On the application of two Gauss–Legendre quadrature rules for composite numerical integration over a tetrahedral region

Abstract

In this paper we first present a Gauss-Legendre quadrature rule for the evaluation of I = f f T ff(x,y,z)dxdydz, where f(X,y,z) is an analytic function in x, y, z and Tis the standard tetrahedral region: {(x,y,z)|0 ≤ x,y,z ≤ I, x + y + z ≤ 1} in three space (x,y,z). We then use a transformation x = x(ξ,η,ζ), y = y(ξ,η,ζ) and z = z(ξ,η,ζ) to change the integral into an equivalent integral I =∫ 1 -1 ∫ 1 -1 ∫ 1 -1 f(x(ξ,η,ζ), y(ξ,η,ζ,)) ∂(x,y,z) ∂(ξ,η,ζ) dξdηdζ over the standard 2-cube in (ξ,η,ζ) space: {(ξ,η,ζ)|-1 ≤ ξ,η,ζ ≤ 1}. We then apply the one-dimensional Gauss-Legendre quadrature rules in ξ, η and ζ variables to arrive at an efficient quadrature rule with new weight coefficients and new sampling points. Then a second Gauss-Legendre quadrature rule of composite type is obtained. This rule is derived by discretising the tetrahedral region T into four new tetrahedra T c i (i= 1,2,3,4) of equal size which are obtained by joining the centroid of T, c = (1/4,1/4,1/4) to the four vertices of T. By use of the affine transformations defined over each T c i and the linearity property of integrals leads to the result: I = Σ4 i=1 ∫∫Tci∫f (x,y,z) dxdydz = 1 4 ∫∫T ∫ G(X,Y,Z)dXdYdZ, where G(X, Y,Z) = 1 p 3 Σ f(x (k) (X, Y,Z), y (k) (X, Y,Z),z (k) (X, Y, Z)), x (k) = x (k) (X,Y,Z), y (k) =y (k) (X,Y,Z) and z (k) = z (k) (X, Y,Z) refer to an affine transformations which map each T c i into the standard tetrahedral region T. We then write I = ∫∫ T ∫ G(X,Y,Z)dXdYdZ = ∫ 1 0 ∫ 1-ξ 0 ∫ 1-ξ-η 0 G(X(ξ,η,ζ), Y(ξ,η, ζ), Z(ξ,η,ζ)) |∂(X,Y,Z)| ∂(ξ,η,ζ) dξdηdζ and a composite rule of integration is thus obtained. We next propose the discretisation of the standard tetrahedral region T into p 3 tetrahedra T i (i = 1(1)p 3 ) each of which has volume equal to 1 /(6p 3 ) units. We have again shown that the use of affine transformations over each T, and the use of linearity property of integrals leads to the result: ∫∫ T ∫f(x,y,z)dxdydz= Σ p3 i=1 ∫∫ T c i ∫f(x,y,z)dxdydz = Σ p3 α=1 ∫ ∫ T[p]α ∫f(x (α,p) , y (α,p) , z (α,p) )dx (α,p) dy (α,p) dz (α,p) = 1 p3 ∫ ∫ T ∫H(X,Y,Z)dXdYdZ, where p3 H(X, Y,Z) = Σ α=1 f(x (α,p) (X Y,Z),y (α,p) (X, Y, Z), z (α,p) (X, Y,Z)), x (α,P) = x (α,P) (X,Y,Z), y (α,P) =y (α,P) (X,Y,Z) and z (α,p) = z (α,P) (X, Y, Z) refer to the affine transformations which map each T i in (x (α,P) ,y (α,P) , z (α,P) ) space into a standard tetrahedron T in the (X, Y,Z) space. We can now apply the two rules earlier derived to the integral ∫∫ T ∫H(X, Y,Z)dXdYdZ, this amounts to the application of composite numerical integration of T into p3 and 4p3 tetrahedra of equal sizes. We have demonstrated this aspect by applying the above composite integration method to some typical triple integrals.