## Unconditional stability of difference formulas.(English)Zbl 0538.65032

The initial-value problem for the equation $dy(t)/dt=A\quad y(t),\quad t>0$ where A is a linear operator in a complex Banach space X, in the case, when it is a partial differential equation of evolution type, is considered. The problem is solved by using a k-step formula ($$k\geq 1)$$ of the form: $(*)\quad p_ 0(\Delta t.A_ n)\quad u_ j=p_ 1\quad(\Delta t.A_ n)u_{j-1}+...+p_ k(\Delta t.A_ n)u_{j-k}$ $$\Delta t>0$$, where $$A_ n:C^{m(n)}\to C^{m(n)}$$, C is the field of complex numbers, m(n) is the number of meshes of the net $$\Omega_ n=\{x^ n_ 1,...,x^ n_{m(n)}\}\subset {\bar \Omega}$$, $$\Omega$$ is a certain domain of the Euclidean space. The matrices $$A_ n$$ have the order m(n), vectors u belong to $$C^{m(n)}$$ and $$p_ 0,p_ 1,...,p_ k$$ are polynomials with no common root. Given a sequence $$A_ n$$, the stability of the formula (*) respectively to the time-step $$\Delta$$ t depends in general on n and $$\Delta$$ t. Usually one considers the case when $$\Delta$$ t is bounded from above with increasing n (conditional stability). Such a restriction impedes the practicability of such formulae, especially for partial differential equations. The unconditional stability, i.e. when $$\Delta$$ t is not bounded from above, is the matter of the paper. The author proves (Theorem 5.1) sufficient conditions for the stability of (*) for sufficiently large n and for $$\Delta$$ $$t\geq \epsilon$$ for an arbitrary $$\epsilon>0$$. These conditions base on a generalization of the notion of A-acceptability known for one- step formulae, applied to the k-step formula (*); some other general assumptions about the operator A(its spectrum, its non-singularity) and about the approximation of A by the matrices $$A_ n$$ are very natural ones. Some examples of $$A_ n$$-acceptable formulae (*) are presented. The results can be applied also for ordinary differential equations.
Reviewer: S.Ząbek

### MSC:

 65J10 Numerical solutions to equations with linear operators 65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs 35G10 Initial value problems for linear higher-order PDEs 35K25 Higher-order parabolic equations 34G10 Linear differential equations in abstract spaces
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### References:

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