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This paper primarily investigates the conservative Swift-Hohenberg (SH) equation with quadratic-cubic nonlinearity, where mass conservation is achieved by introducing a Lagrange multiplier. Based on Fourier spectral method and SSP-RK2 method, we ingeniously utilize operator splitting method to derive an accurate and efficient numerical scheme for solving the SH equation. The theoretical analysis of this numerical scheme, including mass conservation, convergence, and energy stability, is discussed in detail. Finally, a variety of numerical examples are presented to validate the precision and effectiveness of the scheme. [ABSTRACT FROM AUTHOR]

The Swift–Hohenberg equation (SHE) is a partial differential equation that explains how patterns emerge from a spatially homogeneous state. It has been widely used in the theory of pattern formation. Following a recent study by Bramburger and Holzer (SIAM J Math Anal 55(3):2150–2185, 2023), we consider discrete SHE on deterministic and random graphs. The two families of the discrete models share the same continuum limit in the form of a nonlocal SHE on a circle. The analysis of the continuous system, parallel to the analysis of the classical SHE, shows bifurcations of spatially periodic solutions at critical values of the control parameters. However, the proximity of the discrete models to the continuum limit does not guarantee that the same bifurcations take place in the discrete setting in general, because some of the symmetries of the continuous model do not survive discretization. We use the center manifold reduction and normal forms to obtain precise information about the number and stability of solutions bifurcating from the homogeneous state in the discrete models on deterministic and sparse random graphs. Moreover, we present detailed numerical results for the discrete SHE on the nearest-neighbor and small-world graphs. [ABSTRACT FROM AUTHOR]

In this paper we introduce the critical variational setting for parabolic stochastic evolution equations of quasi- or semi-linear type. Our results improve many of the abstract results in the classical variational setting. In particular, we are able to replace the usual weak or local monotonicity condition by a more flexible local Lipschitz condition. Moreover, the usual growth conditions on the multiplicative noise are weakened considerably. Our new setting provides general conditions under which local and global existence and uniqueness hold. In addition, we prove continuous dependence on the initial data. We show that many classical SPDEs, which could not be covered by the classical variational setting, do fit in the critical variational setting. In particular, this is the case for the Cahn–Hilliard equation, tamed Navier–Stokes equations, and Allen–Cahn equation. [ABSTRACT FROM AUTHOR]

This paper presents an unconditional energy-stable method for the Swift-Hohenberg equation over arbitrarily curved surfaces. We directly define the Laplace–Beltrami operator on the triangular mesh and its dual mesh, which are the discretizations of regular surface. The direct discretization method has several advantages including intrinsic geometry, convergence property, and mass conservation. A second-order temporal accuracy method has been proposed with the Crank-Nicolson type formulation. We have applied the stabilized splitting method to solve the governing system. Our proposed scheme is easy to implement since the two elliptic equations are both linear with low computational burden. The proposed scheme is provable to be Unconditional energy-stable. Several computational tests are conducted to demonstrate that the numerical method is efficient. [ABSTRACT FROM AUTHOR]

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 work, we first prove the existence, uniqueness and regularity of the solution of the Swift-Hohenberg equation by applying the Galerkin spectral method. Then we investigate the convergence of a finite element method in the mixed formulation for the Swift-Hohenberg equation, with Crank-Nicolson scheme in time discretization. We prove that our semidiscrete and fully discrete numerical schemes satisfy unique solvability and unconditional energy stability. Moreover, we prove optimal error estimates for the schemes. Finally, numerical tests are given to validate our theoretical results. [ABSTRACT FROM AUTHOR]

This article concerns the dynamical transitions of the stochastic Swift-Hohenberg equation with multiplicative noise on a two-dimensional domain (-L, L) × (-L, L). With α and L regarded as parameters, we show that the approximate reduced system corresponding to the invariant manifold undergoes a stochastic pitchfork bifurcation near the critical points, and the impact of noise on stochastic bifurcation of the Swift-Hohenberg equation. We find the approximation representation of the manifold and the corresponding reduced systems for stochastic Swift-Hohenberg equation when L2 and √ 2L1 are close together. [ABSTRACT FROM AUTHOR]

