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?
- 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)
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What is the forward finite difference operator
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)?
How can you visually check if you have an exponential equation for a graph?
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?
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 ![]()
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?
For the first, dominant error term, this is called first order approximation
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?
- 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?
The question on the page originate from the summary of the following study material:
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