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1.
Silvia Jerez 《Numerical Methods for Partial Differential Equations》2013,29(6):2133-2145
In this work, we present a monotone first‐order weighted (FORWE) method for scalar conservation laws using a variational formulation. We prove theoretical properties as consistency, monotonicity, and convergence of the proposed scheme for the one‐dimensional (1D) Cauchy problem. These convergence results are extended to multidimensional scalar conservation laws by a dimensional splitting technique. For the validation of the FORWE method, we consider some standard bench‐mark tests of bidimensional and 1D conservation law equations. Finally, we analyze the accuracy of the method with L1 and L∞ error estimates. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013 相似文献
2.
Philipp Birken 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2008,59(3):416-433
This article is concerned with the numerical simulation of flows at low Mach numbers which are subject to the gravitational
force and strong heat sources. As a specific example for such flows, a fire event in a car tunnel will be considered in detail.
The low Mach flow is treated with a preconditioning technique allowing the computation of unsteady flows, while the source
terms for gravitation and heat are incorporated via an operator splitting. It is shown that a first order discretization in
space is not able to compute the buoyancy forces properly on reasonable grids. The feasibility of the method is demonstrated
on several test cases. 相似文献
3.
Sebastian Noelle 《Advances in Computational Mathematics》1995,3(1):197-218
We prove convergence to the entropy solution of a general class of higher order finite volume schemes on unstructured, irregular
grids for multidimensional scalar conservation laws. Such grids allow for cells to become flat in the limit. We derive a new
entropy inequality for higher order schemes built on Godunov's numerical flux. Our result implies convergence of suitably
modified versions of MUSCL-type finite volume schemes, ENO schemes and the discontinuous Galerkin finite element method.
Supported by Deutsche Forschungsgemeinschaft, SFB256. 相似文献
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In this paper we shall derive a posteriori error estimates in the -norm for upwind finite volume schemes for the discretization of nonlinear conservation laws on unstructured grids in multi dimensions. This result is mainly based on some fundamental a priori error estimates published in a recent paper by C. Chainais-Hillairet. The theoretical results are confirmed by numerical experiments.
6.
Following Ben-Artzi and LeFloch, we consider nonlinear hyperbolic conservation laws posed on a Riemannian manifold, and we establish an L1-error estimate for a class of finite volume schemes allowing for the approximation of entropy solutions to the initial value problem. The error in the L1 norm is of order h1/4 at most, where h represents the maximal diameter of elements in the family of geodesic triangulations. The proof relies on a suitable generalization of Cockburn, Coquel, and LeFloch's theory which was originally developed in the Euclidian setting. We extend the arguments to curved manifolds, by taking into account the effects to the geometry and overcoming several new technical difficulties. 相似文献
7.
Sebastian Noelle. 《Mathematics of Computation》1996,65(215):1155-1163
Recently, Cockburn, Coquel and LeFloch proved convergence and error estimates for higher-order finite volume schemes. Their result is based on entropy inequalities which are derived under restrictive assumptions on either the flux function or the numerical fluxes. Moreover, they assume that the spatial grid satisfies a standard regularity assumption. Using instead entropy inequalities derived in previous work by Kröner, Noelle and Rokyta and a weaker condition on the grid, we can generalize and simplify the error estimates.
8.
Thomas P.-Y. Yu 《Journal of Mathematical Analysis and Applications》2005,302(1):201-216
It is well known that the critical Hölder regularity of a subdivision schemes can typically be expressed in terms of the joint-spectral radius (JSR) of two operators restricted to a common finite-dimensional invariant subspace. In this article, we investigate interpolatory Hermite subdivision schemes in dimension one and specifically those with optimal accuracy orders. The latter include as special cases the well-known Lagrange interpolatory subdivision schemes by Deslauriers and Dubuc. We first show how to express the critical Hölder regularity of such a scheme in terms of the joint-spectral radius of a matrix pair {F0,F1} given in a very explicit form. While the so-called finiteness conjecture for JSR is known to be not true in general, we conjecture that for such matrix pairs arising from Hermite interpolatory schemes of optimal accuracy orders a “strong finiteness conjecture” holds: ρ(F0,F1)=ρ(F0)=ρ(F1). We prove that this conjecture is a consequence of another conjectured property of Hermite interpolatory schemes which, in turn, is connected to a kind of positivity property of matrix polynomials. We also prove these conjectures in certain new cases using both time and frequency domain arguments; our study here strongly suggests the existence of a notion of “positive definiteness” for non-Hermitian matrices. 相似文献
9.
