Page 170 - Basic Structured Grid Generation

Variational methods and adaptive grid generation 159

6.3.2 Dynamic adaptation

Ideally the numerical solution of the hosted equations should be dynamically coupled
to the process of grid generation, so that, in regions of the physical domain where large
gradients (or higher-order spatial derivatives) of the dependent variables are found to
occur, grid density may be increased to allow more accurate resolution. In two or three
dimensions re-distribution of grid points can result in distortion of grid cells, but in
one dimension the task is simpler.
Here we present a grid generation technique which is equivalent to that defined by
the transformation from computational space to physical space given by eqn (5.90).
This is itself equivalent to the ‘Equidistribution Principle’ proposed by Boor (1974),
which requires that the errors in the numerical solution of a problem should be uni-
formly distributed throughout the solution domain. Given a set of grid points a = x 0 ,
x 1 ,...,x n−1 , x n = b, this may be expressed as

x i+1
W(x) dx = K, constant,i = 0, 1,...,(n − 1), (6.20)
x i
where the weight function W(x) varies with the error at a point x. Clearly W(x) is
equivalent to the reciprocal of the weight function ϕ(x) as given in eqn (5.90) through
the modified form of eqn (6.20) given by
dx
W(x) = K, with x(0) = a, x(1) = b, (6.21)
dξ
so that with an evenly distributed grid in the ξ interval 0 <ξ < 1 the distance between
corresponding points in the x interval a< x < b is inversely proportional to the local
value of W(x).
Integrating eqn (6.21) gives

b 1
W(x) dx = K dξ = K, (6.22)
a 0
and
x ξ
W(x) dx = K dξ = Kξ,
a 0
showing that the transformation from the physical domain to the computational domain
is given by
x
W(x) dx
a . (6.23)
b
ξ(x) =
W(x) dx
a
In one-dimensional adaptive grid generation the weight function will in practice be
taken to depend directly on the derivatives of the solution to the hosted equation,
say u(x) in the physical domain, rather than on some explicit representation of the
error in the solution. Given that the largest numerical errors tend to occur where the
lowest derivatives have high values, we want W(x) to have a high value in regions in