Finite differences are commonly used for discrete approximations of derivatives. Large classess of schemes for the numerical approximation of ordinary differential equations (ODEs) and partial differential equations (PDEs) are derived from finite differences. Formulae for numerical differentiations are generally obtained from a linear combination of Taylor series, which leads to solving a system of linear equationskhan1999closed ; khan2000new ; khan2003taylor ; quarteroni2010
or calculating derivatives of interpolating polynomials (for instance seequarteroni2010 ). References hildebrand1974introduction ; chung2010computational ; dahlquist2008numerical give a number of finite difference formulae, for high order approximation of derivatives, in term of formal power series of finite difference operators.
The purpose of this paper is to provide some basic results on finite difference approximations, which results are required for the numerical analysis of higher order time-stepping schemes for ODEs and PDEs. We introduce a new approach to derive arbitrary high order finite difference formulae which avoids the need for solving a system of linear equations. We provide various formulae for the discrete approximation of any order derivative of an analytic function at a point using arbitrary points evenly spread around . These discrete approximations are of order 1 or 2 (order 2 for centred formulae), with errors explicitly expanded in terms of Taylor series with the derivatives , . Substituting successively by their finite difference approximations in the error term for the discrete approximation of , we improve successively by 1 or 2 the order of the discrete approximation of . An efficient choice of the discrete points minimizes the number of points needed for a given order of accuracy of the discrete approximation of . Our approach can be used to recover the existing finite difference formulae, but it also provides various new formulae. We give three new finite difference formulae which are useful for the construction of new high order time-stepping schemes and their efficient starting procedures via the deferred correction (DC) method. In fact, the use of standard backward and central finite differences in building high order time-stepping schemes via the DC method leads to the computation of starting values for these schemes outside the solution interval while the standard forward finite difference formula leads to unstable schemes (see, e.g., daniel1967interated ; MR2058857 ; kress2002deferred ; koyaguerebo2019arbitrary ; koyaguerebo2020unconditionally ).
The paper is organized as follows: in section 2 we recall the main finite difference operators and prove some of their main properties; section 3 presents general first and second order approximations of derivatives with error terms explicitly expressed as Taylor series; section 4 gives many results for arbitrary high order finite difference approximations, and section 5 deals with a numerical test.
2 Properties of finite difference operators
In this section we recall the standard finite difference operators and provided some of their useful properties.
For a given spacing and a real , we denote and , for each integer . The centered, forward and backward difference operators , and , respectively, related to , and applied to a function from into a Banach space , are defined as follows:
The average operator is denoted by :
The composites of and are defined recursively. They commute, that is
and satisfy the identities
We introduce the double index such that
If is even, then we have
for some integer . For example,
Theorem 1 (Finite difference approximation of a product).
Suppose that is a Banach algebra. Then, for any functions , we have
More generally, for each integer such that exists, we have the formula
The formulae (6)-(8) can be obtained by a straightforward calculation, so we just need to establish (9). We proceed by induction on the positive integer . From the index notation introduced in (4), we can write
These two identities combined with (8) yield
Expanding as in the formula (8), we deduce that
Theorem 2 (Finite difference approximation of a composite).
Consider two functions and with values into Banach spaces such that the composite is defined on and the differential is integrable. Then
As in standard mean value theorem. ∎
3 First and second order discrete approximation of derivatives
In this section we provide various formulae for the finite difference approximation of arbitrarry high order derivatives of analytic functions. The approximations are of order one or two, and the error terms are explicitly expanded i terms of Taylor series. We need the following lemma which proof is an easy induction.
For positive integers and and for any real , we have
In particular, for any nonnegative integer , we have
Suppose that the function is analytic. Let , , be a partition of the interval . For each positive integer , we have
Let be , , and , , be a partition . Let and be two positive integers such that . Then, for each integer such that , is bounded independently of , and we have the estimate
, and we have the estimate
where is a constant depending only on the integer .
According to Remark 1, it is enough to just prove the theorem for or , for suitable positive integer (the case is trivial). As in the previous proof, Taylor expansion of order with integral remainder together with formulae (1) and (14) yields
It follows that
Similar reasoning can be applied in the case of .
4 Arbitrary high order finite difference approximations
There exists a sequence of real numbers such that for any function , where is a positive integer, and a partition , , of , we have
By Taylor expansion we can write
where, for , and , we have
- For we have
- For we have
where is the transpose of the vector
is the transpose of the vector
The following theorem gives a new form of centered finite difference formulae which is useful for efficient starting procedures of high order time-stepping schemes via deferred correction strategy koyaguerebo2019arbitrary ; koyaguerebo2020unconditionally .
Theorem 6 (Interior centered approximations).
Let , where is a positive integer and , , is a real interval. Given a uniform partition of , that is with , and , there exist reals such that
Table 2 gives the coefficients for .