The main objective of this study is to provide a new computational procedure for extracting approximate and exact solutions of the temporal-fractional Swift–Hohenberg (S–H) equations in the context of conformable derivatives using the conformable natural transform (CNT) and Daftardar–Jafari method (DJM). We refer to it as the "natural conformable Daftardar–Jafari method" (CNDJM). The three types of errors are assessed in order to gauge the efficiency and consistency of the proposed method. Furthermore, 2D and 3D graphics are used to compare the exact and approximate solutions. This method offers a considerable benefit over homotopy analysis and Adomian decomposition methods in terms of computational work because it does not require Adomian and He's polynomials. The procedure is quick and easy to use. [ABSTRACT FROM AUTHOR]

For a class of fourth order gradient flow problems, integration of the scalar auxiliary variable (SAV) time discretization with the penalty‐free discontinuous Galerkin (DG) spatial discretization leads to SAV‐DG schemes. These schemes are linear and shown unconditionally energy stable. However, the reduced linear systems are rather expensive to solve due to the dense coefficient matrices. In this paper, we provide a procedure to pre‐evaluate the auxiliary variable in the piecewise polynomial space. As a result the computational complexity of ON2$$ O\left({\mathcal{N}}^2\right) $$ reduces to O(N)$$ O\left(\mathcal{N}\right) $$ when exploiting the conjugate gradient (CG) solver. This hybrid SAV‐DG method is more efficient and able to deliver satisfactory results of high accuracy. This was also compared with solving the full augmented system of the SAV‐DG schemes. [ABSTRACT FROM AUTHOR]

In this paper, we design and analyze an unconditionally energy-stable, second-order-in-time, finite element scheme for the Swift–Hohenberg equation. We prove rigorously that our scheme is unconditionally uniquely solvable and unconditionally energy stable. We also give the boundedness of discrete phase variable for any time and space mesh sizes. Numerical tests are presented to validate the accuracy and energy stability of our scheme. [ABSTRACT FROM AUTHOR]

Swift-Hohenberg equations are frequently used to model the biological, physical and chemical processes that lead to pattern generation, and they can realistically represent the findings. This study evaluates the Elzaki Adomian decomposition method (EADM), which integrates a semi-analytical approach using a novel hybridized fuzzy integral transform and the Adomian decomposition method. Moreover, we employ this strategy to address the fractional-order Swift-Hohenberg model (SHM) assuming gH-differentiability by utilizing different initial requirements. The Elzaki transform is used to illustrate certain characteristics of the fuzzy Atangana-Baleanu operator in the Caputo framework. Furthermore, we determined the generic framework and analytical solutions by successfully testing cases in the series form of the systems under consideration. Using the synthesized strategy, we construct the approximate outcomes of the SHM with visualizations of the initial value issues by incorporating the fuzzy factor ϖ∈[0,1] which encompasses the varying fractional values. Finally, the EADM is predicted to be effective and precise in generating the analytical results for dynamical fuzzy fractional partial differential equations that emerge in scientific disciplines.

In this paper, we design, analyze and numerically validate linearly first- and second-order unconditionally energy stable numerical methods for solving the Phase-Field Crystal equation and Swift–Hohenberg equation which describe a multitude of processes involving spatiotemporal pattern formation. The properties of well-posedness of solution and the decrease of the total energy for the fully discretized schemes are established. Numerical examples are presented to confirm the accuracy, efficiency, and stability of the proposed method. [ABSTRACT FROM AUTHOR]

A Cauchy problem for a modification of the Swift-Hohenberg equation in RN with a mildly integrable potential is considered. Applying the dissipative mechanism of fourth order parabolic equations in unbounded domains, it is shown that the equation generates a semigroup of global solutions possessing a global attractor in the scale of Bessel potential spaces and in H²(RN) in particular. [ABSTRACT FROM AUTHOR]

In this work, we present first- and second-order energy-stable linear schemes for the Swift-Hohenberg equation based on first-order backward Euler and Crank-Nicolson schemes, respectively. We prove rigorously that the schemes satisfy the energy dissipation property. We also derive the error analysis for our schemes. Moreover, we adopt a spectral-Galerkin approximation for the spatial variables and establish error estimates for the fully discrete second-order Crank-Nicolson scheme. Numerical results are presented to validate our theoretical analysis. [ABSTRACT FROM AUTHOR]

A Cauchy problem for a modification of the Swift-Hohenberg equation in RN with a mildly integrable potential is considered. Applying the dissipative mechanism of fourth order parabolic equations in unbounded domains, it is shown that the equation generates a semigroup of global solutions possessing a global attractor in the scale of Bessel potential spaces and in H²(RN) in particular. [ABSTRACT FROM AUTHOR]

