  The recently introduced polynomial time integration framework proposes a
novel way to construct time integrators for solving systems of first-order
ordinary differential equation by using interpolating polynomials in the
complex time plane. In this work we continue to develop the framework by
introducing several additional types of polynomials and proposing a general
class of construction strategies for polynomial block methods with imaginary
nodes. The new construction strategies do not involve algebraic order
conditions and are instead motivated by geometric arguments similar to those
used for constructing traditional spatial finite differences. Moreover, the
newly proposed methods address several shortcomings of previously introduced
polynomial block methods including the ability to solve dispersive equations
and the lack of efficient serial methods when parallelism cannot be used. To
validate our new methods, we conduct two numerical experiments that compare the
performance of polynomial block methods against backward difference methods and
implicit Runge-Kutta schemes.
