Difference Schemes For Approximate Solution Research Articles

The second-order accuracy difference schemes for integral-type time-nonlocal parabolic problems

This is a discussion on the second-order accuracy difference schemes for approximate solution of the integral-type time-nonlocal parabolic problems. The theorems on the stability of r-modified Crank-Nicolson difference schemes and second-order accuracy implicit difference scheme for approximate solution of the integral-type time-nonlocal parabolic problems in a Hilbert space with self-adjoint positive definite operator are established. In practice, stability estimates for the solutions of the second-order accuracy in t difference schemes for the one and multidimensional time-nonlocal parabolic problems are obtained. Numerical results are given.

Second Order Equations in Functional Spaces: Qualitative and Discrete Well-Posedness

The present survey contains the recent results on the local and nonlocal well-posed problems for second order differential and difference equations. Results on the stability of differential problems for second order equations and of difference schemes for approximate solution of the second order problems are presented.

The stability of difference schemes of second-order of accuracy for hyperbolic-parabolic equations

A nonlocal boundary value problem for hyperbolic-parabolic equations in a Hilbert space H is considered. Difference schemes of second order of accuracy difference schemes for approximate solution of this problem are presented. Stability estimates for the solution of these difference schemes are established.

A note on the difference schemes for hyperbolic-elliptic equations

The nonlocal boundary value problem for hyperbolic-elliptic equationd2u(t)/dt2+Au(t)=f(t),(0≤t≤1),−d2u(t)/dt2+Au(t)=g(t),(−1≤t≤0),u(0)=ϕ,u(1)=u(−1)in a Hilbert spaceHis considered. The second order of accuracy difference schemes for approximate solutions of this boundary value problem are presented. The stability estimates for the solution of these difference schemes are established.

Theory of stability and regularization of difference schemes and its application to ill-posed problems of mathematical physics

Solving numerically evolutionary problems of mathematical physics, the problem of stability of difference schemes with respect to initial and boundary data is of great importance. Principal results of the general theory of stability for two- and three-level difference schemes written in the canonical form are presented in this paper. This theory has been developed by the author in 1967. Difference schemes for unsteady problems of mathematical physics are treated in the theory as abstract Cauchy problems for operator-difference problems in Hilbert spaces. The necessary and sufficient conditions for stability in different norms are also formulated for a wide class of difference schemes. To construct stable difference schemes, a general methodological approach based on the regularization principle is employed. In this approach, we can start from any simple scheme (even unstable) as the initial one. By perturbing the initial scheme operators and taking into consideration the stability conditions, absolutely stable schemes are derived. Such an approach is used to obtain the difference schemes for approximate solution of the ill-posed problems for evolutionary equations.