Unconditionally Energy Stable DG Schemes for the Swift–Hohenberg Equation

The Swift-Hohenberg equation as a central nonlinear model in modern physics has a gradient flow structure. Here we introduce fully discrete discontinuous Galerkin (DG) schemes for a class of fourth order gradient flow problems, including the nonlinear Swift-Hohenberg equation, to produce free-energy-decaying discrete solutions, irrespective of the time step and the mesh size. We exploit and extend the mixed DG method introduced in [H. Liu and P. Yin, J. Sci. Comput., 77: 467--501, 2018] for the spatial discretization, and the "Invariant Energy Quadratization" method for the time discretization. The resulting IEQ-DG algorithms are linear, thus they can be efficiently solved without resorting to any iteration method. We actually prove that these schemes are unconditionally energy stable. We present several numerical examples that support our theoretical results and illustrate the efficiency, accuracy and energy stability of our new algorithm. The numerical results on two dimensional pattern formation problems indicate that the method is able to deliver comparable patterns of high accuracy.

[1]  Peter V. E. McClintock,et al.  Pattern formation : an introduction to methods. , 2006 .

[2]  J. Hesthaven,et al.  Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications , 2007 .

[3]  Dee Gt,et al.  Bistable systems with propagating fronts leading to pattern formation. , 1988 .

[4]  Vivi Rottschäfer,et al.  Pattern selection of solutions of the Swift–Hohenberg equation , 2004 .

[5]  Henry S. Greenside,et al.  Pattern Formation and Dynamics in Nonequilibrium Systems , 2004 .

[6]  James D. Gunton,et al.  Numerical solution of the Swift-Hohenberg equation in two dimensions , 1991 .

[7]  Axel Hutt,et al.  Analysis of nonlocal neural fields for both general and gamma-distributed connectivities , 2005 .

[8]  Victor M. Calo,et al.  An energy-stable generalized-α method for the Swift-Hohenberg equation , 2018, J. Comput. Appl. Math..

[9]  Hailiang Liu,et al.  A Mixed Discontinuous Galerkin Method Without Interior Penalty for Time-Dependent Fourth Order Problems , 2018, J. Sci. Comput..

[10]  Nikolai A. Kudryashov,et al.  Exact solutions of the Swift–Hohenberg equation with dispersion , 2011, 1112.5444.

[11]  Edgar Knobloch,et al.  Localized states in the conserved Swift-Hohenberg equation with cubic nonlinearity. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[12]  Francisco Guillén-González,et al.  On linear schemes for a Cahn-Hilliard diffuse interface model , 2013, J. Comput. Phys..

[13]  S. Sánchez Pérez-Moreno,et al.  Numerical Solution of the Swift-Hohenberg Equation , 2014 .

[14]  Proof of Quasipatterns for the Swift–Hohenberg Equation , 2017 .

[15]  Chi-Wang Shu,et al.  Discontinuous Galerkin Methods: General Approach and Stability , 2008 .

[16]  Xesús Nogueira,et al.  A new space–time discretization for the Swift–Hohenberg equation that strictly respects the Lyapunov functional , 2012 .

[17]  L. A. Peletier,et al.  Global Branches of Multi-Bump Periodic Solutions of the Swift-Hohenberg Equation , 2001 .

[18]  Jiang Yang,et al.  The scalar auxiliary variable (SAV) approach for gradient flows , 2018, J. Comput. Phys..

[19]  Xiaofeng Yang,et al.  Linear, first and second-order, unconditionally energy stable numerical schemes for the phase field model of homopolymer blends , 2016, J. Comput. Phys..

[20]  Zhengru Zhang,et al.  On a Large Time-Stepping Method for the Swift-Hohenberg Equation , 2016 .

[21]  N. Magnitskii,et al.  Nonlinear dynamics of laminar-turbulent transition in three dimensional Rayleigh–Benard convection , 2010 .

[22]  Xiaofeng Yang,et al.  Numerical approximations for a phase field dendritic crystal growth model based on the invariant energy quadratization approach , 2017 .

[23]  Manuel G. Velarde,et al.  Implicit time splitting for fourth-order parabolic equations , 1997 .

[24]  Mehdi Dehghan,et al.  The meshless local collocation method for solving multi-dimensional Cahn-Hilliard, Swift-Hohenberg and phase field crystal equations , 2017 .

[25]  Xiaofeng Yang,et al.  Decoupled energy stable schemes for phase-field vesicle membrane model , 2015, J. Comput. Phys..

[26]  William C. Troy,et al.  Spatial patterns described by the extended Fisher-Kolmogorov (EFK) equation: kinks , 1995, Differential and Integral Equations.

[27]  J. Swift,et al.  Hydrodynamic fluctuations at the convective instability , 1977 .

[28]  P. Fife,et al.  A Class of Pattern-Forming Models , 1999 .

[29]  Béatrice Rivière,et al.  Discontinuous Galerkin methods for solving elliptic and parabolic equations - theory and implementation , 2008, Frontiers in applied mathematics.

[30]  Santiago Badia,et al.  Finite element approximation of nematic liquid crystal flows using a saddle-point structure , 2011, J. Comput. Phys..

[31]  W. Bangerth,et al.  deal.II—A general-purpose object-oriented finite element library , 2007, TOMS.

[32]  Christo I. Christov,et al.  Numerical scheme for Swift-Hohenberg equation with strict implementation of lyapunov functional , 2002 .

[33]  C. Doering,et al.  New upper bounds and reduced dynamical modeling for Rayleigh–Bénard convection in a fluid saturated porous layer , 2012 .

[34]  Silvia Bertoluzza,et al.  Numerical Solutions of Partial Differential Equations , 2008 .

[35]  Hyun Geun Lee.  A semi-analytical Fourier spectral method for the Swift-Hohenberg equation , 2017, Comput. Math. Appl..