1. ## Backward Euler Method

Hi,

Can someone please tell me how to use the Backward Euler Method (BEM) manually like the step by step process for the equation, in determining it with a step size of say 0.1 to find the y1 when the inital value of y0 = 2:

dy/dt = 1 - y

I have already done it for Euler's method but I have no idea how to do it for BEM. If someone could refer to a book or something that would be awesome as well

Thanks,
Sam.

2. Some good books on Numerical methods are:

Numerical Recipes in Fortran 77 by Press, Teukolsky et al. (Good on algorithms and their implementation)

A first course in the numerical analysis of differential equations by Iserles. (A good book for the theoretical understanding)

Numerical solution of partial differential equations by Smith (very good on finite difference methods).

The simplest BE method is ofcourse a 1st order scheme which for your problem will be:

$y_{n+1} - y_{n} = \Delta t (1 - y_{n+1})$,

$\Leftrightarrow (1+ \Delta t) \, y_{n+1} -\Delta t = y_{n}$.

As you can see the only difference between the Backwards and Forwards Euler methods is where the derivative function is evaluated. In the forward case it is evaluated at the $n^{\text{th}}$ step whilst for the backwards case it is at the $(n+1)^{\text{th}}$ step. Hence for the backwards case it is an implicit equation that needs to be solved that usually requires the inversion of a matrix.

3. Originally Posted by the_doc
Hence for the backwards case it is an explicit equation that needs to be solved that usually requires the inversion of a matrix.
Implicit

CB

4. Yes, I meant to write implicit.

Thanks for the pedantry - though it doesn't affect the solution.