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Because it's wrong.porcupineman23 said:Hey
I don't understand this backward euler solution, in particular why the f(y,t+h) is equal to y(t+2)
I agree with you there!Chestermiller said:Because it's wrong.
Actually, it should be y(t+h)=y(t)+h*y'(t+h). Given that y'(t)=-y(t), this becomes y(t+h)=y(t)-h*y(t+h), or y(t+h)=y(t)/(1+h).It should be:
y(t+2)=y(t)+y'(t+2)h
Better: ##y(t+2)=\frac {y(t)}{3}##.Solving for y(t+2) gives:
[tex]y(t+2)=\frac{y(t)}{1+2h}[/tex]
Chet
Thanks DH. I'm usually more careful about checking over what I've written before I submit my replies. I hope I didn't confuse the OP too much.D H said:I agree with you there!
Actually, it should be y(t+h)=y(t)+h*y'(t+h). Given that y'(t)=-y(t), this becomes y(t+h)=y(t)-h*y(t+h), or y(t+h)=y(t)/(1+h).
Better: ##y(t+2)=\frac {y(t)}{3}##.
You've already set h=2.
The implicit Euler method is a numerical integration technique used to solve ordinary differential equations. It is a first-order method, meaning that the error decreases linearly with the step size. Unlike the explicit Euler method, it uses the function evaluations at the endpoints of each time step to approximate the solution, making it more stable for stiff systems.
The implicit Euler method is used by first discretizing the continuous time domain into smaller time steps. Then, the differential equation is rewritten in terms of the solution at the end of the time step, instead of the beginning. This results in a nonlinear equation that can be solved iteratively using numerical methods, such as the Newton-Raphson method, to find the solution at the next time step.
There are a few advantages of using the implicit Euler method. Firstly, it is more stable for stiff systems, meaning that it can handle equations with large changes in the solution over small time intervals. Secondly, it is unconditionally stable, meaning that the step size can be chosen arbitrarily without causing the solution to blow up. Lastly, it is relatively simple to implement and can be easily adapted to handle a wide range of differential equations.
While the implicit Euler method has many advantages, there are also some drawbacks to consider. One of the main drawbacks is that it is a first-order method, meaning that it converges to the exact solution at a slower rate compared to higher-order methods. This can lead to a larger number of time steps needed to reach a certain level of accuracy. Additionally, the iterative nature of the method can make it computationally expensive for large systems of equations.
The implicit Euler method is a good choice for solving stiff systems of ordinary differential equations, where the explicit Euler method may fail due to instability. It is also useful when the exact solution is not known or cannot be easily calculated. However, it may not be the best choice for problems with fine time scales or when higher accuracy is required, as other numerical methods may be more efficient in these cases.