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Lec: FD1

Q: What is the difference between the backward method in space and the implicit method in time? What do they have in common?

Implicit: we have to use values we don't know yet, so we have to invert the equaton to figure that out Backward: much simpler in a way, because we already have the full series we need in order to calculate it, so we can calculate the derivative directly as we already have all the points in space. There's no need for implicit type of methods here. The common thing: taylor series is evaluated backwards from the starting point.

16 important questions on Lec: FD1

Explain how the implicit method is executed in the graph in 2 steps

Check out the derivative/slope at the next time step so that it ends up exactly at the starting value
Draw a graph from the next time step, using the slope from this time step, towards the starting point

Name the X most important properties of the implicit method

It is much smoother than the explicit method
It generally overestimates a little (in the example)

Explain how the midpoint method is executed in the graph in steps

First an explicit method is used, where the slope from the starting point is extrapolated to 0.5dt
Then the implicit method is used, where the slope from 1dt to the endpoint of the explicit method in 0.5dt is met
Finally, from the final endpoint at dt, a line is drawn directly back towards the starting point. Thus an 'average' is hereby made.

Name the 3 qualities of the implicit method:

Smooth and stable
Computationally expensive
Difficult for non-linear equations

Name the 3 qualities for the midpoint method

Best of both worlds, especially the Runge-Kutta method
Extra step leads to smaller errors
Computationally more expensive, but often worth it

How can you see if your model results are good enough? Imagining there's no red line or grey slope bars on the general graph?

Smoothness/not too much fluctuation
Calculating the error term (only possible exactly if we have the 2nd and 3rd derivative - which often we do not have)
Convergence!

What is convergence in the determination of model performance following Chiels' two explanations? Does convergence mean you have a good model?

Everytime we reduce the timestep, I would like the solution to get better, up to a point where I don't see the difference anymore between two timesteps = convergence
If model gives a solution that becomes independent of the timestep = convergence

For a model to be converged, it should be on the red line (although this is invisible in most cases).

When a model converges, it means it is a good model

What relation does the midpoint method have with convergence and why?

Midpoint method converges with dt^2 rather than dt.
Thus, if timestep is halved, error gets 4x smaller in the midpoint method. The first order im- and explicit method cancel each other out.

What is a stable model? Does this mean the model is a good model?

In a stable model, the error does not increase over time
A stable model does not necessarily mean the model is good. It only means the model is not exploding.

In unstable models, the error increases with every timestep and blows up exponential over time. This may start with oscillation.

What is the difference between the backward method in space and the implicit method in time? What do they have in common?

Implicit: we have to use values we don't know yet, so we have to invert the equaton to figure that out
Backward: much simpler in a way, because we already have the full series we need in order to calculate it, so we can calculate the derivative directly as we already have all the points in space. There's no need for implicit type of methods here.

The common thing: taylor series is evaluated backwards from the starting point.

Explain based on the derivation of the centered method, why the centered method is more accurate

First error term is cancelled out by subtraction of the forward and backward method, hence the error is now proportional to dx^2 instead of dx. Thus if the error is halved, the centered method becomes 4x as good, compared to only 2x as good for the back- and forward methods.

What are the up/downsides of the back/for/centered methods when compared in a sinus graph?

Forward and backward get the peak right (although this is a bit of luck as well)
Forward estimation is constantly left of the correct solution (as it uses the future state to estimate the current)
Backward estimation is constantly right of the curve
Centered has the right timing, but dampens out at the peaks due to wide step

*derivative of a sin = cos

What happens when with the forward and centered solutions if the stepsize (dx) is reduced in the sin graph?

Halving the stepsize leads to much quicker convergence for the centered method

How can we approximate the second derivative of finite differences space?

Add the forward and backward method so that the first derivative cancels out

Why would it be useful to calculate the second derivative and how can this be done?

2nd derivative can be calculated using Taylor series, it would be useful to approach diffusivity

Which methods of time and space converge fastest and why?

Midpoint (time) and centered (space), because X

The question on the page originate from the summary of the following study material:

Water and Air Flow Numerical Techniques

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Lec: FD1

16 important questions on Lec: FD1

Explain how the implicit method is executed in the graph in 2 steps

Name the X most important properties of the implicit method

Explain how the midpoint method is executed in the graph in steps

Name the 3 qualities of the implicit method:

Name the 3 qualities for the midpoint method

How can you see if your model results are good enough? Imagining there's no red line or grey slope bars on the general graph?

What is convergence in the determination of model performance following Chiels' two explanations? Does convergence mean you have a good model?

What relation does the midpoint method have with convergence and why?

What is a stable model? Does this mean the model is a good model?

What is the difference between the backward method in space and the implicit method in time? What do they have in common?

Explain based on the derivation of the centered method, why the centered method is more accurate

What are the up/downsides of the back/for/centered methods when compared in a sinus graph?

What happens when with the forward and centered solutions if the stepsize (dx) is reduced in the sin graph?

How can we approximate the second derivative of finite differences space?

Why would it be useful to calculate the second derivative and how can this be done?

Which methods of time and space converge fastest and why?

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