Flux Integrals  Elemental Spatial Operators  Elemental Spatial Operators

Integration

We take advantage of the tensor product element coordinate systems to perform integration. The integrations over each element can be performed as a set of one-dimensional integrals using Gauss quadrature. If we used the reference coordinate systems this would be very difficult since the limits of the ``collapsed'' elements are not constant.

We first describe the choice of quadrature type for integrating each direction. We will then motivate the inclusion of quadrature with non-constant weights in order to reduce the number of points we use.

In two-dimensions we consider integrals of the form :

In three-dimensions:

We use the Gauss weights that will perform the discrete integral of a function as a sum:

$$\int_{-1}^{1} (1-z)^{\alpha}(1+z)^{\beta}f(z) dz = \sum_{i=0}^{N-1} f(z_i^{\alpha,\beta})w_i^{\alpha,\beta}$$ This will be used in each of the d directions in the d-dimensional elements. In table 4.1 we show the type of Gaussian quadrature we use in each of the `a',`b' and `c' directions.
 
 

Element `a' `b' `c'

Triangle GLL GRJ_1,0 -

Quadrilateral GLL GLL -

Tetrahedron GLL GRJ_1,0 GRJ_2,0

Pyramid GLL GLL GRJ_2,0

Prism GLL GLL GRJ_1,0

Hexahedron GLL GLL GLL

10/24/1998