Unformatted text preview: 2b), we see that the forward timecentered space scheme
for the heat conduction equation (2.3.1a) is consistent.
Let's emphasize the time dependence of the partial di erential equation by switching
notation from P u = 0 to ut = Lu (3.1.4) where L is a spatial di erential operator, e.g., Lu uxx for the heat conduction equation
(2.3.1a) and Lu ;aux for the kinematic wave equation (2.2.1a). 4 Basic Theoretical Concepts The solution u of the partial di erential equation can be regarded as being an element
of a function space. We'll not be too precise about the nature of such function spaces
at the moment, but simply note that there is a positive scalar, called a norm, that is
associated with many function spaces and which is used to measure the \size" of member
functions and the \distances" between them. De nition 3.1.4. The norm kuk of a function u is a scalar that satis es:
1. kuk 0 and kuk = 0 if and only if u = 0,
2. k uk = j jkuk for any constant , and
3. ku + vk kuk + kv k.
Remark 4. Condition 2 is called the condition of homogeneity and Condition 3 is the
triangular inequality.
The two norms that are most suited to our purposes are the maximum norm kuk1 m2ax ju(x t)j
x
and the Euclidean or L2 norm kuk2 Z u(x t)2dx]1=2 : (3.1.5a) (3.1.5b) In both instances, is the spatial domain of the partial di erential equation. As noted
in Section 1.2, norms often give us useful estimates of the growth or decay of solutions
with time.
While solutions u of the partial di erential equation are regarded as elements of a
function space, we may think of nite di erence solutions as elements of a linear vector
space. To this end, we'll collect all of the unknowns at a given time level n into a vector
Un. For example, the unknowns at the time level n for the forward timecentered space
n
scheme (2.3.2b) for the heatconduction problem (2.3.1) are U1n, U2n, : : : , UJ ;1 hence,
we may de ne the vector Un U1n U2n : : : UJn;1]T : (3.1.6a) 3.1. Consistency, Convergence, Stability 5 The superscript T denotes transposition. Similarly, the unknowns at a time level n for
n
a problem that is periodic in x on an interval (0 X ) might be U0n , U1n, : : : , UJ ;1 and we
would introduce the vector Un U0n U1n : : : UJn;1]T : (3.1.6b) Using this notation, we'll write the nite di erence approximation as a matrix equation of the form Un+1 = L Un: (3.1.7a) An inhomogeneous di erence equation would have the form Un+1 = L Un + f n: (3.1.7b) where the vector f n is independent of Un. We'll ignore this complication at present.
Example 3.1.4. The matrix form of the forward timecentered space di erence scheme
(2.3.3) for the heatconduction problem (2.3.1) is 2 n+1 3 2 1 ; 2r r
U
6 U1n+1 7 6 r 1 ; 2r r
6 2. 7 = 6
r 1 ; 2r r
6 .. 7 6
6
4
56
...
4
n+1
U
J ;1 r 1 ; 2r 32
76
76
76
74
7
5 3 U1n
U2n 7
7 ... 7 :
5 (3.1.8) UJn;1 Elements not shown in the above matrix are zero thus, L is a J ; 1 J ; 1 tridiagonal
matrix.
You may recall the de nition of a linear vector space 4]. De nition 3.1.5. V is a linear vector space if
1. U, V 2 V then U + V 2 V and
2. is a scalar constant and U 2 V then U 2 V . The properties of a norm on a vector space are identical to those for a function space,
but we repeat them for completeness. De nition 3.1.6. The norm kUk of a vector U is a scalar that satis es: 6 Basic Theoretical Concepts
1. kUk 0 and kUk = 0 if and only if U = 0,
2. k Uk = j jkUk for any constant , and
3. kU + Vk kUk + kVk. Again, the two norms that most suit our purposes are the maximum norm kUnk1 mjax jUjnj
and the Euclidean or L2 norm kUnk2 X (Ujn )2]1=2 : (3.1.9a) (3.1.9b) j The range on j is over all components of the vector Un.
Matrix norms provide estimates of the \size" of the discrete operator L and the
growth or decay of Un with n. They are typically de ned by their e ect on vector
norms. De nition 3.1.7. The norm kAk of a matrix A induced by a vector norm is the scalar
Az
kAk kmk6ax0 kkzkk = kmax kAzk:
(3.1.10)
z=
zk=1
The matrix norm satis es the properties of a vector norm as well as kABk kAkkBk (3.1.11a) for any matrices A and B.
Using (3.1.10), we immediately see that kAzk kAkkzk: (3.1.11b) This inequality also follows from (3.1.11a) with B replaced by the vector z.
The maximum and Euclidean norms of a matrix are...
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This document was uploaded on 03/16/2014 for the course CSCI 6840 at Rensselaer Polytechnic Institute.
 Spring '14
 JosephE.Flaherty

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