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


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
Full Text: EuDML


[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.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.