Given a linear ODE y* = 2𝑦. Starting from 𝑦0 = 1, perform three explicit Euler steps with step size ℎ = 0.5!
yi+1 = yi + h*f(yi)
2
4
8
Which quality criteria do we have to evaluate integration schemes?
convergence: do approximations converge to true solutions
accuracy: how fast does the error decrease
stability: is the solution always bounded
efficiency: is a given method a good choice for a given problem
When is an integrator called stable, conditionally-stable or unstable?
If previously introduced discretization errors are not amplified by the integration scheme, then it is stable
If stability is influenced by the time step, the scheme is conditionally stable
If the scheme is stable or unstable for arbitrary time steps, it is unconditionally stable or unstable
Why can't we have arbitrarily small time steps?
not feasible due to round-off errors
What is the difference between local and global integration errors?
global error: accumulated error
local error: error in a single step
How many ODE evaluations does an Euler integrator need per step?
1 evaluation
How many ODE evaluations does a fourth-order Runge-Kutta integrator need per step?
4 evaluations
Is it better to do 4 Euler steps with a quarter of the step size or one large Runge-Kutta step?
difficult to compare:
depends strongly on problem
not a single best method
Show that the explicit Euler method is conditionally-stable!
only stable if time step is restriced
|1+h*lambda|<=1
Show that the implicit Euler method is stable!
stable for any timestep
1/(|1-h*lambda|) <=1
Why is the implicit Euler method more expensive than the explicit Euler method?
because a system of algebraic equations with many unkowns must be solved
What is a stiff ODE?
require small time steps for stable integration
Which integrator would you recommend for a stiff ODE: explicit or implicit?
implicit
What are the problems with the Newton method?
too large step size
local extrema in function
How can we accelerate the Newton method?
semi implicit euler
Name and explain two numerical integration methods for higher-order ODEs!
Leapfrog: alternate between calculating velocity and position with half step size
Symplectic Euler: same as leapfrog but for only full timesteps
What is the difference between symplectic Euler and semi-implicit Euler for second-order ODEs?
there is no difference
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