共查询到20条相似文献,搜索用时 15 毫秒
1.
Christian Rohde 《Numerische Mathematik》1998,81(1):85-123
Summary. Systems of nonlinear hyperbolic conservation laws in two space dimensions are considered which are characterized by the fact
that the coupling of the equations is only due to source terms. To solve these weakly coupled systems numerically a class
of explicit and implicit upwind finite volume methods on unstructured grids is presented. Provided an unique entropy solution
of the system of conservation laws exists we prove that the approximations obtained by these schemes converge for vanishing
discretization parameter to this entropy solution. These results are applied to examples from combustion theory and hydrology
where the existence of entropy solutions can be shown. The proofs rely on an extension of a result due to DiPerna concerning
measure valued solutions to the case of weakly coupled hyperbolic systems.
Received April 29, 1997 相似文献
2.
Yves Renard 《Journal of Computational and Applied Mathematics》2010,234(3):906-923
The purpose of this paper is to present a new family of numerical methods for the approximation of second order hyperbolic partial differential equations submitted to a convex constraint on the solution. The main application is dynamic contact problems. The principle consists in the use of a singular mass matrix obtained by the mean of different discretizations of the solution and of its time derivative. We prove that the semi-discretized problem is well-posed and energy conserving. Numerical experiments show that this is a crucial property to build stable numerical schemes. 相似文献
3.
A model is derived for the coupling of transient free surface and pressurized flows. The resulting system of equations is written under a conservative form with discontinuous gradient of pressure. We treat the transition point between the two types of flows as a free boundary associated to a discontinuity of the gradient of pressure. The numerical simulation is performed by making use of a Roe-like finite volume scheme that we adapted to such discontinuities in the flux. The validation is performed by comparison with experimental results. 相似文献
4.
We study the Cauchy problems for the isentropic 2-d Euler system with discontinuous initial data along a smooth curve. All three singularities are present in the solution: shock wave, rarefaction wave and contact discontinuity. We show that the usual restrictive high order compatibility conditions for the initial data are automatically satisfied. The local existence of piecewise smooth solution containing all three waves is established. 相似文献
5.
In this article one discusses the controllability of a semi-discrete system obtained by discretizing in space the linear 1-D
wave equation with a boundary control at one extremity. It is known that the semi-discrete models obtained with finite difference
or the classical finite element method are not uniformly controllable as the discretization parameter h goes to zero (see [8]).
Here we introduce a new semi-discrete model based on a mixed finite element method with two different basis functions for
the position and velocity. We show that the controls obtained with these semi-discrete systems can be chosen uniformly bounded
in L2(0,T) and in such a way that they converge to the HUM control of the continuous wave equation, i.e. the minimal L2-norm control. We illustrate the mathematical results with several numerical experiments.
Supported by Grant BFM 2002-03345 of MCYT (Spain) and the TMR projects of the EU ``Homogenization and Multiple Scales" and
``New materials, adaptive systems and their nonlinearities: modelling, control and numerical simulations".
Partially Supported by Grant BFM 2002-03345 of MCYT (Spain), Grant 17 of Egide-Brancusi Program and Grant 80/2005 of CNCSIS
(Romania). 相似文献
6.
We introduce the RKGL method for the numerical solution of initial-value problems of the form y′=f(x,y), y(a)=α. The method is a straightforward modification of a classical explicit Runge–Kutta (RK) method, into which Gauss–Legendre (GL) quadrature has been incorporated. The idea is to enhance the efficiency of the method by reducing the number of times the derivative f(x,y) needs to be computed. The incorporation of GL quadrature serves to enhance the global order of the method by, relative to the underlying RK method. Indeed, the RKGL method has a global error of the form Ahr+1+Bh2m, where r is the order of the RK method and m is the number of nodes used in the GL component. In this paper we derive this error expression and show that RKGL is consistent, convergent and strongly stable. 相似文献
7.
Harald Berger 《Numerische Mathematik》1989,56(5):425-447
Summary A finite element formulation for the full potential equation in the case of two-dimensional transonic flow is presented. The formulation is based on an optimal control approach developed by Glowinski and Pironneau. The solution of the full potential equation is obtained by a minimization problem. Using a new compactness result it is possible to prove convergence for the solutions of the minimization problem. The a priori assumption of existence and uniqueness of a weak solution of the full potential equation satisfying an entropy condition implies that the limit function must be the solution. It is possible to extend the convergence result to the case of three-dimensional transonic potential flow.The research reported here was supported by a grant from the Stiftung Volkswagenwerk, Federal Republic of Germany. It is a part of the doctoral thesis of the above author, Universität Stuttgart 1989 相似文献
8.
It is proven that the generalized Riemann problem for a class of quasilinear hyperbolic systems of balance laws admits a unique global piecewise C1 solution u=u(t,x) containing only n shock waves with small amplitude on t?0 and this solution possesses a global structure similar to that of the similarity solution u=U(x/t) of the corresponding homogeneous Riemann problem. As an application of our result, we prove the existence of global shock solutions, piecewise continuous and piecewise smooth solution with shock discontinuities, of the flow equations of a model class of fluids with viscosity induced by fading memory with a single jump initial data. 相似文献
9.
