Lecture17_18wn_NucE309FA11

Lecture17_18wn_NucE309FA11 - NucE 309, Fall 2011, Lectures...

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17.1 NucE 309, Fall 2011, Lectures 17 and 18 21.4 Numerical Methods for Solving Elliptic Differential Equations (was 19.4) NOTE: If an analytical solution can be found, these numerical methods would not be used. These numerical methods are only used for complex equations, boundary conditions, or geometries. We are using simple equations and boundary conditions as an example. A. One-Dimensional Conduction Equation with no Heat Generation We will look at several conduction equations as examples. The first equation is the one-dimensional conduction equation with no heat generation. 2 2 0 T x with boundary conditions: (0) 5, ( ) 10 TT L  The domain from 0to x xL will be discretized into N grid cells. x x = 0 x = L If the grid points are evenly spaced, / x LN  . At each grid point, we will write the finite difference equations: 2 11 22 2 0 ii i i T T xx    This central difference formula for the second derivative is second order accurate. This gives us a set of linear equations. Since all terms are divided by 2 x , we can multiply by 2 x to simplify each equation. 01 2 12 3 23 4 34 5 32 1 21 20 NN N N T T T T T T   X X X
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17.2 This gives us N 1 linear equations to solve for N 1 unknowns. Since T 0 and T N are boundary conditions, we can substitute the value of these variables and write them on the right-hand-side of the equation. 12 0 3 23 4 34 5 32 1 21 2 20 2 NN N NNN TT T TT T T T T TTT      This set of equations can be written in matrix form, . Tr A (I'm using r as the right-hand-side vector since we will use b for a different vector later.) 10 2 3 2 1 2 100000 0 1 0 0 0 0 0 01 2 1000 0 00 0 000 0 0 00001 2 1 000001 2 N T T T           Notice that the matrix has nonzero elements on only three diagonals, the main diagonal and the two next to it. This is called a tridiagonal matrix. It is very easy (efficient) to solve this set of equations since the matrix A has many zeroes. The solution method is called the Thomas Algorithm. Let's compare the FORTRAN program for Gauss elimination and Thomas Algorithm. (The pivoting step is not shown below.) Gauss Elimination Thomas Algorithm c Forward Elimination c Forward Elimination do 5 k=1,n-1 do 5 k=1,n-1 !pivot row do 5 i=k+1,n i=k+1 ! i=row to be
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Lecture17_18wn_NucE309FA11 - NucE 309, Fall 2011, Lectures...

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