"Abel transformation" redirects here. For another transformation, see Abel transform.
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. The summation by parts formula is sometimes called Abel's lemma or Abel transformation.
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 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 ( becomes ) and one which is differentiated ( becomes ).
The process of the Abel transformation is similar, since one of the two initial sequences is summed ( becomes ) and the other one is differenced ( becomes ).
where a is the limit of . As is convergent, is bounded independently of , say by . As go to zero, so go the first two terms. The third term goes to zero by the Cauchy criterion for . The remaining sum is bounded by
by the monotonicity of , and also goes to zero as .
Using the same proof as above, one can show that if
(so that the sum goes to zero as goes to infinity)
In both cases, the sum of the series satisfies:
Summation-by-parts operators for high order finite difference methodsEdit
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.
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^Mattsson, Ken; Nordström, Jan (September 2004). "Summation by parts operators for finite difference approximations of second derivatives". Journal of Computational Physics. 199 (2): 503–540. doi:10.1016/j.jcp.2004.03.001.
^Carpenter, Mark H.; Gottlieb, David; Abarbanel, Saul (April 1994). "Time-Stable Boundary Conditions for Finite-Difference Schemes Solving Hyperbolic Systems: Methodology and Application to High-Order Compact Schemes". Journal of Computational Physics. 111 (2): 220–236. CiteSeerX10.1.1.465.603. doi:10.1006/jcph.1994.1057.