0 and t 0 example 313 using 312b we see that the

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Unformatted text preview: 2b), we see that the forward time-centered 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 time-centered space n scheme (2.3.2b) for the heat-conduction 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 time-centered space di erence scheme (2.3.3) for the heat-conduction 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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