahmen Birgit Gottschlich–Müller Siegfried Müller 《Numerische Mathematik》2001,88(3):399-443
Summary. In recent years a variety of high–order schemes for the numerical solution of conservation laws has been developed. In general,
these numerical methods involve expensive flux evaluations in order to resolve discontinuities accurately. But in large parts
of the flow domain the solution is smooth. Hence in these regions an unexpensive finite difference scheme suffices. In order
to reduce the number of expensive flux evaluations we employ a multiresolution strategy which is similar in spirit to an approach
that has been proposed by A. Harten several years ago. Concrete ingredients of this methodology have been described so far
essentially for problems in a single space dimension. In order to realize such concepts for problems with several spatial
dimensions and boundary fitted meshes essential deviations from previous investigations appear to be necessary though. This
concerns handling the more complex interrelations of fluxes across cell interfaces, the derivation of appropriate evolution
equations for multiscale representations of cell averages, stability and convergence, quantifying the compression effects
by suitable adapted multiscale transformations and last but not least laying grounds for ultimately avoiding the storage of
data corresponding to a full global mesh for the highest level of resolution. The objective of this paper is to develop such
ingredients for any spatial dimension and block structured meshes obtained as parametric images of Cartesian grids. We conclude
with some numerical results for the two–dimensional Euler equations modeling hypersonic flow around a blunt body.
Received June 24, 1998 / Revised version received February 21, 2000 / Published online November 8, 2000 相似文献
6.
《Journal of Computational and Applied Mathematics》2005,175(1):41-62
We present a relaxed scheme with more precise information about local speeds of propagation and a multidimensional construction of the cell averages. The physical domain of dependence is simulated correctly and high resolution is maintained. Relaxation schemes have advantages that include high resolution, simplicity and explicitly no (approximate) Riemann solvers and no characteristic decomposition is necessary. Performance of the scheme is illustrated by tests on two-dimensional Euler equations of gas dynamics. 相似文献
7.
We introduce a high resolution fifth-order semi-discrete Hermite central-upwind scheme for multidimensional Hamilton–Jacobi equations. The numerical fluxes of the scheme are constructed by Hermite polynomials which can be obtained by using the short-time assignment of the first derivatives. The extensions of the proposed semi-discrete Hermite central-upwind scheme to multidimensional cases are straightforward. The accuracy, efficiency and stability properties of our schemes are finally demonstrated via a variety of numerical examples. 相似文献
8.
We consider a numerical scheme for a class of degenerate parabolic equations, including both slow and fast diffusion cases.
A particular example in this sense is the Richards equation modeling the flow in porous media. The numerical scheme is based
on the mixed finite element method (MFEM) in space, and is of one step implicit in time. The lowest order Raviart–Thomas elements
are used. Here we extend the results in Radu et al. (SIAM J Numer Anal 42:1452–1478, 2004), Schneid et al. (Numer Math 98:353–370,
2004) to a more general framework, by allowing for both types of degeneracies. We derive error estimates in terms of the discretization
parameters and show the convergence of the scheme. The features of the MFEM, especially of the lowest order Raviart–Thomas
elements, are now fully exploited in the proof of the convergence. The paper is concluded by numerical examples. 相似文献
9.
Runge–Kutta based convolution quadrature methods for abstract, well-posed, linear, and homogeneous Volterra equations, non
necessarily of sectorial type, are developed. A general representation of the numerical solution in terms of the continuous
one is given. The error and stability analysis is based on this representation, which, for the particular case of the backward
Euler method, also shows that the numerical solution inherits some interesting qualitative properties, such as positivity,
of the exact solution. Numerical illustrations are provided. 相似文献
10.
We present a relaxation system for ideal magnetohydrodynamics (MHD) that is an extension of the Suliciu relaxation system
for the Euler equations of gas dynamics. From it one can derive approximate Riemann solvers with three or seven waves, that
generalize the HLLC solver for gas dynamics. Under some subcharacteristic conditions, the solvers satisfy discrete entropy
inequalities, and preserve positivity of density and internal energy. The subcharacteristic conditions are nonlinear constraints
on the relaxation parameters relating them to the initial states and the intermediate states of the approximate Riemann solver
itself. The 7-wave version of the solver is able to resolve exactly all material and Alfven isolated contact discontinuities.
Practical considerations and numerical results will be provided in another paper. 相似文献
11.
The focus of this paper is to simulate the transport of a passive pollutant by a flow modelled by the two-dimensional shallow water equations. Considering the friction terms, new model for simulating the steady and unsteady transport of pollutant is established. Then the adaptive semi-discrete central-upwind scheme based on central weighted essentially non-oscillatory reconstruction is utilized for simulating the two-dimensional steady and unsteady transport of pollutant. The non-oscillatory behavior and accuracy of the scheme are demonstrated by the numerical result. 相似文献
12.