We study the dynamical transitions of the stochastic Swift-Hohenberg equation with multiplicative noise on a one-dimensional domain (0 , L). With α and the length of the domain L regarded as parameters, we show that the approximate reduced system corresponding to the invariant manifold undergoes a stochastic pitchfork bifurcation near the critical points, and analyze the case when the bifurcation points nearly coincide. We show the impact of noise on stochastic bifurcation of the Swift-Hohenberg equation. [ABSTRACT FROM AUTHOR]

In this work, we propose and analyze a stabilized linear Crank-Nicolson scheme for the Swift-Hohenberg equation. More precisely, we treat the nonlinear term explicitly while two second-order stabilization terms are added to improve the stability of the scheme. We show that our scheme satisfies the discrete energy dissipation. We prove rigorously that our scheme is second-order accurate in time. Moreover, we adopt a spectral-Galerkin approximation for the spacial variables and establish error estimates for the fully discrete scheme. Numerical experiments are presented to show the accuracy and energy stability of our scheme. [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]

Swift-Hohenberg equations are frequently used to model the biological, physical and chemical processes that lead to pattern generation, and they can realistically represent the findings. This study evaluates the Elzaki Adomian decomposition method (EADM), which integrates a semi-analytical approach using a novel hybridized fuzzy integral transform and the Adomian decomposition method. Moreover, we employ this strategy to address the fractional-order Swift-Hohenberg model (SHM) assuming gH-differentiability by utilizing different initial requirements. The Elzaki transform is used to illustrate certain characteristics of the fuzzy Atangana-Baleanu operator in the Caputo framework. Furthermore, we determined the generic framework and analytical solutions by successfully testing cases in the series form of the systems under consideration. Using the synthesized strategy, we construct the approximate outcomes of the SHM with visualizations of the initial value issues by incorporating the fuzzy factor ϖ∈[0,1] which encompasses the varying fractional values. Finally, the EADM is predicted to be effective and precise in generating the analytical results for dynamical fuzzy fractional partial differential equations that emerge in scientific disciplines. [ABSTRACT FROM AUTHOR]

The very recently introduced Virtual Element Method (VEM) is a numerical method for solving partial differential equations that was created out of the mimetic difference method, but was later reformulated into the Galerkin framework. It is a generalization of the standard Finite Element Method (FEM) to general meshes made up by arbitrary polyhedra. The greatest advantage of VEM is its ability to deal with very complex geometries, i.e., made up by elements of any number of edges not necessarily convex, hanging nodes, flat angles, collapsing nodes, etc., while retaining the same approximation properties of FEM. In this article, the C 0 -virtual element method is formulated and analyzed to solve generalized Swift-Hohenberg equation on polygonal meshes. The Swift-Hohenberg equation as a central nonlinear model in modern physics has a gradient flow structure. Here, we present the spatial VE discretization based on a mixed formulation for the nonlinear Swift-Hohenberg equation as a class of fourth-order gradient flow problems. For time discretization, we use Crank-Nicolson so that the resulting scheme is unconditionally stable and second-order in time. By following the algebraic implementation of the discrete system, we provide numerical tests validating the theoretical estimates and plotting two-dimensional pattern formation problems. [ABSTRACT FROM AUTHOR]

In the paper, an unconditional energy-stable scheme is developed with the help of the scalar auxiliary variable (SAV) approach and the leapfrog time-discretization. The fully discrete scheme gives a linear and decoupled system, whereas previous SAV-type methods usually gave a coupled linear system. Therefore, the proposed scheme is decoupled and thus easier to be implemented. Moreover, it is proved that the proposed scheme has second-order time accuracy in the temporal direction. Several numerical examples are provided to support the proposed theoretical results. • An unconditional energy-stable scheme is developed to numerically solve the Swift-Hohenberg equations. • The proposed scheme is linearly implicit and decoupled and thus easier to be implemented. • Convergence of the semi-discrete scheme is obtained. [ABSTRACT FROM AUTHOR]

The present research paper is related to the analytical solution of fractional-order nonlinear Swift-Hohenberg equations using an efficient technique. The presented model is related to the temperature and thermal convection of fluid dynamics which can also be used to explain the formation process in liquid surfaces bounded along a horizontally well-conducting boundary. In this work Laplace Adomian decomposition method is implemented because it require small volume of calculations. Unlike the variational iteration method and Homotopy pertubation method, the suggested technique required no variational parameter and having simple calculation of fractional derivative respectively. Numerical examples verify the validity of the suggested method. It is confirmed that the present method's solutions are in close contact with the solutions of other existing methods. It is also investigated through graphs and tables that the suggested method's solutions are almost identical with different analytical methods.