Hua-zhong Tang Gerald Warnecke 《计算数学(英文版)》2006,24(2):121-140
This paper presents a class of high resolution local time step schemes for nonlinear hyperbolic conservation laws and the closely related convection-diffusion equations, by projecting the solution increments of the underlying partial differential equations (PDE) at each local time step. The main advantages are that they are of good consistency, and it is convenient to implement them. The schemes are L^∞ stable, satisfy a cell entropy inequality, and may be extended to the initial boundary value problem of general unsteady PDEs with higher-order spatial derivatives. The high resolution schemes are given by combining the reconstruction technique with a second order TVD Runge-Kutta scheme or a Lax-Wendroff type method, respectively.
The schemes are used to solve a linear convection-diffusion equation, the nonlinear inviscid Burgers' equation, the one- and two-dimensional compressible Euler equations, and the two-dimensional incompressible Navier-Stokes equations. The numerical results show that the schemes are of higher-order accuracy, and efficient in saving computational cost, especially, for the case of combining the present schemes with the adaptive mesh method [15]. The correct locations of the slow moving or stronger discontinuities are also obtained, although the schemes are slightly nonconservative. 相似文献
10.
In this article, we propose a new discontinuous finite volume element (DFVE) method for the second‐order elliptic problems. We treat the DFVE method as a perturbation of the interior penalty method and get a superapproximation estimate in a mesh dependent norm between the solution of the DFVE method and that of the interior penalty method. This reveals that the DFVE method is much closer to the interior penalty method than we have known. By using this superapproximation estimate, we can easily get the optimal order error estimates in the L2 ‐norm and in the maximum norms of the DFVE method.© 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 425–440, 2012 相似文献
11.
Numerical schemes are presented for a class of fourth order diffusion problems. These problems arise in lubrication theory for thin films of viscous fluids on surfaces. The equations being in general fourth order degenerate parabolic, additional singular terms of second order may occur to model effects of gravity, molecular interactions or thermocapillarity. Furthermore, we incorporate nonlinear surface tension terms. Finally, in the case of a thin film flow driven by a surface active agent (surfactant), the coupling of the thin film equation with an evolution equation for the surfactant density has to be considered. Discretizing the arising nonlinearities in a subtle way enables us to establish discrete counterparts of the essential integral estimates found in the continuous setting. As a consequence, the resulting algorithms are efficient, and results on convergence and nonnegativity or even strict positivity of discrete solutions follow in a natural way. The paper presents a finite element and a finite volume scheme and compares both approaches. Furthermore, an overview over qualitative properties of solutions is given, and various applications show the potential of the proposed approach. 相似文献
12.
A high order finite volume method for one dimensional nonlocal reactive flows of parabolic type
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In this paper, we consider a high order finite volume approximation of one‐dimensional nonlocal reactive flows of parabolic type. The method is obtained by discretizing in space by arbitrary order vertex‐centered finite volumes, followed by a modified Simpson quadrature scheme for the time stepping. Compared to the existed finite volume methods, this new finite volume scheme could achieve the desired accuracy with less data storage by employing higher‐order trial spaces. The finite volume approximations are proved to possess optimal order convergence rates in the H1‐norm and L2‐norm, which are also confirmed by numerical tests. Copyright © 2015 John Wiley & Sons, Ltd. 相似文献
13.
Xiu Ye 《Numerical Methods for Partial Differential Equations》2011,27(5):1165-1178
We establish a posteriori error analysis for finite volume methods of a second‐order elliptic problem based on the framework developed by Chou and Ye [SIAM Numer Anal, 45 (2007), 1639–1653]. This residual type estimators can be applied to different finite volume methods associated with different trial functions including conforming, nonconforming and totally discontinuous trial functions. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1165–1178, 2011 相似文献
14.
New conservative finite volume element schemes for the modified Korteweg–de Vries equation
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Jinliang Yan Qian Zhang Zhiyue Zhang 《Mathematical Methods in the Applied Sciences》2016,39(18):5149-5161
In this paper, three conservative finite volume element schemes are proposed and compared for the modif ied Korteweg–de Vries equation, especially with regard to their accuracy and conservative properties. The schemes are constructed basing on the discrete variational derivative method and the finite volume element method to inherit the properties of the original equation. The theoretical analysis show that three schemes are conservative under suitable boundary conditions as well as unconditionally linear stability. Numerical experiments are given to confirm the theoretical results and the capacity of the proposed methods for capturing the solitary wave phenomena. Copyright © 2016 John Wiley & Sons, Ltd. 相似文献
15.
