# Solving Backward Euler with Newton's Method

• jacki
In summary, the conversation discusses implementing backward euler implicit method using Newton-raphson iteration for a simple planar pendulum. The differential equation for the pendulum is given by (1) and the update function for implicit integration is given by (2). The question is about the f in Newton's method and whether it refers to the residual of the pendulum function or the update step. Clarification is also sought about the notation used for u_ and velocity_.
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)?

Need to be clear about the notation first.

Is your u_ the approximation for angle ?

Is your velocity_ the approximation for angularVelocity (angle) ?

## What is Backward Euler method?

Backward Euler method is a numerical method used for solving ordinary differential equations. It is an implicit method where the derivative at the current time step is approximated using the value at the next time step.

## What is Newton's Method?

Newton's Method is an algorithm for finding the roots of a function. It involves an iterative process of using a tangent line to approximate the root of a function and then refining the approximation until a desired level of accuracy is achieved.

## How does Backward Euler with Newton's Method work?

Backward Euler with Newton's Method combines the Backward Euler method with Newton's Method to solve ordinary differential equations. The Backward Euler method is used to approximate the derivative at the next time step, and then Newton's Method is used to find the root of the resulting equation. This process is repeated until the desired level of accuracy is achieved.

## What are the advantages of using Backward Euler with Newton's Method?

One advantage of using Backward Euler with Newton's Method is that it can handle stiff equations, which are equations with rapidly changing solutions. It also has good stability properties and can handle a wide range of initial conditions.

## What are the limitations of Backward Euler with Newton's Method?

One limitation of Backward Euler with Newton's Method is that it can be computationally expensive, especially for large systems of equations. It also requires the initial guess for the root to be close to the actual root in order to converge to the correct solution.

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