In this paper, we propose a second-order BDF time marching scheme for the Swift–Hohenberg gradient flows (Swift–Hohenberg equation and phase-field crystal equation) with quadratic–cubic nonlinearity and vacancy potential. Based on the multiple scalar auxiliary variables (MSAV) approach and stabilization technique, an efficient, second-order accurate, and unconditionally energy stable numerical scheme is constructed. In this scheme, two new scalar auxiliary variables are introduced to deal with the nonlinear terms, and a linear stabilization term is added to enhance the stability and keep the required accuracy while using the large time steps. The unique solvability and unconditional energy stability are proved. Some numerical experiments are presented to demonstrate the accuracy and energy stability of the proposed scheme and show the effect of quadratic term and vacancy potential on pattern formation. [ABSTRACT FROM AUTHOR]

In this work, we develop linear, first- and second-order accurate, and energy-stable numerical scheme for the Swift–Hohenberg (SH) equation. An auxiliary variable (Lagrange multiplier) is used to control the nonlinear term so that the linear temporal scheme can be easily constructed. To further achieve the accuracy with large time steps, a proper stabilized term is adopted. For the first-order time-accurate scheme, the backward Euler approximation is adopted. The Crank–Nicolson (CN) and explicit Adams–Bashforth (AB) approximations are applied to achieve temporally second-order accuracy. We analytically perform the estimations of the semi-discrete solvability, the energy stability with respect to the original and pseudo-energy functionals, and the convergence error. To numerically solve the resulting discrete system of equations, we use an efficient linear multigrid method. We present various two- (2D) and three-dimensional (3D) computational examples to demonstrate the accuracy and energy stability of the proposed scheme. [ABSTRACT FROM AUTHOR]

The Swift-Hohenberg equation is ubiquitous in the study of bistable dynamics. In this paper, we study the dynamic transitions of the Swift-Hohenberg equation with a third-order dispersion term in one spacial dimension with a periodic boundary condition. As a control parameter crosses a critical value, the trivial stable equilibrium solution will lose its stability, and undergoes a dynamic transition to a new physical state, described by a local attractor. The main result of this paper is to fully characterize the type and detailed structure of the transition using dynamic transition theory [ 7 ]. In particular, employing techniques from center manifold theory, we reduce this infinite dimensional problem to a finite one since the space on which the exchange of stability occurs is finite dimensional. The problem then reduces to analysis of single or double Hopf bifurcations, and we completely classify the possible phase changes depending on the dispersion for every spacial period. [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]

In this paper, the nonlinear problem of the 1 D Swift-Hohenberg equation (S-HE) has been solved by using five reliable iterative methods. The first one is the Daftardar-Jafari method namely (DJM), the second method is the Temimi-Ansari method namely (TAM), the third method is the Banach contraction method namely (BCM), the fourth method is the Adomian decomposition method namely (ADM) and finally the fifth method is the Variational iteration method (VIM) to obtain the approximate solutions. In this work, we discussed and applied these iterative methods to solve the S-HE and compared them. In addition, the fixed-point theorem was given to illustrate the convergence of the five methods. To illustrate the accuracy and efficiency of the five methods, the maximum error remainder was calculated since the exact solution is unknown. The results showed that the five iterative methods are accurate, reliable, time saver and effective. All the iterative processes in this paper implemented in MATHEMATICA®11.

We study stripe formation in two-dimensional systems under directional quenching in a phase-diffusion approximation including nonadiabatic boundary effects. We find stripe formation through simple traveling waves for all angles relative to the quenching line using an analytic continuation procedure. We also present comprehensive analytical asymptotic formulas in limiting cases of small and large angles as well as small and large quenching rates. Of particular interest is a regime of small angle and slow quenching rate which is well described by the glide motion of a boundary dislocation along the quenching line. A delocalization bifurcation of this dislocation leads to a sharp decrease of strain created in the growth process at small angles. We complement our results with numerical continuation reliant on a boundary-integral formulation. We also compare results in the phase-diffusion approximation numerically to quenched stripe formation in an anisotropic Swift--Hohenberg equation. [ABSTRACT FROM AUTHOR]

