A note on the difference schemes for hyperbolic equations

Abstract

The initial value problem for hyperbolic equationsHilbert space H is considered. The first and second order accuracy difference schemes generated by the integer power of A approximately solving this initial value problem are presented. The stability estimates for the solution of these difference schemes are obtained.

On the numerical solution of hyperbolic PDEs with variable space operator

“…Nevertheless, Au = A 0 u + Bu and A 0 is a self-adjoint positive definite operator in H and BA −1 0 is bounded in H . The proof of this statement is based on the abstract results of [14] and difference analogy of integral inequality.…”

“…The stability estimates for the solution of these difference schemes are also established. In this article the applications of [14] to the numerical solutions of the initial-boundary value problem for the multidimensional hyperbolic equations…”

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

“…Such type of stability inequalities for the solutions of the first order of accuracy difference scheme for the differential equations of hyperbolic type were established for the first time in [62]. The first and second order of accuracy difference schemes approximately solving the abstract initial value problem for hyperbolic equations in Hilbert spaces were presented in [63]. Applying the operator approach, the stability estimates for the solution of these difference schemes were obtained.…”

An operator-difference scheme for abstract Cauchy problems

“…In numerical techniques for solving these equations, the problem of stability in various functional spaces has received a great deal of importance and attention (see [13][14][15][16][17][18][19][20][21]). Especially, a proper difference scheme with a time dependent unbounded operator provides a suitable model for analyzing the stability.…”

“…have been established in [16]. In [17,18], for the same problem, the high-order two-step difference methods generated by an exact difference scheme, and by the Taylor expansion on three points have been discussed; here the stability estimates for approximate solutions by these difference methods are also discussed.…”