We study the large time behavior of solutions of a one-dimensional hyperbolic relaxation system that may be written as a nonlinear damped wave equation. First, we prove the global existence of a unique solution and their decay properties for sufficiently small initial data. We also show that for some large initial data, solutions blow-up in finite time. For quadratic nonlinearities, we prove that the large time behavior of solutions is given by the fundamental solution of the viscous Burgers equation. In some other cases, the convection term is too weak and the large time behavior is given by the linear heat kernel. 相似文献
10.
Summary.
An explicit finite element method for numerically solving
the drift-diffusion semiconductor device equations in two space dimensions
is analyzed.
The method is based on the use of a mixed finite element method for the approximation
of the electric field and a discontinuous
upwinding finite element method for the approximation
of the electron and hole concentrations. The mixed method gives an approximate electric
field in the precise form needed by the discontinuous method, which is trivially
conservative and fully parallelizable. It is proven that the method produces
uniformly bounded concentrations and electric fields and that it converges
to the exact solution provided there is a convergent subsequence of the electron
concentrations. Numerical simulations are presented that display the
performance of the method and indicate the behavior of the solution.
Received
September 9, 1993 / Revised version received May 25,
1994 相似文献
11.
Chun Shen 《Nonlinear Analysis: Theory, Methods & Applications》2010,73(10):3284-3294
We prove that the Riemann solutions are stable for a nonstrictly hyperbolic system of conservation laws under local small perturbations of the Riemann initial data. The proof is based on the detailed analysis of the interactions of delta shock waves with shock waves and rarefaction waves. During the interaction process of the delta shock wave with the rarefaction wave, a new kind of nonclassical wave, namely a delta contact discontinuity, is discovered here, which is a Dirac delta function supported on a contact discontinuity and has already appeared in the interaction process for the magnetohydrodynamics equations [M. Nedeljkov and M. Oberguggenberger, Interactions of delta shock waves in a strictly hyperbolic system of conservation laws, J. Math. Anal. Appl. 344 (2008) 1143-1157]. Moreover, the global structures and large time asymptotic behaviors of the solutions are constructed and analyzed case by case. 相似文献
12.
Ying-Chin Su 《Journal of Differential Equations》2011,250(9):3668-3700
This research explores the Cauchy problem for a class of quasi-linear wave equations with time dependent sources. It can be transformed into the Cauchy problem of hyperbolic integro-differential systems of nonlinear balance laws. We introduce the generalized Glimm scheme in new version and study its stability which is proved by Glimm-type interaction estimates in a dissipativity assumption. The generalized solutions to the perturbed Riemann problems, the building blocks of generalized Glimm scheme, are constructed by Riemann problem method modeled on the source free equations. The global existence for the Lipschitz continuous solutions and weak solutions to the systems is established by the consistency of scheme and the weak convergence of source. Finally, the weak solutions are also the entropy solutions which satisfy the entropy inequality. 相似文献
13.
The two-grid method is studied for solving a two-dimensional second-order nonlinear hyperbolic equation using finite volume element method. The method is based on two different finite element spaces defined on one coarse grid with grid size H and one fine grid with grid size h, respectively. The nonsymmetric and nonlinear iterations are only executed on the coarse grid and the fine grid solution can be obtained in a single symmetric and linear step. It is proved that the coarse grid can be much coarser than the fine grid. A prior error estimate in the H1-norm is proved to be O(h+H3|lnH|) for the two-grid semidiscrete finite volume element method. With these proposed techniques, solving such a large class of second-order nonlinear hyperbolic equations will not be much more difficult than solving one single linearized equation. Finally, a numerical example is presented to validate the usefulness and efficiency of the method. 相似文献
14.
In this paper we study the existence of global solutions to the Euler equations of compressible isothermal gas dynamics with
semiconductor devices. We construct the approximate solutions by Lax–Friedrichs scheme. The convergence and consistency are
obtained by using the compensated compactness framework for γ = 1. The global entropy solutions in L∞ are obtained. We deal with the initial data containing unbounded velocity which is different from the isentropic case.
Received: November 18, 2003 相似文献
15.
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
16.
17.
18.
Summary.
We prove convergence of a class of higher order upwind
finite
volume schemes on unstructured grids for scalar conservation laws in
several space dimensions. The result is applied to the discontinuous
Galerkin method due to Cockburn, Hou and Shu.
Received
April 15, 1993 / Revised version received March 13, 1995 相似文献
19.
This paper considers a stabilized method based on the difference between a consistent and an under-integrated mass matrix of the pressure for the Stokes equations approximated by the lowest equal-order finite element pairs (i.e., the P1–P1 and Q1–Q1 pairs). This method only offsets the discrete pressure space by the residual of the simple and symmetry term at element level in order to circumvent the inf–sup condition. Optimal error estimates are obtained by applying the standard Galerkin technique. Finally, the numerical illustrations agree completely with the theoretical expectations. 相似文献
20.
Global solution for a one-dimensional model problem in thermally radiative magnetohydrodynamics 总被引:1,自引:0,他引:1
The dynamics of gaseous stars is often described by magnetic fields coupled to self-gravitation and radiation effects. In this paper we consider an initial-boundary value problem for nonlinear planar magnetohydrodynamics (MHD) in the case that the effect of self-gravitation as well as the influence of radiation on the dynamics at high temperature regimes are taken into account. Based on the fundamental local existence results and global-in-time a priori estimates, we establish the global existence of a unique classical solution with large initial data to the initial-boundary value problem under quite general assumptions on the heat conductivity. 相似文献