By Hernandez D. B., Spigler R.

Numerical balance of either specific and implicit Runge-Kutta equipment for fixing usual differential equations with an additive noise time period is studied. the idea that of numerical balance of deterministic schemes is prolonged to the stochastic case, and a stochastic analogue of Dahlquist's A-stability is proposed. it really is proven that the discretization of the float time period by myself controls the A-stability of the entire scheme. The quantitative influence of implicitness upon A-stability is usually investigated, and balance areas are given for a family members of implicit Runge-Kutta tools with optimum order of convergence.

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**Extra info for A -stability of Runge-Kutta methods for systems with additive noise**

**Sample text**

Then the set Bf ∈ F misses A ∈ F ⊥ , which is a contradiction. The following proposition gives a partial answer to the Minimal Tower problem. 4. If p ≤ max{td , tp+ }, then p = t. In particular, p = t under p = ℵ1 . Proof. Assume that p ≤ max{td , tp+ }. 9. So assume that d > p = ω1 . Then p+ ≤ d and td ≤ tp+ = max{td , tp+ } ≥ p. By the deﬁnition of p there is a Centered family F = {Fα }α

If F is a semiﬁlter with χ(F ) < d, then t(F ⊥ ) ≥ ℵ1 . Proof. We should show that each decreasing sequence (An )n∈ω in F ⊥ has a pseudointersection in F ⊥ . For any f : ω → ω consider the pseudointersection An ∩ [0, f (n)] Af = n∈ω of the sequence (An ). We claim that Af ∈ F ⊥ for some f . Fix a subfamily B ⊂ F of size |B| = ℵ0 · χ(F ) < d such that each F ∈ F contains some B ∈ B. For each B ∈ B let fB (n) = min B ∩ An , n ∈ ω. The function set {fB : B ∈ B} has size < d and thus is not dominating in (ω ω , ≤).

5 is a simultaneous generalization of a result of Posp´ıˇ sil [1937] (asserting that there is an ultraﬁlter U with χ(U) = c) and a result of Bell, Kunen [1981] (establishing the existence of an ultraﬁlter V with πχ(V) ≥ cf(c)). The latter paper also describes a model of ZFC containing no ultraﬁlter V with πχ(V) = c. The existence of ultarﬁlters with character c was also proved by Kunen [1972]. 1 essentially is due to Rothberger [1948]. a] who indicates that a similar statement was proved also by Ketonen [1976].