We study the bifurcation analysis of the Swift–Hohenberg equation (SHE) with the odd-periodic condition as period parameter λ moves. Motivated by Peletier and Rottschäfer [Pattern selection of solutions of the Swift–Hohenberg equations. Phys D. 2004;194:95–126] and Peletier and Williams [Some canonical bifurcations in the Swift–Hohenberg equation. SIAM J Appl Dyn Syst. 2007;6:208–235], with the complete proof based on center manifold reduction, we show how the periodic SHE bifurcates from the trivial solution to an attractor when λ passes a critical number, and mainly when a gap collapsed to a point, and an overlapped interval emerges. Peletier and Williams provided the local behavior about nontrivial solutions of the SHE depending on critical lengths or the overlapped interval based on L 2 -norm. In this paper, by dropping the symmetric condition given in the paper by Peletier and Williams, we extend their results and find the explicit stationary solution of the SHE. We also present several numerical results explaining our results. [ABSTRACT FROM AUTHOR]

In this study, we develop linear and energy stable numerical schemes for the Swift–Hohenberg equation with quadratic-cubic nonlinearity. A modified scalar auxiliary variable (SAV) approach is used to construct the temporally first- and second-order accurate discretizations. Different from the classical SAV approach, the proposed schemes permit us to solve the governing equations in a step-by-step manner, i.e., the calculation of inner product is not needed. We analytically prove the energy stability. We solve the resulting system of discrete equations using the linear multigrid method. We perform various numerical examples to show the accuracy and energy stability of the proposed method. The pattern formations in two- and three-dimensional spaces are also simulated. [ABSTRACT FROM AUTHOR]

We propose a phase-field theory for enriched continua. To generalize classical phase-field models, we derive the phase-field gradient theory based on balances of microforces, microtorques, and mass. We focus on materials where second gradients of the phase field describe long-range interactions. By considering a nontrivial interaction inside the body, described by a boundary-edge microtraction, we characterize the existence of a hypermicrotraction field, a central aspect of this theory. On surfaces, we define the surface microtraction and the surface-couple microtraction emerging from internal surface interactions. We explicitly account for the lack of smoothness along a curve on surfaces enclosing arbitrary parts of the domain. In these rough areas, internal-edge microtractions appear. We begin our theory by characterizing these tractions. Next, in balancing microforces and microtorques, we arrive at the field equations. Subject to thermodynamic constraints, we develop a general set of constitutive relations for a phase-field model where its free-energy density depends on second gradients of the phase field. A priori, the balance equations are general and independent of constitutive equations, where the thermodynamics constrain the constitutive relations through the free-energy imbalance. To exemplify the usefulness of our theory, we generalize two commonly used phase-field equations. We propose a 'generalized Swift–Hohenberg equation'—a second-grade phase-field equation—and its conserved version, the 'generalized phase-field crystal equation'—a conserved second-grade phase-field equation. Furthermore, we derive the configurational fields arising in this theory. We conclude with the presentation of a comprehensive, thermodynamically consistent set of boundary conditions. [ABSTRACT FROM AUTHOR]

This article concerns the optimal distributed control for the modified Swift-Hohenberg equation. Using the Dubovitskii and Milyutin functional analytical approach, we prove the Pontryagin maximum principle of the controlled modified Swift-Hohenberg equation. A necessary optimality condition is established for the problem in fixed final time horizon case. Also we indicate how to utilize the obtained results.

In this paper, a new scheme based on the exponential fitting technique is presented for solving the nonlinear time-fractional Swift-Hohenberg equation, where the first and second-order derivatives are replaced by Caputo fractional derivative. The exponential fitting technique depends on a parameter that led to getting high-order accuracy. The convergence and unconditional stability will be discussed using the Fourier method and the analysis is built on uniform and nonuniform time steps, to solve initial singularity by using the graded meshes. Applicability and theoretical results will be demonstrated and enhanced with numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, the nonlinear problem of the 1 D Swift-Hohenberg equation (S-HE) has been solved by using five reliable iterative methods. The first one is the Daftardar-Jafari method namely (DJM), the second method is the Temimi-Ansari method namely (TAM), the third method is the Banach contraction method namely (BCM), the fourth method is the Adomian decomposition method namely (ADM) and finally the fifth method is the Variational iteration method (VIM) to obtain the approximate solutions. In this work, we discussed and applied these iterative methods to solve the S-HE and compared them. In addition, the fixed-point theorem was given to illustrate the convergence of the five methods. To illustrate the accuracy and efficiency of the five methods, the maximum error remainder was calculated since the exact solution is unknown. The results showed that the five iterative methods are accurate, reliable, time saver and effective. All the iterative processes in this paper implemented in MATHEMATICA®11. [ABSTRACT FROM AUTHOR]

