Z. Gegechkori, J. Rogava, and M. Tsiklauri

I. Javakhishvili Tbilisi State University Tbilisi, Georgia

THE THIRD AND FOURTH ORDER ACCURACY DECOMPOSITION FORMULAS FOR A SEMIGROUP

The first decomposition formula for an exponential matrix function was constructed by Lie in 1875. Namely, he showed that

lim_{n→∞} (exp(A_{1}/n) exp(A_{2}/n))^{n} = exp(A_{1} + A_{2}),

where A_{1} and A_{2} are finite dimensional matrices. In 1959, Trotter generalized this formula for exponential operator function – semigroup exp(t(A_{1} + A_{2})) (A_{1} and A_{2} are self-adjoined positively defined operators). Resolvent analog to this formula for the first time was constructed by Chernoff in 1968.

Note that Lie’s formula is of the first order accuracy decomposition formula, that is: (exp(A_{1}/n) exp(A_{2}/n))^{n} − exp(A_{1} + A_{2}) = Op(1/n), where the norm of the matrix Op(1/n) is of the O(1/n) order. There are also known the second order accuracy decomposition formulas. The third and fourth order accuracy decomposition formulas for two-dimensional splitted operator was constructed by B. O. Dia and M. Schatzman in 1996. Note that the formulas constructed by these authors are not automatically stable decomposition formulas. Decomposition formula is called automatically stable if the sum of absolute values of the split coefficients is equal to one. Q. Sheng has proved that in the real number field there does not exist such automatically stable splitting of the exponential operator function-semigroup whose accuracy order is higher than two.

We introduce the notation: T_{1}(λ) = exp(−τλA_{1}) ⋯ exp(−τλA_{m}) and T_{2}(λ) = exp(−τλA_{m}) ⋯ exp(−τλA_{1}), where λ is a scalar, and A_{1}, …, A_{m} (m ≥ 2) are linear (generally unbounded) operators; A = A_{1} + ⋯ + A_{m}, τ = t/n (t ≥ 0, n – is a natural number). The following formulas are true:

exp(−tA) − (\frac{1}{2}(T_{1}(α)T_{2}(\overline{α}) + T_{2}(α)T_{1}(\overline{α})))^{n} = Op(τ^{3}),
exp(−tA) − [\frac{1}{2}(T_{1}(\frac{α}{2})T_{2}(\frac{\overline{α}}{2})T_{1}(\frac{\overline{α}}{2})T_{2}(\frac{α}{2})+

+ T_{2}(\frac{α}{2})T_{1}(\frac{\overline{α}}{2})T_{2}(\frac{\overline{α}}{2})T_{1}(\frac{α}{2}))]^{n} = Op(τ^{4}),

where α = \frac{1}{2} ± i\frac{1}{2\sqrt{3}}; operators (−αA_{j}), (−\overline{α}A_{j}) (j = 1, …, m) and (−A) generate strongly continuous semigroups; Op(τ^{k}) is an operator whose norm is of the fourth order with respect to τ (more precisely, in case of unbounded operators ∥Op(τ^{k})φ∥ = O(τ^{k}) for any φ from the domain of definition of the operator Op(τ^{k})).

Note that the decomposition formulas constructed by us are automatically stable.