Summary. Based on Nessyahu and Tadmor's nonoscillatory central difference schemes for one-dimensional hyperbolic conservation laws
[16], for higher dimensions several finite volume extensions and numerical results on structured and unstructured grids have
been presented. The experiments show the wide applicability of these multidimensional schemes. The theoretical arguments which
support this are some maximum-principles and a convergence proof in the scalar linear case. A general proof of convergence,
as obtained for the original one-dimensional NT-schemes, does not exist for any of the extensions to multidimensional nonlinear
problems. For the finite volume extension on two-dimensional unstructured grids introduced by Arminjon and Viallon [3,4] we
present a proof of convergence for the first order scheme in case of a nonlinear scalar hyperbolic conservation law.
Received April 8, 2000 / Published online December 19, 2000 相似文献
13.
In this article, a class of nonlinear evolution equations – reaction–diffusion equations with time delay – is studied. By
combining the domain decomposition technique and the finite difference method, the results for the existence, convergence
and the stability of the numerical solution are obtained in the case of subdomain overlap and when the time-space is completely
discretized.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
14.
Non-oscillatory schemes are widely used in numerical approximations of nonlinear conservation laws. The Nessyahu–Tadmor (NT) scheme is an example of a second order scheme that is both robust and simple. In this paper, we prove a new stability property of the NT scheme based on the standard minmod reconstruction in the case of a scalar strictly convex conservation law. This property is similar to the One-sided Lipschitz condition for first order schemes. Using this new stability, we derive the convergence of the NT scheme to the exact entropy solution without imposing any nonhomogeneous limitations on the method. We also derive an error estimate for monotone initial data. 相似文献
15.
The determination of boundary conditions for the Euler equations of gas dynamics in a pipe with partially open pipe ends is considered. The boundary problem is formulated in terms of the exact solution of the Riemann problem and of the St. Venant equation for quasi-steady flow so that a pressure-driven calculation of boundary conditions is defined. The resulting set of equations is solved by a Newton scheme. The proposed algorithm is able to solve for all inflow and outflow situations including choked and supersonic flow.Received: August 7, 2002; revised: November 11, 2002 相似文献
16.
This paper focuses on the numerical analysis of a finite element method with stabilization for the unsteady incompressible
Navier–Stokes equations. Incompressibility and convective effects are both stabilized adding an interior penalty term giving
L
2-control of the jump of the gradient of the approximate solution over the internal faces. Using continuous equal-order finite
elements for both velocities and pressures, in a space semi-discretized formulation, we prove convergence of the approximate
solution. The error estimates hold irrespective of the Reynolds number, and hence also for the incompressible Euler equations,
provided the exact solution is smooth. 相似文献
17.
Summary. We analyze a fully discrete numerical scheme approximating the evolution of n–dimensional graphs under anisotropic mean curvature. The highly nonlinear problem is discretized by piecewise linear finite
elements in space and semi–implicitly in time. The scheme is unconditionally stable und we obtain optimal error estimates
in natural norms. We also present numerical examples which confirm our theoretical results.
Received October 2, 2000 / Published online July 25, 2001 相似文献
18.
Summary.
We consider the positivity preserving property of first and
higher order finite volume schemes for one and two
dimensional Euler equations of gas dynamics.
A general framework is established which shows the positivity
of density and pressure whenever the underlying one
dimensional first order building block based on an exact
or approximate
Riemann solver and the reconstruction are both positivity
preserving.
Appropriate limitation to achieve a high order
positivity preserving reconstruction is described.
Received May 20, 1994 相似文献
19.
We propose and analyze a numerical scheme for nonlinear degenerate parabolic convection–diffusion–reaction equations in two or three space dimensions. We discretize the diffusion term, which generally involves an inhomogeneous and anisotropic diffusion tensor, over an unstructured simplicial mesh of the space domain by means of the piecewise linear nonconforming (Crouzeix–Raviart) finite element method, or using the stiffness matrix of the hybridization of the lowest-order Raviart–Thomas mixed finite element method. The other terms are discretized by means of a cell-centered finite volume scheme on a dual mesh, where the dual volumes are constructed around the sides of the original mesh. Checking the local Péclet number, we set up the exact necessary amount of upstream weighting to avoid spurious oscillations in the convection-dominated case. This technique also ensures the validity of the discrete maximum principle under some conditions on the mesh and the diffusion tensor. We prove the convergence of the scheme, only supposing the shape regularity condition for the original mesh. We use a priori estimates and the Kolmogorov relative compactness theorem for this purpose. The proposed scheme is robust, only 5-point (7-point in space dimension three), locally conservative, efficient, and stable, which is confirmed by numerical experiments.This work was supported by the GdR MoMaS, CNRS-2439, ANDRA, BRGM, CEA, EdF, France. 相似文献
20.
This paper deals with finite-difference approximations of Euler equations arising in the variational formulation of image
segmentation problems. We illustrate how they can be defined by the following steps: (a) definition of the minimization problem
for the Mumford–Shah functional (MSf), (b) definition of a sequence of functionals Γ-convergent to the MSf, and (c) definition
and numerical solution of the Euler equations associated to the kth functional of the sequence. We define finite difference approximations of the Euler equations, the related solution algorithms,
and we present applications to segmentation problems by using synthetic images. We discuss application results, and we mainly
analyze computed discontinuity contours and convergence histories of method executions.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献