A new embedded variable stepsize, variable order family of low computational complexity
It addresses the computational complexity and implementation difficulty of VSVO methods for solving ODEs in complex applications.
The paper introduces a family of implicit, embedded, variable stepsize variable order (VSVO) methods that require only one BDF solve per time step, achieving orders two to four with the same computational complexity as variable stepsize BDF3.
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.