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

Piskarev, S. I.

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.

MSC:

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

Keywords:

stability of difference schemes; Cauchy problems; first-order evolution equation; almost periodic solutions