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NUEN-430 Lecture 4

NUEN-430 Lecture 4 - NUEN 430 Lecture 4 Theme of...

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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 finite-dimensional 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 finite-element 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 non-zero only
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