Lec: FD2
12 important questions on Lec: FD2
Define the advec/diff eq in a different way, showing the divergence of two fluxes (q). Why is this useful?
Change at certain point is in a way the derivative of the flux, which means it is the balance of what comes in/out. Thus flux q is displacing the state variable, which is visible in the displaced peak.
What happens to the peak of the advection/diffusion eq if only diffusion is increased?
What happens to the peak of the advection/diffusion eq if both diffusion and advection are increased?
Advection thus only displaces
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What is the minus sign in the advection/diffusion equation?
What are i and n used for?
n = time
Give the definition and properties of the Crank-Nicolson method. Also write down the stencil and the formula
Combines properties of im- and explicit methods: smoothness of implicit method and CN gives more accuracy while remaining stable (in contrast to explicit)
Explain the definitions and difference between unstability, and oscillation
Oscillatnig: fluctuating around the correct solution, but never becomes the correct solution; may have varying amplitudes.
DIfference: Error is not blowing up over time, it is just consistently wrong.
Why use matrix-vector algebra and matrix inversion?
Give the formula for a matrix inversion using the terms M, v and s
M^-1 *M*s = M^-1 * v
I * s = M^-1 * v
s = M^-1 * v
Why BCs only at corners of matrix?
When solving system, the end solution will look like the graphs (see notes).
BCs just apply at one single point, but it propagates into the system when you start solving it.
What is needed in order to solve a model (6)?
- BCs (Dirichlet, Neumann, Robin)
- Initial conditions
- Parameter values
- Numerical method
- Duration
- Domain description
Explain and give the formulas of the Dirichlet, Neumann and Robin BCs?
Neumann: no flow/gradient of the state at the boundary has a fixed value. Neumann is used when we want to control flux over the boundary
Robin: linear combi of Dirichlet and Neumann
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