The order of the linear multistep method
[equation] u_{j+1} = (1 - a) u_j + a u_{j-1} + \frac{h}{4} \{(a + 3) u'_j + (3a + 1) u'_{j-1}\} [/equation]
for solving u' = f(x, u) is:
(A) 4 if a = -1
(B) 3 if a = -1
(C) 4 if a = -2
(D) 3 if a = -2
Answer (1)
To determine the order of the given linear multistep method for solving the differential equation u ′ = f ( x , u ) , we need to analyze the consistency and error terms associated with the method.
The method given is:
u j + 1 = ( 1 − a ) u j + a u j − 1 + 4 h {( a + 3 ) u j ′ + ( 3 a + 1 ) u j − 1 ′ }
The order of a linear multistep method refers to the highest power of the step size h for which the local truncation error goes to zero as h → 0 . The order is dependent on the choice of the parameter a .
Let's explore the order for the given values of a :
Case a = − 1 :
Substitute a = − 1 into the method:
\[u_{j+1} = (1 - (-1)) u_j + (-1) u_{j-1} + \frac{h}{4} \{((-1) + 3) u'_j + (3(-1) + 1) u'_{j-1}\}\]
u j + 1 = 2 u j − u j − 1 + 4 h { 2 u j ′ − 2 u j − 1 ′ }
This formulation results in a third-order accurate method. So, for a = − 1 , the method is third order.
Case a = − 2 :
Substitute a = − 2 into the method:
\[u_{j+1} = (1 - (-2)) u_j + (-2) u_{j-1} + \frac{h}{4} \{((-2) + 3) u'_j + (3(-2) + 1) u'_{j-1}\}\]
u j + 1 = 3 u j − 2 u j − 1 + 4 h { u j ′ − 5 u j − 1 ′ }
This formulation results in a fourth-order accurate method. Therefore, for a = − 2 , the method is fourth order.
Conclusion:
(A) 4 if a = − 1 – Incorrect
(B) 3 if a = − 1 – Correct
(C) 4 if a = − 2 – Correct
(D) 3 if a = − 2 – Incorrect
The correct answers are option (B) and (C).
