Summary of initial value problems#
Explicit methods are:
Easy to calculate.
Parallelize well.
Suffer from numerical instability.
Require small step sizes for stiff equations.
Implicit methods are:
Computationally intensive (general require root finding / linear systems).
Don’t parallelize well.
Are much more numerically stable.
Can take significantly larger step sizes without diverging.
Systems of equations are natural and ready extensions of the methods.
Reduction of order can be applied to higher order derivatives.
Adaptive time stepping is very important and can be achieved through clever (or brute force) methods without much additional expense.
The explicit Runge-Kutta methods efficiently achieve accurate estimates if numerical instability isn’t a factor (or small time steps are feasible).
Implicit RK methods require simultaneous solution of several equations which can exponenetially increase the computaitonal cost.
Explicit linear multistep methods have conditions which make them impractical.
Implicit linear multistep (Backward Differential Formulas: BDF) methods have excellent properties while exploiting the solution history.
Dr. Mike’s tips:#
Know your physics. If you don’t, go with an adaptive BDF method (the default of most software).
If you know your system is not stiff, RK45 is the go-to and you will substantially benefit in computer time.
If you know your physics has abrupt changes (e.g.: steps / pulses) consider keeping with Backward Euler - no higher-order accuracy is possible.