  Immersed boundary methods simplify mesh generation by embedding the domain of
interest into an extended domain that is easy to mesh, introducing the
challenge of dealing with cells that intersect the domain boundary. Combined
with explicit time integration schemes, the finite cell method introduces a
lower bound for the critical time step size. Explicit transient analyses
commonly use the spectral element method due to its natural way of obtaining
diagonal mass matrices through nodal lumping. Its combination with the finite
cell method is called the spectral cell method. Unfortunately, a direct
application of nodal lumping in the spectral cell method is impossible due to
the special quadrature necessary to treat the discontinuous integrand inside
the cut cells. We analyze an implicit-explicit (IMEX) time integration method
to exploit the advantages of the nodal lumping scheme for uncut cells on one
side and the unconditional stability of implicit time integration schemes for
cut cells on the other. In this hybrid, immersed Newmark IMEX approach, we use
explicit second-order central differences to integrate the uncut degrees of
freedom that lead to a diagonal block in the mass matrix and an implicit
trapezoidal Newmark method to integrate the remaining degrees of freedom (those
supported by at least one cut cell). The immersed Newmark IMEX approach
preserves the high-order convergence rates and the geometric flexibility of the
finite cell method. We analyze a simple system of spring-coupled masses to
highlight some of the essential characteristics of Newmark IMEX time
integration. We then solve the scalar wave equation on two- and
three-dimensional examples with significant geometric complexity to show that
our approach is more efficient than state-of-the-art time integration schemes
when comparing accuracy and runtime.
