共查询到20条相似文献,搜索用时 15 毫秒
1.
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
2.
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
3.
This paper provides an analysis of a fractional-step projection method to compute incompressible viscous flows by means of
finite element approximations. The analysis is based on the idea that the appropriate functional setting for projection methods
must accommodate two different spaces for representing the velocity fields calculated respectively in the viscous and the
incompressible half steps of the method. Such a theoretical distinction leads to a finite element projection method with a
Poisson equation for the incremental pressure unknown and to a very practical implementation of the method with only the intermediate
velocity appearing in the numerical algorithm. Error estimates in finite time are given. An extension of the method to a problem
with unconventional boundary conditions is also considered to illustrate the flexibility of the proposed method.
Received October 2, 1995 / Revised version received July 9, 1997 相似文献
4.
In this work we propose and apply a numerical method based on finite volume relaxation approximation for computing the bed-load sediment transport in shallow water flows, in one and two space dimensions. The water flow is modeled by the well-known nonlinear shallow water equations which are coupled with a bed updating equation. Using a relaxation approximation, the nonlinear set of equations (and for two different formulations) is transformed to a semilinear diagonalizable problem with linear characteristic variables. A second order MUSCL-TVD method is used for the advection stage while an implicit–explicit Runge–Kutta scheme solves the relaxation stage. The main advantages of this approach are that neither Riemann problem solvers nor nonlinear iterations are required during the solution process. For the two different formulations, the applicability and effectiveness of the presented scheme is verified by comparing numerical results obtained for several benchmark test problems. 相似文献
5.
In this work we design and analyze an efficient numerical method to solve two dimensional initial-boundary value reaction–diffusion
problems, for which the diffusion parameter can be very small with respect to the reaction term. The method is defined by
combining the Peaceman and Rachford alternating direction method to discretize in time, together with a HODIE finite difference
scheme constructed on a tailored mesh. We prove that the resulting scheme is ε-uniformly convergent of second order in time
and of third order in spatial variables. Some numerical examples illustrate the efficiency of the method and the orders of
uniform convergence proved theoretically. We also show that it is easy to avoid the well-known order reduction phenomenon,
which is usually produced in the time integration process when the boundary conditions are time dependent.
This research has been partially supported by the project MEC/FEDER MTM2004-01905 and the Diputación General de Aragón. 相似文献
6.
We study dislocation dynamics with a level set point of view. The model we present here looks at the zero level set of the solution of a non local Hamilton Jacobi equation, as a dislocation in a plane of a crystal. The front has a normal speed, depending on the solution itself. We prove existence and uniqueness for short time in the set of continuous viscosity solutions. We also present a first order finite difference scheme for the corresponding level set formulation of the model. The scheme is based on monotone numerical Hamiltonian, proposed by Osher and Sethian. The non local character of the problem makes it not monotone. We obtain an explicit convergence rate of the approximate solution to the viscosity solution. We finally provide numerical simulations.This work has been supported by funds from ACI JC 1041 “Mouvements d’interfaces avec termes non-locaux”, from ACI-JC 1025 “Dynamique des dislocations” and from ONERA, Office National d’Etudes et de Recherches. The second author was also supported by the ENPC-Région Ile de France. 相似文献
7.
Stefano Berrone 《Numerische Mathematik》2002,91(3):389-422
Summary. We derive a residual-based a posteriori error estimator for a stabilized finite element discretization of certain incompressible Oseen-like equations. We focus our
attention on the behaviour of the effectivity index and we carry on a numerical study of its sensitiveness to the problem
and mesh parameters. We also consider a scalar reaction-convection-diffusion problem and a divergence-free projection problem
in order to investigate the effects on the robustness of our a posteriori error estimator of the reaction-convection-diffusion phenomena and, separately, of the incompressibility constraint.
Received March 21, 2001 / Revised version received July 30, 2001 / Published online October 17, 2001 相似文献
8.
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
9.
Summary. A finite element formulation is developed for the two dimensional nonlinear time dependent compressible Navier–Stokes equations
on a bounded domain. The existence and uniqueness of the solution to the numerical formulation is proved. An error estimate
for the numerical solution is obtained.
Received September 9, 1997 / Revised version received August 12, 1999 / Published online July 12, 2000 相似文献
10.
A one-dimensional free surface problem is considered. It consists in Burgers’ equation with an additional diffusion term on a moving interval. The well-posedness of the problem is investigated and existence and uniqueness results are obtained locally in time. A semi-discretization in space with a piecewise linear finite element method is considered. A priori and a posteriori error estimates are given for the semi-discretization in space. A time splitting scheme allows to obtain numerical results in agreement with the theoretical investigations.Supported by the Swiss National Science Foundation 相似文献
11.
N. Yu. Bakaev 《BIT Numerical Mathematics》1997,37(2):237-255
The maximum norm error estimates of the Galerkin finite element approximations to the solutions of differential and integro-differential
multi-dimensional parabolic problems are considered. Our method is based on the use of the discrete version of the elliptic-Sobolev
inequality and some operator representations of the finite element solutions. The results of the present paper lead to the
error estimates of optimal or almost optimal order for the case of simplicial Lagrangian piecewise polynomial elements. 相似文献
12.
