16.920J/SMA 5212 Numerical Methods for PDEs Let d be represented by u at the nodes j-1, J, and j*l with a-1, d and a being the coefficients to be determined ie 11-+a0u+a1u O(△xP d Seek Taylor Expansions for u;-,u, and u +r about u, and present them in a (Note that p is not known a priori but is determined at the end of the analysis when the as are made known. This column consists of all the terms on the LHS of (1) 0 a1|-△xa △x3a 0 0 a1 x·C1 △x-a1 a S S2 Each cell in this row comprises the sum of its corresponding column.16.920J/SMA 5212 Numerical Methods for PDEs 4 1. Let j du dx ￾ ✁ ✂ ✄ ☎ ✆ be represented by u at the nodes j−1, j, and j+1 with α−1 , α0 and α1 being the coefficients to be determined, i.e. 1 1 0 1 1 ( ) p j j j j du u u u O x dx α− − α α + ✝ ✞ + + + = ∆ ✟ ✠ ✡ ☛ 2. Seek Taylor Expansions for j 1 u − , j u and j 1 u + about j u and present them in a table as shown below. (Note that p is not known a priori but is determined at the end of the analysis when the α’s are made known.) uj uj ′ uj ′′ uj ′′′ uj ′ 0 1 0 0 α−1uj−1 α−1 1 −∆x α− ⋅ 2 1 1 2 ∆x α− ⋅ 3 1 1 6 − ∆x α− ⋅ α0uj α0 0 0 0 1 j 1 α u + α1 1 ∆x ⋅α 2 1 1 2 ∆x ⋅α 3 1 1 6 ∆x ⋅α 1 1 k j k j k k u α u = + =− ′ + ☞ S1 S2 S3 S4 ( 1 ) This column consists of all the terms on the LHS of (1). Each cell in this row comprises the sum of its corresponding column