A mixed discontinuous Galerkin method without interior penalty for time-dependent fourth order problems
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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 +(Δ 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.
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