# BDF methods for implicit-explicit method

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.

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

## 1 Answer

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.