Search

In this article, we propose and analyze an energy stable, linear, second-order in time, exponential time differencing multi-step (ETD-MS) method for solving the Swift–Hohenberg equation with quadratic–cubic nonlinear term. The ETD-based explicit multi-step approximations and Fourier collocation spectral method are applied in time integration and spatial discretization of the corresponding equation, respectively. In particular, a second-order artificial stabilizing term, in the form of A τ 2 ∂ (Δ 2 + 1) u ∂ t , is added to ensure the energy stability. The long-time unconditional energy stability of the algorithm is established rigorously. In addition, error estimates in ℓ ∞ (0 , T ; ℓ 2) -norm are derived, with a careful estimate of the aliasing error. Numerical examples are carried out to verify the theoretical results. The long-time simulation demonstrates the stability and the efficiency of the numerical method. [ABSTRACT FROM AUTHOR]

In this paper, we develop a local discontinuous Galerkin (LDG) method for the Swift–Hohenberg equation. The energy stability and optimal error estimates in L 2 norm of the semi-discrete LDG scheme are established. To avoid the severe time step restriction of explicit time marching methods, a first-order linear scheme based on the scalar auxiliary variable (SAV) method is employed for temporal discretization. Coupled with the LDG spatial discretization, we achieve a fully-discrete LDG method and prove its energy stability and optimal error estimates. To improve the temporal accuracy, the semi-implicit spectral deferred correction (SDC) method is adapted iteratively. Combining with the SAV method, the SDC method can be linear, high-order accurate and energy stable in our numerical tests. Numerical experiments are presented to verify the theoretical results and to show the efficiency of the proposed methods. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyze a second-order energy stable numerical scheme for the Swift–Hohenberg equation, with a mixed finite element approximation in space. We employ second-order backward differentiation formula scheme with a second-order stabilized term, which guarantees the long time energy stability. We prove that our two-step scheme is unconditionally energy stable and uniquely solvable. Furthermore, we present an optimal error estimate for the scheme. In the end, several numerical experiments are presented to support our theoretical analysis. [ABSTRACT FROM AUTHOR]

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 Liu and 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. [ABSTRACT FROM AUTHOR]

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 L2 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 L2 error estimate of O(hk+1) for polynomials of degree k for semi-discrete DG schemes, and the L2 error of O(hk+1+(Δt)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. [ABSTRACT FROM AUTHOR]