In this paper, we propose a new method to compute the numerical flux of a finite volume scheme, used for the approximation
of the solution of the nonlinear partial differential equation ut+div(qf(u))−ΔΦ(u)=0 in a 1D, 2D or 3D domain. The function Φ is supposed to be strictly increasing, but some values s such that Φ′(s)=0 can exist. The method is based on the solution, at each interface between two control volumes, of the nonlinear elliptic
two point boundary value problem (qf(υ)+(Φ(υ))′)′=0 with Dirichlet boundary conditions given by the values of the discrete approximation in both control volumes. We prove
the existence of a solution to this two point boundary value problem. We show that the expression for the numerical flux can
be yielded without referring to this solution. Furthermore, we prove that the so designed finite volume scheme has the expected
stability properties and that its solution converges to the weak solution of the continuous problem. Numerical results show
the increase of accuracy due to the use of this scheme, compared to some other schemes. 相似文献
13.
Olaf Steinbach 《Numerische Mathematik》2001,88(2):367-379
Summary. In this paper we prove the stability of the projection onto the finite element trial space of piecewise polynomial, in particular, piecewise linear basis functions in
for . We formulate explicit and computable local mesh conditions to be satisfied which depend on the Sobolev index s. In conclusion we prove a stability condition needed in the numerical analysis of mixed and hybrid boundary element methods
as well as in the construction of efficient preconditioners in adaptive boundary and finite element methods.
Received October 14, 1999 / Revised version received March 24, 2000 / Published online October 16, 2000 相似文献
14.
Summary. We introduce two classes of monotone finite volume schemes for Hamilton-Jacobi equations. The corresponding approximating
functions are piecewise linear defined on a mesh consisting of triangles. The schemes are shown to converge to the viscosity
solution of the Hamilton–Jacobi equation.
Received February 25, 1998 / Published online: June 29, 1999 相似文献
15.
Saulo P. Oliveira Alexandre L. Madureira Frederic Valentin 《Journal of Computational and Applied Mathematics》2009
We discuss the numerical integration of polynomials times non-polynomial weighting functions in two dimensions arising from multiscale finite element computations. The proposed quadrature rules are significantly more accurate than standard quadratures and are better suited to existing finite element codes than formulas computed by symbolic integration. We validate this approach by introducing the new quadrature formulas into a multiscale finite element method for the two-dimensional reaction–diffusion equation. 相似文献
16.
Daniel Bouche Jean-Michel Ghidaglia 《Journal of Computational and Applied Mathematics》2011,235(18):5394-5410
In this paper we estimate the error of upwind first order finite volume schemes applied to scalar conservation laws. As a first step, we consider standard upwind and flux finite volume scheme discretization of a linear equation with space variable coefficients in conservation form. We prove that, in spite of their lack of consistency, both schemes lead to a first order error estimate. As a final step, we prove a similar estimate for the nonlinear case. Our proofs rely on the notion of geometric corrector, introduced in our previous paper by Bouche et al. (2005) [24] in the context of constant coefficient linear advection equations. 相似文献
17.
In this paper we obtain convergence results for the fully discrete projection method for the numerical approximation of the
incompressible Navier–Stokes equations using a finite element approximation for the space discretization. We consider two
situations. In the first one, the analysis relies on the satisfaction of the inf-sup condition for the velocity-pressure finite
element spaces. After that, we study a fully discrete fractional step method using a Poisson equation for the pressure. In
this case the velocity-pressure interpolations do not need to accomplish the inf-sup condition and in fact we consider the
case in which equal velocity-pressure interpolation is used. Optimal convergence results in time and space have been obtained
in both cases. 相似文献
18.
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
19.
Summary. For the high-order numerical approximation of hyperbolic systems of conservation laws, we propose to use as a building principle
an entropy diminishing criterion instead of the familiar total variation diminishing criterion introduced by Harten for scalar equations. Based on this new
criterion, we derive entropy diminishing projections that ensure, both, the second order of accuracy and all of the classical discrete entropy inequalities. The resulting scheme
is a nonlinear version of the classical Van Leer's MUSCL scheme. Strong convergence of this second order, entropy satisfying
scheme is proved for systems of two equations. Numerical tests demonstrate the interest of our theory.
Received March 28, 1995 / Revised version received June 17, 1995 相似文献
20.
A finite volume scheme for the global shallow water model on the cubed-sphere mesh is proposed and studied in this paper. The new cell-centered scheme is based on Osher’s Riemann solver together with a high-order spatial reconstruction. On each patch interface of the cubed-sphere only one layer of ghost cells is needed in the scheme and the numerical flux is calculated symmetrically across the interface to ensure the numerical conservation of total mass. The discretization of the topographic term in the equation is properly modified in a well-balanced manner to suppress spurious oscillations when the bottom topography is non-smooth. Numerical results for several test cases including a steady-state nonlinear geostrophic flow and a zonal flow over an isolated mountain are provided to show the flexibility of the scheme. Some parallel implementation details as well as some performance results on a parallel supercomputer with more than one thousand processor cores are also provided. 相似文献