Numerical Solutions to PDEs Flashcards
(25 cards)
What is exponential decay?
When a quantity decreases at a rate which is proportional to the quantity.
What are some examples of exponential decay?
- Radioactive Decay
- Rates of chemical transfer
- Heat transfer
How are PDEs found using numerical approximation?
By approximating the derivative using small but finite differences
How is radioactive decay described?
Using a differential equation which gives rate of change of number of nuclei
What is advection?
The transport of a quantity by a velocity field
Why are boundary conditions required?
In order to calculate a value for the gradient at t=1 extra information is required.
What is the advection equation?
Its is a partial differential equation containing a time derivative and one or more spatial derivatives
What is time stepping used for?
It is used to calculate a solution from initial conditions such as for radioactive decay
How is a spatial derivative calculated?
Using the finite difference approximation
What is diffusion?
The net movement of a quantity from a region of high concentration to a region of lower concentration.
How is diffusion caused?
It is caused by the random movement of molecules in liquids and gases
How are decay equations made more accurate?
Decreasing the time steps when approximating
How is the error in a finite difference expression quantified?
Using the Taylor Series
What is the truncation error?
The error present in a finite difference expression as found using a Taylor series. A truncation error is first order, second order, etc depending on the power of the first Δx expression.
What does the truncation error depend on?
The size of the truncation error depends on the spacing between grid points
What makes a solution stable?
If the solution is guaranteed to remain finite
What make a solution unstable?
If errors in the solution can grow without bound
What are three factors affecting stability?
- The equation being solved
- How the spatial derivatives are calculated
- How the time steps are calculated
What are the three types of stability?
- Unconditionally Stable
- Unconditionally Unstable
- Conditionally Stable
What is an unconditionally stable solution?
A solution where errors are damped and the solution is always stable
What is an Unconditionally unstable solution?
A solution where errors are amplified and the solution is always unstable
What is a conditionally stable solution?
A solution that is stable subject to a constraint
What happens if a forward-Euler timestep is combined with a centred differences for the spatial derivetive?
The solution becomes unconditionally unstable when applied toa advection equations
How can an advection equation be kept stable with a forward-Eular timestep?
Using a one-sided spatial difference results in stable solution assuming the size of the time step is less than C(Δx/Vmax).
