A Mixed Discontinuous Galerkin Method Without Interior Penalty for Time-Dependent Fourth Order Problems

Abstract

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 + (∆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.

Unconditionally Energy Stable DG Schemes for the Swift–Hohenberg Equation

“…Symmetrization. The idea in [19] is to apply the mixed DG discretization without interior penalty to a symmetrized mixed formulation. For the fourth order PDE (1.1), we let L = − ∆ + a 2 so that the model admits the following form…”

“…DG discretization. The mixed semi-discrete DG scheme (2.2) was presented in [19] for one and two dimensional rectangular meshes. Here we extend it to a unified form valid for more general meshes and different boundary conditions, and further study its energy dissipation property.…”

“…To extend the results in [19] to general meshes we need to recall some conventions. Let the domain Ω be a union of shape regular meshes T h = {K}, with the mesh size h K = diam{K} and h = max K h K .…”

“…, where t n = n∆t with ∆t being the time step. This scheme is shown in [19] to preserve the energy dissipation law in the sense that…”

“…For the spatial discretization, we exploit and further extend the mixed discontinuous Galerkin method introduced in [19]. The method involves three ingredients: (a) rewriting the scalar equation into a symmetric system called mixed formulation; (b) applying the DG discretization to the mixed formulation using only central fluxes on interior cell interfaces; and (c) weakly enforcement of boundary conditions of types as listed in (1.3) through both u and the auxiliary variable q = − ∆ + a 2 u.…”