Question

Derivation of the 2D Rabinowitz Equation?

Answer 1

Integration rules (5.7.5, (5.7.11) and (5.7.14) are fully rules. An integration rule over an FS region B is said to be fully symmetric if it is of the form etric (FS) Σ" Σ ί0) w(x)f (x) dx > (5.7.15) y(Nirs where wx) is an FS weight function. Such a rule is determined by n weights w and s of degree 2s, then it is of rule generators x Clearly, if such de Since any point ye xs determines the set x, we arbitrarily call the generator of the set that point x(x, x2, ... x) for which x x 2* 2x20. In such a generator, there will be p distinct nonzero components , 0 spsd, each one with a multiplicity n, 2 1. The class of a generator 1,n, , where- d oesponds to the generator (0, 0, 0 If we x is the set of positive integers empty se dafin sum of positive integers, then the number of classes of generators in 372 APPROXIMATE INTEGHATION IN Two OR MORE DIMENSIONS dimensiondis given by M then the number of points -, pr).If x is of class the set xs is given by n,,n,.n, 24-d! | (d- )! n, ! -*- m, - Σ. is the number of zeros in where We shall now assume that we have assigned an ordering to the various classes of generators so that they are indexed by the integers M. For example, in three dimensions, we could order the seven classes of generators as follows: A , N2 = {1} N3 = 2), N4 = {1, 1}, Ng -(3 , N6=2,1 N,= {1, 1 1 The structure of an FS integration rule is deter- mined by an M-vector of nonnegative integers K = (K\, ..., KM) which have decided on a particular structure, we can write down nonlinear moment equations similar to (5.7.12) to determine the weights w and generators x, of the rule. However, not all structures K lead system of consistent systems of equations. A necessary condition for consistency is linear consistency which means that no linear combination of equations the system can yield the equation0.Inis imposes ceriain linear constraints Rabinowitz in the two- and three-dimensional case. Subsequently, using a different approach, Keast and Lyness rederived the sets of constraints for these two cases as well as for the four-dimensional case. Once we have these constraints, we can derive structures leading to rules with a minimal number of points by solving the integer programming problem Σκ Μ| minimize subject to the consistency constraints mentioned above and the nonnegativity constraints K, 2 0, i-1,., M. In addition to optimal solutions to this problem, one can also find suboptimal solutions. These are necessary since a consistent structure does not always yield an integration rule because ne nonnea system oment equao olution real solut f the lav points may lie outside the integration region and the weights may have too large a figure of demerit as defined below (page 377). Thus if a particular structure does not lead to an acceptable solution, we proceed to the next best structure This procedure was followed by Mantel and Rabinowitz who derived th FS regions. Previously, Rabinowitz and Richter, following the same