Chapter 5: Integration of Ordinary Differential Equations Flashcards
Why are single step methods often preferred?
1) They are self-starting. They do not need initial conditions for times earlier than t0. In fact, to predict the solution at time t1, one would only need y0, y0_dot, whereas a multistep method wold also need y-1,…,y-1_dot,…
2) They tend to produce better quality results when integrating problems with a ceratin amount of discontuinity, since the dependence on several previous times tends to introduce some sort of “regularization” in the numerical solution when discontuinities in the input are present.W
When is a method implicit or explicit?
When the solution yk at time tk depends on the value of the derivative yk at time tk, namely yk_dot, the method is implicit. An implicit method required b0 to be different than zero. Otherwise, when b0 = 0, the method is explicit.
Define accuracy.
With accuracy we indicate the capability of a numerical integration scheme to integrate a polynomial of a given order. A method with accuracy n can exactly integrate polynomials up to the nth order.
Describe stability.
The term stability refers to the stability of the numerical process that estimates the solution. A method that is not algorithmically stable could result in a divergent solution when the physical problem that is being solved would instead result in a non-diverging one. In other terms, it could yield a false positive.
There is no guarantee for the opposite case: a method that is algorithmically stable may provide a non-divergent solution when the physical problem is actually unstable. In other terms, algorithmic stability may result in a false negative.
Stability may be conditional or unconditional. Conditional stability occurs when it is verified only for a range of time steps that extends from zero to a limit value. This is the case of explicit integration schemes. When stability is only conditional, the size of the time step is limited by stability considerations irrespective of any accuracy consideration, as algorithmic stability is mandatory for a successful integration.
Implicit integration schemes on the contrary, can have unconditional stability. When stability is unconditional, the size of the time step can be chosen also according to accuracy considerations, as algorithmic stability is always guaranteed.
What is zero-stability, A-stability and L-stability?
1) Zero-Stability
We have zero stability when the numerical method, applied to a stable proble, is not divergent for h->0. This means that stability (ρ<=1) must exist for an interval of time steps h that as a lower bound extend to 0. Zero stability formalizes the notion of condition stability. Zero-stability means that it must be possible to achieve algorithmic stability by reducing the size of the time-step.
2) A-stability
We have A-stability when the numerical method, applied to a stable problem, is zero-stable and not divergent irrespective of the value of h, namely |ρ|<=1 for |hλ| ->+oo. A-stability formalizes the notion of unconditional stability. A-stability means that algorithmic stability is guaranteed regardless of the time step size.
3) L-stability
We have L-stability when the numerical method, applied to a stable problem, is A-stable and the norm of the spectral radius tends to zero as |hλ| ->+oo. The rationale is that the norm of the spectral radius must reduce and eventually and tend to zero as the time step h grows enough for the method to be no longer able to yield an accurate solution. L-stability means that not only algorithic stability is guaranteed regardless of the time step size, but also the larger the time step is, the faster any peturbation vanishes.
What are the first and second Dahlquist barriers?
1) First Dahlquist barrier
A zero-stable n-step method cannot be more accurate than n+1, for n odd, or than n+2 for n even. Moreover, if the method is explicit, it cannot be more accurate than n.
2) Second Dahlquist barrier
Explicit multistep methods cannot be A-stable, only implicit ones can. Those that are, can only achieve order 2 accuracy. Among them, the A-stable method with the smallest residual error ir the one named after Crank-Nicolson (a1 = 1, b0=b1=1/2)
What are some useful integration methods?
1) Crank-Nicolson
This methos is only used when the highest possible accuracy is needed and the accuracy requirement can be satisfied for the fastest possible dynamics. It cannot be used to integrate system of DAEs, as is the case of constrained dynamics formulated according to the RCS approach.
2) Two-step BDF
This method is a second-order accurate and L-stable. It can be used to integrates systems of DAEs. It provides a considerable amount of algorithmic dissipation, at the cost of reduced accuracy.
3) Two-step method with tunable algorithmic dissipation.
Two-step, second-order accurate algorithm with tunable algorithmic dissipation. A reasonable trade-off is ρ00 = 0.6, which corresponds to a minimum in the third-order residual.
What is the predictor-corrector?
The so-called predictor-corrector approach can be used to transform the integration of the implicit differential problem into a solution of a nonlinear algrebraic one.
First of all, the derivative of the state is predicted, ie extrapoalted to time tk based on the solution at previous times yielding y_dot(0). Different strategies can be used. Then the selected integration methos is used to predict the state y(0).
