Implicit Methods for Drag-Dependent Acceleration in Euler Integration

In summary, the conversation discusses the use of the backwards Euler method to integrate the equation of motion and the challenges posed by the dependence of acceleration on the new position and velocity. The method is iterative, with the first estimate usually being the forward Euler method. The conversation also covers different approaches to calculating the acceleration at time "i+1" and the potential use of extrapolation.
  • #1
RH10
6
0
I'm trying to write a code to implement he backwards Euler method to integrate the equation of motion. The sticking point seems to be that the acceleration is due to drag, and thus is dependent on the new position and velocity.

I understand the method to be:

[itex]v_{i+1}=v_{i}+a_{i+1}δ[/itex]
[itex]x_{i+1}=x_{i}+v_{i+1}δ[/itex]

With only the current conditions I can’t evaluate [itex]a_{i+1}[/itex] and am stuck.

Any help on how the implicit methods work would be really appreciated.


I’ve considered using [itex]a_{i+1}= a_{i}+ (a_{i}-a_{i-1}){δ} [/itex] but then that’s not really an implicit method is it? – you’re simply reusing the acceleration from last time around.

Thanks in advance,
 
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  • #2
The backwards Euler method is iterative. The first guess is usually the forward Euler method, so the right-hand side is determined using the information at time "i". You now have a first estimate of the properties at time "i+1". You can now use these estimates in the right-hand side to get a better estimate of the properties at time "i+1".

so for your example, first use a and v at time "i", then you have v and x at time "i+1". Now calculate a at "i+1" and recalculate v an x, but using your new estimate of a and v at "i+1":

first iteration:
[itex]v^1_{i+1}=v_{i}+a_{i}\Delta t[/itex]
[itex]x^1_{i+1}=x_{i}+v_{i}\Delta t[/itex]
calculate [itex]a_{i+1}=f(v^1_{x+1})\Delta t[/itex]
second iteration:
[itex]v^2_{i+1}=v_{i}+a^1_{i+1}\Delta t[/itex]
[itex]x^2_{i+1}=x_{i}+v^1_{i+1}\Delta t[/itex]
 
  • #3
You take a guess at the starting value, and (with luck) the interations will converge to a better value. If they don't converge, try a smaller time step.

RH10 said:
I’ve considered using [itex]a_{i+1}= a_{i}+ (a_{i}-a_{i-1}){δ} [/itex]

That's not right, because the "dimensions" don't make sense. Multiplying ##a## by ##\delta## gives you a quantity that is a ##v##, not an ##a##.

You could just start with ##a_{i+1} = a_i##, or if you want to extrapolate, the right fomula is ##a_{i+1} = a_i + (a_i-a_{i-1}) = 2a_i - a_{i-1}##.

Whether it's "better" to extrapolate or not will depend on the problem you are solving and the size of the time steps.
 

1. What is Implicit Euler Integration?

Implicit Euler Integration is a numerical method used to approximate the solution of ordinary differential equations (ODEs). It is an implicit method, meaning that it takes into account the future values of the ODEs in order to approximate the next step.

2. How does Implicit Euler Integration differ from Explicit Euler Integration?

Explicit Euler Integration is a numerical method for solving ODEs that only considers the current values of the ODEs to approximate the next step. Implicit Euler Integration, on the other hand, also takes into account the future values, making it more accurate but also more computationally intensive.

3. What are the advantages of using Implicit Euler Integration?

Implicit Euler Integration has several advantages, including stability, accuracy, and the ability to handle stiff ODEs. It also allows for larger time steps compared to other implicit methods, which can significantly reduce computational time.

4. What are the limitations of Implicit Euler Integration?

One limitation of Implicit Euler Integration is that it requires the solution of a nonlinear equation at each time step, which can be computationally expensive. It can also introduce some numerical errors, especially if the time step is too large.

5. How can I choose the best time step for Implicit Euler Integration?

The optimal time step for Implicit Euler Integration depends on the specific ODE being solved. Generally, a smaller time step will result in a more accurate solution, but it also increases the computational time. It is recommended to test different time steps and choose the one that balances accuracy and computational efficiency.

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