16.920J/SMA 5212 Numerical Methods for PDEs 4. Substituting the ax's into u, +2au +=S+S2+S,+S+.yields 2△x(1“ △x2·l,+ higher- order terms In other words, l--+0(△x2)+ dx 0(△x2),p i.e. the above representation is accurate up to O(4r) Some useful points to note These 4 steps are the general procedure used to obtain the representation of the spatial operator up to the order of accuracy O(ArP) d2u For other spatial operators, say we simply replace/du in(1)with dx d the said spatial operator 3. For one-sided representations, one can choose nodal points u,k,k>0. This may be important especially for representations on a boundary. For example +a+am+a2+…=0(△) dt One possibility is dx 2Ax which is also second-order accurate (We can also use a similar procedure to construct the finite difference scheme of Hermitian type for a spatial operator. This is not covered here) 616.920J/SMA 5212 Numerical Methods for PDEs 6 4. Substituting the αk ’s into 1 1 2 3 4 1 .... k j k j k k u α u S S S S = + =− ′ + = + + + + ￾ yields ( ) 2 1 1 1 1 2 6 j j j j u u u x u x + − ′ ′′′ − − = ∆ ⋅ + ∆ higher-order terms In other words, ( ) 1 1 2 .... 2 j j j j du u u u O x dx x + − − ✁ ✂ ′ = = + ∆ + ✄ ☎ ∆ ✆ ✝ i.e. the above representation is accurate up to ( ) 2 O ∆x . Some useful points to note: 1. These 4 steps are the general procedure used to obtain the representation of the spatial operator up to the order of accuracy ( ) p O ∆x . 2. For other spatial operators, say 2 2 j d u dx ✞ ✟ ✠ ✡ ☛ ☞ , we simply replace j du dx ✌ ✍ ✎ ✏ ✑ ✒ in (1) with the said spatial operator. 3. For one-sided representations, one can choose nodal points uj+k , k ≥ 0 . This may be important especially for representations on a boundary. For example 0 1 1 2 2 .... ( ) p j j j j du u u u O x dx α α + α + ✓ ✔ + + + + = ∆ ✕ ✖ ✗ ✘ One possibility is ( ) 1 2 2 3 4 2 j j j j du u u u O x dx x − + + + ✙ ✚ + = ∆ ✛ ✜ ∆ ✢ ✣ which is also second-order accurate. (We can also use a similar procedure to construct the finite difference scheme of Hermitian type for a spatial operator. This is not covered here). ( ), 2 p O ∆x p =