Solving Backward Euler with Newton's Method

[jacki]

Greetings, I am trying to implement backward euler implicit method by Newton-raphson iteration. The differential equation is for a simple planar pendulum. So the function for the pendulum is :

(1) angularAcceleration (angle) = ( -gravity/length ) * sin(angle);

and the update function for implicit integration is:

(2) u_(t+1) = u_t + deltaTime * velocity_(t+1)

velocity_(t+1) = velocity_t + deltaTime * angularAcceleration ( u_(t+1) )

My question is: given that Newton's method is stated as x_(i+1) = x_i - f( x_i) / f ' (x_i) ,
what is the f in Newton's method? Is it the residual of the pendulum function given by (1) or the residual of the update step given by (2)?

Thanks in advance!

 

[matematikawan]

Need to be clear about the notation first.

Is your u_ the approximation for angle ?

Is your velocity_ the approximation for angularVelocity (angle) ?