共查询到20条相似文献,搜索用时 703 毫秒
1.
A. Sh. Akysh 《Numerical Analysis and Applications》2013,6(2):111-118
Convergence of a splitting method scheme for the nonlinear Boltzmann equation is considered. Using the splitting method scheme, boundedness of the positive solutions in a space of continuous functions is obtained. By means of the solution boundedness and some a priori estimates, convergence of the splitting method scheme and uniqueness of the limiting element are proved. The limiting element satisfies an equivalent integral Boltzmann equation. Thereby global in time solvability of the nonlinear Boltzmann equation is shown. 相似文献
2.
Daisuke Furihata 《Numerische Mathematik》2001,87(4):675-699
Summary. We propose a stable and conservative finite difference scheme to solve numerically the Cahn-Hilliard equation which describes
a phase separation phenomenon. Numerical solutions to the equation is hard to obtain because it is a nonlinear and nearly
ill-posed problem. We design a new difference scheme based on a general strategy proposed recently by Furihata and Mori. The
new scheme inherits characteristic properties, the conservation of mass and the decrease of the total energy, from the equation.
The decrease of the total energy implies boundedness of discretized Sobolev norm of the solution. This in turn implies, by
discretized Sobolev's lemma, boundedness of max norm of the solution, and hence the stability of the solution. An error estimate
for the solution is obtained and the order is . Numerical examples demonstrate the effectiveness of the proposed scheme.
Received July 22, 1997 / Revised version received October 19, 1999 / Published online August 2, 2000 相似文献
3.
In this paper, numerical solution of the Burgers–Huxley (BH) equation is presented based on the nonstandard finite difference (NSFD) scheme. At first, two exact finite difference schemes for BH equation obtained. Moreover an NSFD scheme is presented for this equation. The positivity, boundedness and local truncation error of the scheme are discussed. Finally, the numerical results of the proposed method with those of some available methods compared. 相似文献
4.
In this paper, we design a semi-implicit scheme for the scalar time fractional reaction-diffusion equation. We theoretically prove that the numerical scheme is stable without the restriction on the ratio of the time and space stepsizes, and numerically show that the convergence orders are 1 in time and 2 in space. As a concrete model, the subdiffusive predator-prey system is discussed in detail. First, we prove that the analytical solution to the system is positive and bounded. Then, we use the provided numerical scheme to solve the subdiffusive predator-prey system, and theoretically prove and numerically verify that the numerical scheme preserves the positivity and boundedness. 相似文献
5.
We develop a finite-difference scheme for approximation of a system of nonlinear PDEs describing the Q-switching process.
We construct it by using staggered grids. The transport equations are approximated along characteristics, and quadratic nonlinear
functions are linearized using a special selection of staggered grids. The stability analysis proves that a connection between
time and space steps arises only due to approximation requirements in order to follow exactly the directions of characteristics.
The convergence analysis of this scheme is done in two steps. First, some estimates of the uniform boundedness of the discrete
solution are proved. This part of the analysis is done locally, in some neighborhood of the exact solution. Second, on the
basis of the obtained estimates, the main stability inequality is proved. The second-order convergence rate with respect to
the space and time coordinates follows from this stability estimate. Using the obtained convergence result, we prove that
the local stability analysis in the selected neighborhood of the exact solution is sufficient. 相似文献
6.
《Journal of Computational and Applied Mathematics》2002,146(2):213-230
This paper provides Galerkin and Inertial Algorithms for solving a class of nonlinear evolution equations. Spatial discretization can be performed by either spectral or finite element methods; time discretization is done by Euler explicit or Euler semi-implicit difference schemes with variable time step size. Moreover, the boundedness and stability of these algorithms are studied. By comparison, we find that the boundedness and stability of Inertial Algorithm are superior to the ones of Galerkin Algorithm in the case of explicit scheme and the boundedness and stability of two algorithms are same in the case of semi-implicit scheme. 相似文献
7.
Eric R. Kaufmann 《Journal of Difference Equations and Applications》2013,19(7):731-740
A nonstandard discretization scheme is applied to continuous Volterra integro-differential equations. We will show that under our discretization scheme the stability of the zero solution of the continuous dynamical system is preserved. Also, under the same discretization, using a combination of Lyapunov functionals, Laplace transforms and z-transforms, we show that the boundedness of solutions of the continuous dynamical system is preserved. 相似文献
8.
构造了非正交网格上扩散方程新的非线性单元中心型有限体积格式,证明了该格式满足离散极值原理,且在适当条件下具有强制性、以及在离散H1范数下解的有界性和一阶收敛性. 相似文献
9.
Qifeng
Zhang Xuping Wang Zhi-zhong Sun 《Numerical Methods for Partial Differential Equations》2020,36(6):1611-1628
In this article, we are concerned with the numerical analysis of a nonlinear implicit difference scheme for Burgers' equation. A priori estimation of the analytical solution is provided in the sense of L∞ -norm when the initial value is bounded in H1-norm. Conservation, boundedness, and unique solvability are proved at length. Inspired by the method of the priori estimation for the analytical solution, we prove the convergence and stability of the difference scheme in L∞ -norm. Finally, numerical examples are carried out to verify our theoretical results. 相似文献
10.
