2
$\begingroup$

Are there BDF formulas like the ones given here but one that can be used with implicit-explicit discretization? The right hand side in those formulas is supposed to be implicitly discretized at the current time step. Can I, for instance, replace that slope with multistage Runge-Kutta approximation of the slope and expect it to work? For example,

  • BDF2: $$\frac{3}{2}[y_n - \frac{4}{3} y_{n-1} + \frac{1}{3} y_{n-2}] = h[f(t_{n},y_{n})], $$
  • Combined BDF2-RK: $$\frac{3}{2}[y_n - \frac{4}{3} y_{n-1} + \frac{1}{3} y_{n-2}] = h\big[G\big(f(t_{n-1},y_{n-1}),...,f(t_{n-1}+\Delta t,y_{n-1}+\Delta y))\big],$$ where G does the Runge-Kutta approximation of the slope at the current time step.

Also, are there higher-order BDF formulas for second derivates? I could only find a first-order formula.

$\endgroup$
  • $\begingroup$ Welcome to Scicomp.SE! If you have more than one question (like the above), it's usually best to ask each one separately. $\endgroup$ – David Ketcheson Nov 5 '15 at 6:06
2
$\begingroup$

IMEX methods based on BDF methods have been around for a long time; for a relatively recent discussion take a look at this paper by Ascher, Ruuth, & Spiteri.

There are higher-order accurate BDF methods, but I don't think that is what you mean. The term BDF is generally understood to refer to a class of methods for first-order differential equations. If you are interested in multistep methods for second-order differential equations, take a look at Section III.10 of the text by Hairer, Norsett, & Wanner (vol. 1). Another related but different class of methods are the "second derivative BDF methods", which use both the first and second derivative; these are discussed in Section V.3 of the same book.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.