This article investigates the semi-analytical method coupled with a new hybrid fuzzy integral transform and the Adomian decomposition method via the notion of fuzziness known as the Elzaki Adomian decomposition method (briefly, EADM). In addition, we apply this method to the time-fractional Swift–Hohenberg equation (SHe) with various initial conditions (IC) under gH-differentiability. Some aspects of the fuzzy Caputo fractional derivative (CFD) with the Elzaki transform are presented. Moreover, we established the general formulation and approximate findings by testing examples in series form of the models under investigation with success. With the aid of the projected method, we establish the approximate analytical results of SHe with graphical representations of initial value problems by inserting the uncertainty parameter 0≤℘≤1 with different fractional orders. It is expected that fuzzy EADM will be powerful and accurate in configuring numerical solutions to nonlinear fuzzy fractional partial differential equations arising in physical and complex structures.

This work studies the effects of rapid oscillations (with respect to time) of the forcing term on the long-time behaviour of the solutions of the Swift-Hohenberg equation. More precisely, we establish three kinds of averaging principles for the Swift-Hohenberg equation, they are averaging principle in a time-periodic problem, averaging principle on a finite time interval and averaging principle on the entire axis. [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]

In this article, we use orthogonal spline collocation methods (OSCM) for the fourth-order linear and nonlinear boundary value problems (BVPs). Cubic monomial basis functions and piecewise Hermite cubic basis functions are used to approximate the solution. We establish the existence and uniqueness solution to the discrete problem. We discuss dynamics of the stationary Swift–Hohenberg equation for different values of α. Finally, we perform some numerical experiments and using grid refinement analysis, we compute the order of convergence of the numerical method. Comparative to existing methods, we show that the OSCM gives optimal order of convergence for | | u − u h | | L p , p = 2 , ∞ norms and superconvergent result for | | u ′ − u ′ h | | L ∞ -norm at the knots with minimal computational cost. [ABSTRACT FROM AUTHOR]

We analyze a first order in time Fourier pseudospectral scheme for Swift-Hohenberg equation. One major challenge for the higher order diffusion non-linear systems is how to ensure the unconditional energy stability and we propose an efficient scheme for the equation based on the convex splitting of the energy. Theoretically, the energy stability of the scheme is proved. Moreover, following the derived aliasing error estimate, the convergence analysis in the discrete l2-norm for the proposed scheme is given. [ABSTRACT FROM AUTHOR]

A collocation Fourier scheme for Swift-Hohenberg equation based on the convex splitting idea is implemented. To ensure an efficient numerical computation, we propose a general framework with linear iteration algorithm to solve the non-linear coupled equations which arise with the semi-implicit scheme. Following the contraction mapping theorem, we present a detailed convergence analysis for the linear iteration algorithm. Various numerical simulations, including verification of accuracy, dissipative property of discrete energy and pattern formation, are presented to demonstrate the efficiency and the robustness of proposed method. [ABSTRACT FROM AUTHOR]

Localized roll patterns are structures that exhibit a spatially periodic profile in their center. When following such patterns in a system parameter in one space dimension, the length of the spatial interval over which these patterns resemble a periodic profile stays either bounded, in which case branches form closed bounded curves ("isolas"), or the length increases to infinity so that branches are unbounded in function space ("snaking"). In two space dimensions, numerical computations show that branches of localized rolls exhibit a more complicated structure in which both isolas and snaking occur. In this paper, we analyze the structure of branches of localized radial roll solutions in dimension 1+ε, with 0 < ε ≪ 1, through a perturbation analysis. Our analysis sheds light on some of the features visible in the planar case. [ABSTRACT FROM AUTHOR]

In this paper we derive rigorously the amplitude equation, using the natural separation of time-scales near a change of stability, for the stochastic generalized Swift–Hohenberg equation with quadratic and cubic nonlinearity in this form du=-(1+∂x2)2u+νεu+γu2-u3dt+σεdW, where W(t)is a Wiener process. For deterministic PDE it is known that the quadratic term generates an additional cubic term, which is unstable. We consider two cases depending on γ2. If γ2