PDEs Flashcards
(26 cards)
What is a differential equation (DE)?
What do they describe?
- A differential equation is an equation that contains one or more derivatives
- Describe the rate of change of a variable with respect to another.
What is a partial differential equation (PDE)?
A PDE is a differential equation involving derivatives with respect to more than one variable
What is the difference between time derivatives and spatial derivates?
- Time derivates represent rate of change with respect to time
- Spatial derivates represent rate of change with respect to position
What is exponential decay?
When a quantity N(t) decreases at a rate proportional to itself, described by dN/dt = -λN.
Where λ is a positive constant
How is the radioactive decay rate mathematically modeled?
dN/dt = -λN, where λ = ln(2)/halflife
What is numerical approximation in the context of PDEs? Why is it used?
- We don’t know how the rate of change will be over a large time period
- It approximates derivatives using small finite differences over a discrete time step
What is advection? Name two examples.
The transport of a quantity by a velocity field.
1. Dust in the atmosphere
2. Silt in a river
What is the 1D advection equation?
The advection of a scalar field u(x,t) by a constant velocity c is:
∂u/∂t = -c ∂u/∂x
What is the 2D advection equation?
The advection of a scalar field u(x,y,t) by a constant velocity c=(c{x}, c{y}) is:
∂u/∂t = -(c{x}∂u/∂x + c{y}∂u/∂y)
What is the Gaussian distribution’s behavior under the 1D advection equation?
The shape remains the same but shifts horizontally with time.
What is a finite difference approximation?
A numerical method that approximates derivatives using discrete values
What would be the finite difference expression to calculate du/dx at x=x{i}?
(u{i} - u{i-1})/(x{i} - x{i-1})
What would be the finite difference expression to calculate d2u/dx2 at x=x{i}?
(u{i+1} - 2u{i} + u{i-1}) / (dx)^2
What is diffusion?
The movement of a quantity from high to low concentration, modelled by the diffusion equation.
What is the 1D diffusion equation?
The diffusion of a scalar field u(x,t) is:
∂u/∂t = K ∂²u/∂x², where K is a constant.
What is the 2D diffusion eqauation?
The diffusion of a scalar field u(x,y,t) is:
∂u/∂t = -K(∂2u/∂x2 + ∂2u/∂y2), where K is a constant
What is the Gaussian’s distribution’s behaviour under the 1D diffusion equation?
The ‘bump’ ‘spreads out’ as we timestep
What is truncation error in finite difference approximations?
The error introduced by approximating a derivative with a finite difference formula.
How does reducing step size affect accuracy?
Smaller step sizes reduce truncation error, improving accuracy.
What is accuracy in PDEs?
How exact (close) your solution is to the solution
What is stable/unstable in PDEs?
- Stable - The solution must be guaranteed to remain finite
- Unstable - Errors in the solution can grow without bounds
What factors affect stability?
- The (type of) equation being solved
- How to the spatial derivatives are calculated
- How the timesteps are calculated
What are the types of stability?
Explain each
- Unconditionally Stable - errors are damped and solution is always stable
- Unconditionally Unstable - errors are amplified and the solution is always unstable
- Conditionally Stable - solution is stable subject to a constraint
What is CFL?
When solving the advection equation, the resulting scheme is stable provided:
Δt <= C (Δx / v{max}), where 0 < C < 1 is a constant
