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(Solved): Implicit Trapezoid Method: We would like to show that the implicit Trapezoid method for numerical OD ...
Implicit Trapezoid Method: We would like to show that the implicit
Trapezoid method for numerical ODEs,
y_(i+1)=y_(i)+(h)/(2)(f(t_(i),y_(i))+f(t_(i+1),y_(i+1)))
has a local error that is O(h^(3)) and a global error of O(h^(2)). (Lecture 11 will help with
following the procedure for this.)
(a) Take a Taylor series of y^(')(t_(i)) at t=t_(i+1).
(b) Find the local error and global error of the implicit Trapezoid method. Note that
you will need to assume y_(i)=y(t_(i)),f(t_(i),y_(i))=y^(')(t_(i)), and f(t_(i+1),y_(i+1))=y^(')(t_(i+1)).
You will need to take a Taylor series of y_(i)=y(t_(i)) and y_(i)^(')=y^(')(t_(i)).