This paper details our note [6] and it is an extension of our previous works and which dealt with first order (both in time and space) and second order time accurate (second order in time and first order in space) implicit finite volume schemes for second order hyperbolic equations with Dirichlet boundary conditions on general nonconforming multidimensional spatial meshes introduced recently in [14]. We aim in this work (and some forthcoming studies) to get higher order (both in time and space) finite volume approximations for the exact solution of hyperbolic equations using the class of spatial generic meshes introduced recently in [14] on low order schemes from which the matrices used to compute the discrete solutions are sparse. We focus in the present contribution on the one dimensional wave equation and on one of its implicit finite volume schemes described in [4]. The implicit finite volume scheme approximating the one dimensional wave equation we consider (hereafter referred to as the basic finite volume scheme) yields linear systems to be solved successively. The matrices involved in these linear systems are tridiagonal, symmetric and definite positive. The finite volume approximate solution of the basic finite volume scheme is of order h+k, where h (resp. k ) is the mesh size of the spatial (resp. time) discretization. We construct a new finite volume approximation of order (h+k)2 in several discrete norms which allow us to get approximations of order two for the exact solution and its first derivatives. This new high-order approximation can be computed using linear systems whose matrices are the same ones used to compute the discrete solution of the basic finite volume scheme while the right hand sides are corrected. The construction of these right hand sides includes the approximation of some high order spatial derivatives of the exact solution. The computation of the approximation of these high order spatial derivatives can be performed using the same matrices stated above with another two tridiagonal matrices. The manner by which this new high-order approximation is constructed can be repeated to compute successively finite volume approximations of arbitrary order using the same matrices stated above. These high-order approximations can be obtained on any one dimensional admissible finite volume mesh in the sense of [13] without any condition. To reach the above results, a theoretical framework is developed and some numerical examples supporting the theory are presented. Some of the tools of this framework are new and interesting and they are stated in the one space dimension but they can be extended to several space dimensions. In particular a new and useful a prior estimate for a suitable discrete problem is developed and proved. The proof of this a prior estimate result is based essentially on the decomposition of the solution of the discrete problem into the solutions of two suitable discrete problems. A new technique is used in order to get a convenient finite volume approximation whose discrete time derivatives of order up to order two are also converging towards the solution of the wave equation and their corresponding time derivatives. 相似文献
16.
Michael Breuß Dominik Dietrich 《Numerical Methods for Partial Differential Equations》2013,29(3):884-896
A classic strategy to obtain high‐quality discretizations of hyperbolic partial differential equations is to use flux limiter (FL) functions for blending two types of approximations: a monotone first‐order scheme that deals with discontinuous solution features and a higher order method for approximating smooth solution parts. In this article, we study a new approach to FL methods. Relying on a classification of input data with respect to smoothness, we associate specific basis functions with the individual smoothness notions. Then, we construct a limiter as a linear combination of the members of parameter‐dependent families of basis functions, and we explore the possibility to optimize the parameters in interesting model situations to find a corresponding optimal limiter. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013 相似文献
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A stabilized Hermite spectral method, which uses the Hermite polynomials as trial functions, is presented for the heat equation and the generalized Burgers equation in unbounded domains. In order to overcome instability that may occur in direct Hermite spectral methods, a time‐dependent scaling factor is employed in the Hermite expansions. The stability of the scheme is examined and optimal error estimates are derived. Numerical experiments are given to confirm the theoretical results.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007 相似文献
19.
Philippe Blanc Robert Eymard Raphale Herbin 《Numerical Methods for Partial Differential Equations》2004,20(6):907-918
We give here an error estimate for a finite volume discretization of the Stokes equations in two space dimensions on equilateral triangular meshes. This work was initiated by an analogous result presented by Alami‐Idrissi and Atounti for general triangular meshes. However, in this latter article, the result is not actually proven. We state here the restricting assumptions (namely equilateral triangles) under which the error estimate holds, using the tools which were introduced by Eymard, Gallouet and Herbin and used by Alami‐Idrissi and Atounti. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004 相似文献
20.
Reiner Vanselow HansPeter Scheffler 《Numerical Methods for Partial Differential Equations》1998,14(2):213-231
The article is devoted to the study of convergence properties of a Finite Volume Method (FVM) using Voronoi boxes for discretization. The approach is based on the construction of a new nonconforming Finite Element Method (FEM), such that the system of linear equations coincides completely with that for the FVM. Thus, by proving convergence properties of the FEM, we obtain similar ones of the FVM. In this article, the investigations are restricted to the Poisson equation. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14:213–231, 1998 相似文献