Summation by parts

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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.

StatementEdit

Suppose   and   are two sequences. 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.

Newton seriesEdit

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.

MethodEdit

For two given sequences   and  , with  , one wants to study the sum of the following series:
 

If we define    then for every       and

 
 

Finally   

This process, called an Abel transformation, can be used to prove several criteria of convergence for   .

Similarity with an integration by partsEdit

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   ).

ApplicationsEdit

Proof of Abel's test. Summation by parts gives

 

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
  1. the partial sums   form a bounded sequence independently of   ;
  2.   (so that the sum   goes to zero as   goes to infinity)
  3.  
then   converges.

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.[2][3] The boundary conditions are usually imposed by the Simultaneous-Approximation-Term (SAT) technique.[4] 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.

See alsoEdit

ReferencesEdit

  1. ^ Edmonds, Sheila M. (1957). "Sums of powers of the natural numbers". The Mathematical Gazette. 41 (337): 187–188. doi:10.2307/3609189. JSTOR 3609189. MR 0096615.
  2. ^ Strand, Bo (January 1994). "Summation by Parts for Finite Difference Approximations for d/dx". Journal of Computational Physics. 110 (1): 47–67. doi:10.1006/jcph.1994.1005.
  3. ^ 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.
  4. ^ 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. CiteSeerX 10.1.1.465.603. doi:10.1006/jcph.1994.1057.