An oscillation-free high order scheme is presented for convection discretization by using the normalized-variable formulation
in the finite volume framework. It adopts the cubic upwind interpolation scheme as the basic scheme so as to obtain high order
accuracy in smooth solution domain. In order to avoid unphysical oscillations of numerical solutions, the present scheme is
designed on the TVD (total variational diminishing) constraint and CBC (convection boundedness criterion) condition. Numerical
results of several linear and nonlinear convection equations with smooth or discontinuous initial distributions demonstrate
the present scheme possesses second-order accuracy, good robustness and high resolution. 相似文献
11.
Wendi Qin Deqiong Ding Xiaohua Ding 《Mathematical Methods in the Applied Sciences》2015,38(15):3308-3321
In this note, a non‐standard finite difference (NSFD) scheme is proposed for an advection‐diffusion‐reaction equation with nonlinear reaction term. We first study the diffusion‐free case of this equation, that is, an advection‐reaction equation. Two exact finite difference schemes are constructed for the advection‐reaction equation by the method of characteristics. As these exact schemes are complicated and are not convenient to use, an NSFD scheme is derived from the exact scheme. Then, the NSFD scheme for the advection‐reaction equation is combined with a finite difference space‐approximation of the diffusion term to provide a NSFD scheme for the advection‐diffusion‐reaction equation. This new scheme could preserve the fixed points, the positivity, and the boundedness of the solution of the original equation. Numerical experiments verify the validity of our analytical results. Copyright © 2014 JohnWiley & Sons, Ltd. 相似文献
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13.
Huimin Yu 《Nonlinear Analysis: Theory, Methods & Applications》2010,73(11):3586-3597
This paper deals with the uniform boundedness (as well as the existence) and large time behavior of the weak entropy solutions to a kind of compressible Euler equation with dissipation effect. The existence and uniform boundedness in time of weak solutions are proved by using the Lax-Friedrichs scheme and compensate compactness. Time asymptotically, the density is showed to satisfy a kind of nonlinear Fokker-Planck equation and the momentum obeys to the Darcy’s law. As a by product, the exponentially decay rate is obtained. 相似文献
14.
J.K. Djoko 《Numerical Methods for Partial Differential Equations》2008,24(6):1371-1387
In this work we examine the stability of a finite difference approximation for Burgers' equation. More precisely, we consider a Backward Euler discretization scheme in time and approximated the nonlinear term by a linear expression using techniques from 15 . The boundedness of the solution sequence with respect to Δt for t ε [0,∞) is proved with the help of the discrete Gronwall lemmas. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008 相似文献
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16.
Thomas G. Hallam 《Annali di Matematica Pura ed Applicata》1970,85(1):307-325
Summary The usual definition of the stability of a solution of a system of ordinary differential equations is extended by introducing
two positive control functions. These functions are used to control the rate of growth of the in?tial position of the solution
and the rate of growth of the solution. Definitions and results are also given for the corresponding analogues of boundedness,
weak boundedness, and uniform properties of the sotions of differential equations. The problem of determining when solutions
of certain linear and weakly nonlinear differential equations lie in a modified Lp-space is also considered.
This research was supported by the National Science Foundation under grant GP-8921.
Entrata in Redazione il 13 maggio 1969. 相似文献
17.
In this paper, the authors use Glimm scheme to study the global existence of BV solutions to Cauchy problem of the pressure-gradient system with large initial data. To this end, some important properties of the shock curves of the pressure-gradient system in the Riemann invariant coordinate system and verify that the shock curves satisfy Diperna’s conditions (see [Diperna, R. J., Existence in the large for quasilinear hyperbolic conservation laws, Arch. Ration. Mech. Anal., 52(3), 1973, 244–257]) are studied. Then they construct the approximate solution sequence through Glimm scheme. By establishing accurate local interaction estimates, they prove the boundedness of the approximate solution sequence and its total variation. 相似文献
18.
Ratio-dependent predator-prey models have been increasingly favored by field ecologists where predator-prey interactions have to be taken into account the process of predation search. In this paper we study the conditions of the existence and stability properties of the equilibrium solutions in a reaction-diffusion model in which predator mortality is neither a constant nor an unbounded function, but it is increasing with the predator abundance. We show that analytically at a certain critical value a diffusion driven (Turing type) instability occurs, i.e. the stationary solution stays stable with respect to the kinetic system (the system without diffusion). We also show that the stationary solution becomes unstable with respect to the system with diffusion and that Turing bifurcation takes place: a spatially non-homogenous (non-constant) solution (structure or pattern) arises. A numerical scheme that preserve the positivity of the numerical solutions and the boundedness of prey solution will be presented. Numerical examples are also included. 相似文献
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20.
Zvi Drezner 《Mathematical Methods of Operations Research》2010,72(2):205-216
Facility layout problems involve the location of facilities in a planar arrangement such that facilities that are strongly
connected to one another are close to each other and facilities that are not connected may be far from one another. Pairs
of facilities that have a negative connection should be far from one another. Most solution procedures assume that the optimal
arrangement is bounded and thus do not incorporate constraints on the location of facilities. However, especially when some
of the coefficients are negative, it is possible that the optimal configuration is unbounded. In this paper we investigate
whether the solution to the facility layout problem is bounded or not. The main Theorem is a necessary and sufficient condition
for boundedness. Sufficient conditions that prove boundedness or unboundedness are also given. 相似文献