  A novel discontinuous Galerkin (DG) method is developed to solve
time-dependent bi-harmonic type equations involving fourth derivatives in one
and multiple space dimensions. We present the spatial DG discretization based
on a mixed formulation and central interface numerical fluxes so that the
resulting semi-discrete schemes are $L^2$ stable even without interior penalty.
For time discretization, we use Crank-Nicolson so that the resulting scheme is
unconditionally stable and second order in time. We present the optimal $L^2$
error estimate of $O(h^{k+1})$ for polynomials of degree $k$ for semi-discrete
DG schemes, and the $L^2$ error of $O(h^{k+1} +(\Delta t)^2)$ for fully
discrete DG schemes. Extensions to more general fourth order partial
differential equations and cases with non-homogeneous boundary conditions are
provided. Numerical results are presented to verify the stability and accuracy
of the schemes. Finally, an application to the one-dimensional Swift-Hohenberg
equation endowed with a decay free energy is presented.
