16.920J/SMA 5212 Numerical Methods for PDEs where S=(a1+a+a)u S2=(1-△xa1+△xa1) +-△ S4=-△x2a1+△xa1| S1+S2+S3+S4+ 3. Make as many S,'s as possible vanish by choosing appropriate as In this instance, since we have three unknowns a_, a and a, we can therefore set (Note that in the Taylor Series expansion, one starts off with the lower-order terms and progressively obtain the higher-order terms. We have deliberately set the s pertaining to the lower-order terms to zero, thereafter followed by ncreasingly higher-order terms) 0 0 0 Solving the 2△16.920J/SMA 5212 Numerical Methods for PDEs 5 where ∴ 1 1 2 3 4 1 .... k j k j k k u α u S S S S = + =− ′ + = + + + + ￾ 3. Make as many Si ’s as possible vanish by choosing appropriate αk ’s. In this instance, since we have three unknowns α−1 , α0 and α1 , we can therefore set: 1 2 3 0 0 0 S S S = = = (Note that in the Taylor Series expansion, one starts off with the lower-order terms and progressively obtain the higher-order terms. We have deliberately set the i S pertaining to the lower-order terms to zero, thereafter followed by increasingly higher-order terms.) Hence, 1 0 1 0 1 1 1 1 1 0 1 1 0 1 0 x α α α − ✁ ✂ ✁ ✂✄✁ ✂ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ − = − ☎ ✆ ☎ ✆ ∆ ☎ ✆ ☎ ✆ ☎ ✆ ✝ ✞✄✝ ✞ ☎ ✆ ✝ ✞ Solving the system of equations, we obtain 1 0 1 1 2 0 1 2 x x α α α − = ∆ = = − ∆ ( ) ( ) 1 1 0 1 2 1 1 2 2 3 1 1 3 3 4 1 1 1 1 1 2 2 1 1 6 6 j j j j S u S x x u S x x u S x x u α α α α α α α α α − − − − = + + ′ = −∆ ⋅ + ∆ ⋅ ✟ ✠ ′′ = ∆ ⋅ + ∆ ⋅ ✡ ☛ ☞ ✌ ✟ ✠ ′′′ = − ∆ ⋅ + ∆ ⋅ ✡ ☛ ☞ ✌