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Lec: Introduction + FD

18 important questions on Lec: Introduction + FD

Write down the finite difference definition of a derivative

See

Write down the Taylor series to derive finite difference approximations and errors

See

Why is the Taylor series used in FD?

Extrapolation of a function
Allow for the construction of operators
Approximation of the accuracy of the forward finite differences operator (allow for the estimation of errors)

What is the forward finite difference operator

Or could also be the more specific case of just the right hand side with s(t+dt)-s(t)/dt

What are 2 characteristics of the FD method?

Computation of gradients using finite steps
Valid for a single point in time (or space)

What are 3 characteristics of the explicit method (Euler forward)?

Computationally cheap
Straightforward for non-linear equations
Instabilities can develop for too large steps
Calculates future based on now

Write down the analytical solution of the ODE used for the simple reservoir (only outlet)?

See

How can you visually check if you have an exponential equation for a graph?

Plot it on a semilogarithmic plot!
X: normal axis
Y: log axis

The graph approaches zero but never reaches it

What does explicit mean in this case? How is it also known?

We take values of the current timestep in order to predict the next timestep, so we only use information we know.

Euler forward

Given are the equation (dh/dt = alpha*h) and the general equation for the calculation of the derivative. Write this down into a specific case of Euler forward

See

Give the general equation to calculate the derivative, if no analytical solution would be available (Euler forward)

See

Write down the general Taylor series and the Euler forward version, using states (s) and time (t)

See

What are the consecutive terms in the Taylor series called?

Approximation of derivative; derivative; error of approximation, incl dominant error term (first one)

The derivative should be larger/smaller than the error? Why?

Larger, if the error would be larger the resulting value would not contain much information.

What is the relation between the error terms and dt? What is the name of this relationship?

The first error term is proportional to dt, thus if the timestep is halved, the error term is halved as well

For the first, dominant error term, this is called first order approximation

Why are errors introduced by the FD definition of a derivative?

Because in FD, finite steps are taken instead of infitesimal small steps. This means errors will be introduced by taking finite steps.

Explain in words what the steps are of the FD approach that occur in a graph?

Check the derivative/slope at t=0
Linear expansion (Multiply by timestep)

Why can we not use the mathematical definition of a derivative, that gives the perfect answer?

It assumes you will take infetesimal stepsizes (in this case in time), which is impossible in a model, because it would take to infinity to solve it.

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Water and Air Flow Numerical Techniques

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Lec: Introduction + FD

18 important questions on Lec: Introduction + FD

Write down the finite difference definition of a derivative

Write down the Taylor series to derive finite difference approximations and errors

Why is the Taylor series used in FD?

What is the forward finite difference operator

What are 2 characteristics of the FD method?

What are 3 characteristics of the explicit method (Euler forward)?

Write down the analytical solution of the ODE used for the simple reservoir (only outlet)?

How can you visually check if you have an exponential equation for a graph?

What does explicit mean in this case? How is it also known?

Given are the equation (dh/dt = alpha*h) and the general equation for the calculation of the derivative. Write this down into a specific case of Euler forward

Give the general equation to calculate the derivative, if no analytical solution would be available (Euler forward)

Write down the general Taylor series and the Euler forward version, using states (s) and time (t)

What are the consecutive terms in the Taylor series called?

The derivative should be larger/smaller than the error? Why?

What is the relation between the error terms and dt? What is the name of this relationship?

Why are errors introduced by the FD definition of a derivative?

Explain in words what the steps are of the FD approach that occur in a graph?

Why can we not use the mathematical definition of a derivative, that gives the perfect answer?

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