Stability of an implicit difference scheme for a linear differential-algebraic system of partial differential equations

An implicit difference scheme is considered for approximating the initial-boundary value problem for a linear differential-algebraic system of partial differential equations with variable matrix coefficients of special structure. The conditional and asymptotic stability of the difference scheme with respect to the initial and boundary conditions and the right-hand side is proved.     

一阶双曲型方程组初边值问题的差分格式及其稳定性

本文比较系统地讨论了有关数值求解两个自变量的一阶双曲型方程组初边值问题的某些问题,给出了几种能用于任何类型的初边值问题的差分格式,并在很宽的条件下证明了其中的某些变系数的初边值问题的差分格式对初值和边值是稳定的、差分格式所立出的方程组是良态的.其中的某些格式已用于解决某些复杂的实际问题(应用部分见[16]).     

A Nonstandard Finite Difference Scheme for Nonlinear Heat Transfer in a Thin Finite Rod

A nonstandard finite difference scheme is constructed to solve an initial-boundary value problem involving a quartic nonlinearity that arises in heat transfer involving conduction with thermal radiation. It is noted that the positivity condition is equivalent to the usual linear stability criteria and it is shown that the representation of the nonlinear term in the finite difference scheme, in addition to the magnitudes of the equation parameters, has a direct bearing on the scheme's stability. Finally, solution profiles are plotted and avenues of further inquiry are discussed.     

Reliable Solution of Parabolic Obstacle Problems with Respect to Uncertain Data

A class of parabolic initial-boundary value problems is considered, where admissible coefficients are given in certain intervals. We are looking for maximal values of the solution with respect to the set of admissible coefficients. We give the abstract general scheme, proposing how to solve such problems with uncertain data. We formulate a general maximization problem and prove its solvability, provided all fundamental assumptions are fulfilled. We apply the theory to certain Fourier obstacle type maximization problem.     

Stability and monotonicity of a conservative difference scheme for a multidimensional nonlinear scalar conservation law

We prove the existence, uniqueness, and monotonicity of the solution of an upwind conservative explicit difference scheme approximating an initial-boundary value problem for a many-dimensional nonlinear scalar conservation law with a quadratic nonlinearity under some specific conditions imposed only on the input data of the problem. We show that the resulting solution is not necessarily stable. Under some additional conditions on the input data, which provide the absence of shock waves, we prove the stability of the unique solution of the difference scheme for any finite time.     

On an upwind difference scheme for strongly degenerate parabolic equations modelling the settling of suspensions in centrifuges and non-cylindrical vessels

We prove the convergence of an explicit monotone finite difference scheme approximating an initial-boundary value problem for a spatially one-dimensional quasilinear strongly degenerate parabolic equation, which is supplied with two zero-flux boundary conditions. This problem arises in a model of sedimentation–consolidation processes in centrifuges and vessels with varying cross-sectional area. We formulate the definition of entropy solution of the model in the sense of Kružkov and prove the convergence of the scheme to the unique BV entropy solution of the problem. The scheme and the model are illustrated by numerical examples.     

Average implicit linear difference scheme for generalized Rosenau-Burgers equation

In this paper, a linear three-level average implicit finite difference scheme for the numerical solution of the initial-boundary value problem of Generalized Rosenau-Burgers equation is presented. Existence and uniqueness of numerical solutions are discussed. It is proved that the finite difference scheme is convergent in the order of O(τ² + h²) and stable. Numerical simulations show that the method is efficient.     

Application of wavelet transforms to the solution of boundary value problems for linear parabolic equations

A method based on wavelet transforms is proposed for finding weak solutions to initial-boundary value problems for linear parabolic equations with discontinuous coefficients and inexact data. In the framework of multiresolution analysis, the general scheme for finite-dimensional approximation in the regularization method is combined with the discrepancy principle. An error estimate is obtained for the stable approximate solution obtained by solving a set of linear algebraic equations for the wavelet coefficients of the desired solution.     

双曲-双曲奇异摄动问题的指数型拟合差分格式

本文讨论带有关于x的一阶导数项的双曲奇异摄动初边值问题,在较弱的相容性条件下构造了问题的渐近解并证明了解的一致有效性.然后我们对原问题构造一个指数型拟合差分格式并建立了离散能量不等式.最后我们证明差分问题的解一致收敛于原问题的精确解.     

An efficient method for solving nonlocal initial-boundary value problems for linear and nonlinear first-order hyperbolic partial differential equations

In this paper, we present a new approach to solve nonlocal initial-boundary value problems of linear and nonlinear hyperbolic partial differential equations of first-order subject to initial and nonlocal boundary conditions of integral type. We first transform the given nonlocal initial-boundary value problems into local initial-boundary value problems. Then we apply a modified Adomian decomposition method, which permits convenient resolution of these problems. Moreover, we prove this decomposition scheme applied to such nonlocal problems is convergent in a suitable Hilbert space, and then extend our discussion to include systems of first-order linear equations and other related nonlocal initial-boundary value problems.     

