A novel discontinuous Galerkin (DG) method is developed to solve timedependent 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 L2
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.