  Variable Stepsize Variable Order (VSVO) methods are the methods of choice to
efficiently solve a wide range of ODEs with minimal work and assured accuracy.
However, VSVO methods have limited impact in timestepping methods in complex
applications due to their computational complexity and the difficulty to
implement them in legacy code. We introduce a family of implicit, embedded,
VSVO methods that require only one BDF solve at each time step followed by
adding linear combinations of the solution at previous time levels. In
particular, we construct implicit and linearly implicit VSVO methods of orders
two, three and four with the same computational complexity as variable stepsize
BDF3. The choice of changing the order of the method is simple and does not
require additional solves of linear or nonlinear systems.
