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.
[1] Suppose
Then, Using the forward difference operator
, 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.
The formula is sometimes given in one of these - slightly different - forms which represent a special case (
) of the more general rule both result from iterated application of the initial formula.
The auxiliary quantities are Newton series: and A particular (
) result is the identity Here,
is the binomial coefficient.
For two given sequences
, one wants to study the sum of the following series:
This process, called an Abel transformation, can be used to prove several criteria of convergence for
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 (
The process of the Abel transformation is similar, since one of the two initial sequences is summed (
Proof of Abel's test.
Summation by parts gives
{\displaystyle {\begin{aligned}S_{M}-S_{N}&=a_{M}B_{M}-a_{N}B_{N}-\sum _{n=N}^{M-1}B_{n}(a_{n+1}-a_{n})\\&=(a_{M}-a)B_{M}-(a_{N}-a)B_{N}+a(B_{M}-B_{N})-\sum _{n=N}^{M-1}B_{n}(a_{n+1}-a_{n}),\end{aligned}}}
is bounded independently of
The third term goes to zero by the Cauchy criterion for
The remaining sum is bounded by
Using the same proof as above, one can show that if then
In both cases, the sum of the series satisfies:
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.
[3][4] The boundary conditions are usually imposed by the Simultaneous-Approximation-Term (SAT) technique.
[5] 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.