NUEN
–
430
Lecture 4
Theme of Discretization
Given a differential equation, the purpose of discretization is to:
a.
Approximate the solution to the differential equation in terms of a finite number of
unknowns associated with a mesh of the variable domain associated with the
differential equation.
b.
Ensure that the approximate solution becomes “exact” in some mathematically
meaningful way in the limit as the number of unknowns is increased without bound
c.
“Replace” the differential equation with an algebraic sy
stem of equations for the
unknowns that can be solved to desired accuracy via a computational algorithm.
There are two major approaches to discretization:
1.
Finite difference
2.
Finite Element
Finite difference
schemes are generally based upon the use of Taylor series expansions to derive
difference approximations to derivatives.
Unknowns are point values of the solution.
Finite volume
schemes are a subclass of finite difference schemes that are applied to balance or
conservation equations.
Most finite difference approximations used in nuclear engineering are
of this type.
The basic idea behind these schemes is to ensure that the discrete approximation
makes a statement of conservation on each mesh cell in analogy with the fact that the analytic
equation makes a statement of conservation over each differential volume.
Finite element
methods are based upon the use of finitedimensional function spaces to
approximate the solution to the differential equations.
This sounds sophisticated, but an
element of a function space is just a function,
̃
( )
, that is defined in terms of a set of Basis
functions,
{
(
⃗)}
and an associated set of coefficients,
{
}
.
Thus,
̃
( ) ∑
(
⃗)
A finiteelement basis set is always related to a mesh structure, and the expansion coefficients
often take the form of discrete values of
̃
( )
on the mesh.
Furthermore, the Basis functions
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always have the property of local support.
This means that each basis function is nonzero only
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 Fall '10
 Morel
 Numerical Analysis, Partial differential equation, finite difference, discretization, approximate solution

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