Numerical Solutions to the Wave Equation Flashcards
What is used in complex wave solutions?
Numerical methods e.g. finite difference method
How is a finite difference method achieved?
Place a regular grid of points over the domain of interest
The solution to the wave equation will be obtained at the grid points
The continuous derivatives δ^2/ δt^2 and ∇^2 are replaced with the finite difference approximations
What is first-order accurate backward difference scheme?
by looking at f_j - f_ j -1 over Δx
What is first-order accurate central difference scheme?
f_j+1 - f_j -1 over Δx
What is first-order accurate forward difference scheme?
f_j+1 - f_j over Δx
How are finite difference schemes derived?
Via a Taylor series
What does the number of terms kept in the Taylor series define?
The accuracy of the scheme
What is Big-O notation?
O (Δx^n)
It is used to indicate truncation error
What is the error bound by?
const multiplied by Δx^n
What happens if Δx is reduced by a factor of 2?
The truncation error will reduce by a factor of 2^n
where n is equal to the order accurate e.g. if n = 1 it is first order
What is the finite difference scheme?
It incorporates an approximation to the space and time derivatives in the wave equation
How do we know when the finite difference scheme is accurate?
When it converges (doesn’t change any more) : we can gradually decrease Δx and Δt until the simulation result
Why does convergence work?
δ^2f(x)/ δx^2 = f(x - Δx) - 2f(x) + f(x + Δx) / Δx^2
What is the rule of thumb about the amount of grid points which ensure convergence?
10 - 20 grid points per wavelength (Δx = c_0 /PPW_max)
When does numerical dispersion occur?
When the finite difference approximation introduces a frequency-dependent sound speed (c(k)) (truncation error)
As the wave equation is built up from sum of plane waves with different frequencies, different components of wave will travel at different sound speeds, which changes the wave shape over time
What is one way of reducing numerical dispersion?
Use a higher order finite difference formulae
Reducing grid spacing (Δx) and time step (Δt)
Why do numerical methods become unstable?
Due to the restriction on the size of the time step
What is the CFL number?
CFL = c_0 Δt / Δx
(applies in 1D only)
What is the stability condition for higher dimensions?
c_0 Δt / Δx <= 1/√D
D = number of dimensions (e.g 2 for 2D)
What does the Lax’s equivalence theorem state?
That a numerical scheme will be convergent if it is also consistent and stable:
consistency (reduce to the continuous equations in the limit) + stability (doesn’t grow without bound) = convergence
When are numerical methods used in medical physics?
Material properties are not uniform
Wave propagation is nonlinear
Acoustic absorption follows a frequency power law
