Stability of difference schemes in Cauchy problems with almost periodic solutions. (English. Russian original) Zbl 0551.65063

Differ. Equations 20, 525-530 (1984); translation from Differ. Uravn. 20, No. 4, 689-695 (1984).
It is known that in the approximation of an abstract parabolic equation, an explicit difference scheme is stable when the discrete time and space steps satisfy the conditions \(\tau_ n\leq ch^ 2_ n\), where c is a constant. We prove that, in the solution of a first-order evolution equation with almost periodic solutions (a hyperbolic equation), the conditions \(\tau_ n\leq ch_ n\) must be satisfied if we are to have stability.


65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65J10 Numerical solutions to equations with linear operators
35K55 Nonlinear parabolic equations
34G10 Linear differential equations in abstract spaces