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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:

[1] I. Babuška M.Práger, E. Vitásek: Numerical Processes in Differential Equations. SNTL, Prague, 1966. · Zbl 0156.16003
[2] B. L. Ehle: A-stable Methods and Pade Approximations to the Exponential. SIAM J. Math. Anal., Vol. 4 (1973), No. 4, pp. 671-680. · Zbl 0236.65016
[3] A. Iserles: On the A-acceptability of Pade Approximations. SIAM J. Math. Anal., Vol. 10 (1979), No. 5, pp. 1002-1007. · Zbl 0441.41010
[4] E. Hille, R. S. Phillips: Functional Analysis and Semi-groups. Amer. Math. Soc., Vol. 31., rev., Waverly Press, Baltimore, 1957.
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