OPTIMAL DIFFERENCE FORMULAS
Abstract
In this paper we consider the problem of the local discretization of a given differential operator by means of a difference formula. Specifically, we address the problem of finding best difference formulas in a worst case error sense for a given class of functions. We show that the HODIE difference formulas for ordinary differential equations are interpolatory and strongly optimal. This result is generalized in Theorem 4.4 so that the same holds for the Hodie difference formulas of any differential operator. This provides a new interpretation of the Hodie method. Furthermore, in Theorem 4.6 we show that optimal difference formulas for ordinary differential equations are really (asymptotically) a discretization of the inverse of the differential operator subject to some multipoint initial condition. This gives a new view point of the discretization of a differential operator. For the operators D('2) + dD + e and D('m), we explicitly find difference formulas that minimize the truncation error for certain classes of functions. These formulas satisfy the condition of being strongly optimal. We also address the following problem: It is known that nine-point difference formulas exact for high order polynomials for the Laplace's operator (DELTA) do not exist. This is due to the fact that some polynomials belong to the kernal of (DELTA). This raises the question of what happens if none of the basis functions belong to the kernel of (DELTA). Is it possible to construct difference formulas by making them exact for a large set of "good" approximating functions? Do the resulting formulas have coefficients different from zero (for the Laplace's operator and nine-point formulas, the coefficients of polynomial based formulas are identically zero)? If these answers are yes, are these formulas any better? The first two questions are answered affirmatively and we present a new family of nine point formulas which is exact on tensor product splines. The final question is answered negatively.
Degree
Ph.D.
Subject Area
Mathematics
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