In mathematics, summation by parts transforms the summation of products of sequences into other summations, often simplifying the computation or (especially) estimation of certain types of sums. It is also called Abel's lemma or Abel transformation, named after Niels Henrik Abel who introduced it in 1826.

Statement

Suppose {f_k} and {g_k} are two sequences. Then, Using the forward difference operator \Delta, it can be stated more succinctly as Summation by parts is an analogue to integration by parts: or to Abel's summation formula: An alternative statement is which is analogous to the integration by parts formula for semimartingales. Although applications almost always deal with convergence of sequences, the statement is purely algebraic and will work in any field. It will also work when one sequence is in a vector space, and the other is in the relevant field of scalars.

Newton series

The formula is sometimes given in one of these - slightly different - forms which represent a special case (M = 1) of the more general rule both result from iterated application of the initial formula. The auxiliary quantities are Newton series: and A particular (M=n+1) result is the identity Here, is the binomial coefficient.

Method

For two given sequences (a_n) and (b_n), with n \in \N, one wants to study the sum of the following series: If we define then for every n>0, and Finally This process, called an Abel transformation, can be used to prove several criteria of convergence for S_N.

Similarity with an integration by parts

The formula for an integration by parts is. Beside the boundary conditions, we notice that the first integral contains two multiplied functions, one which is integrated in the final integral (g' becomes g) and one which is differentiated (f becomes f'). The process of the Abel transformation is similar, since one of the two initial sequences is summed (b_n becomes B_n) and the other one is differenced (a_n becomes ).

Applications

Proof of Abel's test. Summation by parts gives where a is the limit of a_n. As \sum_n b_n is convergent, B_N is bounded independently of N, say by B. As a_n-a go to zero, so go the first two terms. The third term goes to zero by the Cauchy criterion for \sum_n b_n. The remaining sum is bounded by by the monotonicity of a_n, and also goes to zero as. Using the same proof as above, one can show that if then converges. In both cases, the sum of the series satisfies:

Summation-by-parts operators for high order finite difference methods

A summation-by-parts (SBP) finite difference operator conventionally consists of a centered difference interior scheme and specific boundary stencils that mimics behaviors of the corresponding integration-by-parts formulation. The boundary conditions are usually imposed by the Simultaneous-Approximation-Term (SAT) technique. The combination of SBP-SAT is a powerful framework for boundary treatment. The method is preferred for well-proven stability for long-time simulation, and high order of accuracy.