Summation by parts on arvau

Wikipedia / Summation by parts

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

are two sequences. Then,

Using the forward difference operator, it can be stated more succinctly as

Or, equivalently, using backward difference operator:

Both can be used to obtain a more symmetric version:

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

) of the more general rule

both result from iterated application of the initial formula. The auxiliary quantities are Newton series:

and

) result is the identity

is the binomial coefficient.

Method

Similarity with an integration by parts

Beside the boundary conditions).

The process of the Abel transformation).

Applications

It is used to prove Kronecker's lemma, which in turn, is used to prove a version of the strong law of large numbers under variance constraints.
It may be used to prove Nicomachus's theorem positive integers.
Summation by parts is frequently used to prove Abel's theorem and Dirichlet's test.
One can also use this technique to prove Abel's test is a convergent series a bounded monotone sequence converges.

Proof of Abel's test. where a go to zero, so go the first two terms. The third term goes to zero by the Cauchy criterion.

Using the same proof as above, one can show that if

form a bounded sequence;
goes to infinity)

converges.

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.