On the choice of initial conditions of difference schemes for parabolic equations

We study finite difference schemes to approximate the first initial-boundary value problem for linear second order parabolic equations and obtain some convergence rate estimates. When difference schemes are constructed for such problems, in the process of obtaining convergence rate estimates compatible with smoothness of the solution, various authors assume that the solution of the problem can be extended to the exterior of the domain of integration, preserving the Sobolev class. Our investigations show that this restriction can be removed if, instead of using the exact initial condition, we use certain approximations of the initial conditions. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical algorithm for solving diffusion-type equations on the basis of multigrid methods

A new numerical algorithm based on multigrid methods is proposed for solving equations of the parabolic type. Theoretical error estimates are obtained for the algorithm as applied to a two-dimensional initial-boundary value model problem for the heat equation. The good accuracy of the algorithm is demonstrated using model problems including ones with discontinuous coefficients. As applied to initial-boundary value problems for diffusion equations, the algorithm yields considerable savings in computational work compared to implicit schemes on fine grids or explicit schemes with a small time step on fine grids. A parallelization scheme is given for the algorithm.     

Stability with respect to the initial data and monotonicity of an implicit difference scheme for a homogeneous porous medium equation with a quadratic nonlinearity

We study the stability and monotonicity of a conservative difference scheme approximating an initial-boundary value problem for a porous medium equation with a quadratic nonlinearity under certain conditions imposed only on the input data of the problem. We prove a grid analog of the Bihari lemma, which is used to obtain a priori estimates for higher derivatives; these estimates are needed both in the proof of the continuous dependence of the solution on small perturbations in the input data and for the analysis of monotonicity in the nonlinear case. We show that, regardless of the smoothness of the initial condition, the higher derivatives can become infinite in finite critical time. We give an example in which there arises a runningwave solution, which justifies the theoretical conclusions.     

A Volterra integral type method for solving a class of nonlinear initial-boundary value problems

It is observed that the one-dimensional heat equation with certain nonlinear boundary conditions can be reformulated as a system of coupled Volterra integral equations. A product trapezoidal scheme is proposed for the numerical solution of this integral equation system, and some numerical experiments are given to compare the performances of this integral equation approach and the Crank-Nicholson method applied to the original initial-boundary value problem. © 1996 John Wiley & Sons, Inc.     

具有非光滑边界层函数的线性变系数抛物型方程初边值问题的差分格式

本文利用非均匀网格和指数型拟合差分方法给出了具有非光滑边界层函数的线性抛物型方程关于小参数ε一致收敛的差分格式.文章还给出了误差估计和数值结果.     

Stability with respect to the input data and monotonicity of an implicit finite-difference scheme for a quasilinear parabolic equation

We analyze the stability and monotonicity of a conservative difference scheme approximating an initial-boundary value problem for a quasilinear parabolic equation under specific conditions imposed solely on the problem input data. We prove some kinds of the maximum principle for the nonlinear equations that are used in the derivation of a priori estimates for the solution; we also prove estimates for some kinds of recursion inequalities that are used in the derivation of a priori estimates for higher-order derivatives, these estimates being necessary for proving the continuous dependence of the solution on small perturbations of the input data and for analyzing monotonicity in the nonlinear case. We show that, depending on the properties of the input data, higher derivatives can become infinite in finite critical time. We obtain conditions on the input data guaranteeing the stability of the difference scheme on the entire time interval.     

Coefficient Stability of the Solution of a Difference Scheme Approximating a Mixed Problem for a Semilinear Parabolic Equation

We study the coefficient stability of a difference scheme approximating a mixed problem for a one-dimensional semilinear parabolic equation. We obtain sufficient conditions on the input data under which the solutions of the differential and difference problems are bounded. We also obtain estimates of perturbations of the solution of a linearized difference scheme with respect to perturbations of the coefficients; these estimates agree with the estimates for the differential problem.     

具有初始跳跃的双曲型方程奇异摄动问题的数值解法

本文考虑具有初始跳跃的二阶双曲型方程初边值问题.首先给出解的导数估计.然后在一非均匀网格上构造了一个差分格式,最后在能量范数意义下证明了差分格式解的一致收敛性.     

Initial-boundary value problems for incomplete singular perturbations of hyperbolic systems

The general problem studied has as a prototype the full non-linear Navier-Stokes equations for a slightly viscous compressible fluid including the heat transfer. The boundaries are of inflow-outflow type, i.e. non-characteristic, and the boundary conditions are the most general ones with any order of derivatives. It is assumed that the uniform Lopatinsky condition is satisfied. The goal is to prove uniform existence and boundedness of solution as the viscosity tends to zero and to justify the boundary layer asymptotics. The paper consists of two parts. In Part I the linear problem is studied. Here, uniform lower and higher order tangential estimates are derived and the existence of a solution is proved. The higher order estimates depend on the smoothness of coefficients; however this smoothness does not exceed the smoothness of the solution. In Part II the quasilinear problem is studied. It is assumed that for zero viscosity the overall initial-boundary value problem has a smooth solutionu ₀ in a time interval 0≦t≦T ₀. As a result the boundary laye, is weak and is uniformlyC ¹ bounded. This makes the linear theory applicable. an iteration scheme is set and proved to converge to the viscous solution. The convergence takes place for small viscosity and over the original time interval 0≦t≦T ₀.     

Well-posedness of difference schemes for semilinear parabolic equations with weak solutions

The well-posedness of difference schemes approximating initial-boundary value problem for parabolic equations with a nonlinear power-type source is studied. Simple sufficient conditions on the input data are obtained under which the weak solutions of the differential and difference problems are globally stable for all 0 ⩽ t ⩽ +∞. It is shown that, if the condition fails, the solution can blow up (become infinite) in a finite time. A lower bound for the blow-up time is established. In all the cases, the method of energy inequalities is used as based on the application of the Chaplygin comparison theorem, Bihari-type inequalities, and their difference analogues. A numerical experiment is used to illustrate the theoretical results and verify two-sided blow-up time estimates.