I'm solving a system of stiff ODEs, at first I wanted to implement BDF, but it seem to be a quite complicated method, so I decided to start with Backward Euler method. Basically it says that you can solve an ODE: y′=f(t,y), with initial guess y(t_0)=y_0 Using the following approximation: y_(k+1)=y_k+hf(t_(k+1),y_(k+1)), where h is a step size on parameter t

gemauert79 2022-09-23 Answered
I'm solving a system of stiff ODEs, at first I wanted to implement BDF, but it seem to be a quite complicated method, so I decided to start with Backward Euler method. Basically it says that you can solve an ODE:
y = f ( t , y ), with initial guess y ( t 0 ) = y 0
Using the following approximation:
y k + 1 = y k + h f ( t k + 1 , y k + 1 ), where h is a step size on parameter t
Wikipedia article says that you can solve this equation using Newton-Raphson method, which is basically a following iteration:
x n + 1 = x n g ( x n ) g ( x n )
So, the question is how to correctly mix them together? What initial guess x 0 and function g should be?
Also f is quite complex in my case and I'm not sure if it possible to find another derivative of it analytically. I want to write an implementation of it by myself, so pblackefined Mathematica functions wouldn't work.
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Answers (1)

Nancy Phillips
Answered 2022-09-24 Author has 12 answers
Your aim is to solve the following equation for y k + 1 :
g ( y k + 1 ) := y k + 1 y k h f ( t k + 1 , y k + 1 ) = 0 ,
where f is a known function and y k , t k + 1 and h are known values. This gives you the g for a Newton-Raphson method. As an initial guess, I'd suggest that you use a 1-step forward Euler method to explicitly calculate
y ^ k + 1 := y k + h f ( t k + 1 , y k ) ,
and use y ^ k + 1 as your initial guess for y k + 1 .
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