In this article, we discuss the nonlinear stability and convergence of a fully discrete Fourier pseudospectral method coupled with specially designed second‐order time‐stepping for numerical solution “good” Boussinesq equation. Our analysis improves existing results presented in earlier literature two ways. First, time‐derivative are obtained instead time‐derivative, given De Frutos, et al., Math Comput 57 (1991), 109–122. addition, prove that is unconditionally stable convergent time step...

In this paper we propose and analyze an energy stable numerical scheme for the square phase field crystal (SPFC) equation, a gradient flow modeling dynamics at atomic scale in space but on diffusive scales time. particular, modification of free potential to standard model leads composition 4-Laplacian regular Laplacian operators. To overcome difficulties associated with highly nonlinear operator, design algorithms based structures individual terms. A Fourier pseudo-spectral approximation is...

The long-time stability properties of a few multistep numerical schemes for the two-dimensional incompressible Navier--Stokes equations (formulated in vorticity-stream function) are investigated this article. These semi-implicit use combination explicit Adams--Bashforth extrapolation nonlinear convection term and implicit Adams--Moulton interpolation viscous diffusion term, up to fourth order accuracy time. As result, only two Poisson solvers needed at each time step achieve desired temporal...

Some Gronwall-Bellman-Gamidov type integral inequalities with power nonlinearity and their weakly singular analogues are established, which can give the explicit bound on solution of a class nonlinear fractional equations. An example is presented to show application for qualitative study solutions equation Riemann-Liouville operator.

We present an efficient second-order finite difference scheme for solving the 2D sine-Gordon equation, which can inherit discrete energy conservation undamped model theoretically.Due to semi-implicit treatment nonlinear term, it leads a sequence of coupled equations.We use linear iteration algorithm, solve them efficiently, and contraction mapping property is also proven.Based on truncation errors numerical scheme, convergence analysis in l 2 -norm investigated detail.Moreover, we carry out...

Some linear and nonlinear Gamidov type integral inequalities in two variables are established, which can give the explicit bounds on solutions to a class of Volterra-Fredholm equations. examples application presented show boundedness uniqueness equation.

We present a second-order in time linearized semi-implicit Fourier pseudospectral scheme for the generalized regularized long wave equation.Based on consistency analysis, nonlinear stability and convergence of are discussed, along with priori assumption an aliasing error control estimate.The numerical examples demonstrate features proposed scheme, including order, conservative properties, evolution unstable wave.

In this paper, we discuss the nonlinear stability and convergence of a fully discrete Fourier pseudospectral method coupled with specially designed second order time-stepping for numerical solution "good" Boussinesq equation. Our analysis improves existing results presented in earlier literature two ways. First, an $l_\infty(0, T^*; H2)$ l_2)$ time-derivative are obtained instead H^{-2})$ time-derivative, given [17]. addition, is shown to be unconditional time step terms spatial grid size,...

<p style='text-indent:20px;'>The existence and uniqueness of Gevrey regularity solutions for the functionalized Cahn-Hilliard (FCH) Cahn-Hilliard-Willmore (CHW) equations are established. The energy dissipation law yields a uniform-in-time <inline-formula><tex-math id="M1">\begin{document}$ H^2 $\end{document}</tex-math></inline-formula> bound solution, polynomial patterns nonlinear terms enable one to derive local-in-time solution with regularity. A careful calculation reveals that time...

Abstract We propose and analyze a second order accurate in time, energy stable numerical scheme for the strongly anisotropic Cahn–Hilliard system, which biharmonic regularization has to be introduced make equation well‐posed. A convexity analysis on interfacial is necessary overcome an essential difficulty associated with its highly nonlinear singular nature. The backward differentiation formula temporal approximation applied, combined Fourier pseudo‐spectral spatial discretization. surface...

The global well-posedness analysis for the three dimensional dynamic Cahn-Hilliard-Stokes (CHS) model is provided in this paper.In model, velocity vector determined by phase variable both Darcy law and Stokes equation.Based on of weak solutions to CHS equation standard Galerkin method, we present a time strong solution model.Moreover, existence uniqueness are also proven.

In this paper we propose and analyze a (temporally) third order accurate exponential time differencing (ETD) numerical scheme for the no-slope-selection (NSS) equation of epitaxial thin film growth model, with Fourier pseudo-spectral discretization in space. A linear splitting is applied to physical an ETD-based multistep approximation used integration corresponding equation. addition, Douglas-Dupont regularization term, form $-A \dt^2 \phi_0 (L_N) \Delta_N^2 ( u^{n+1} - u^n)$, added scheme....

Some new weakly singular versions of discrete nonlinear inequalities are established, which generalize some existing and can be used in the analysis Volterra type difference equations with kernels. A few applications to upper bound uniqueness solutions also involved.

We present an efficient second-order finite difference scheme for solving the 2D sine-Gordon equation, which can inherit discrete energy conservation undamped model theoretically. Due to semi-implicit treatment nonlinear term, it leads a sequence of coupled equations. use linear iteration algorithm, solve them efficiently, and contraction mapping property is also proven. Based on truncation errors numerical scheme, convergence analysis in $l^2$-norm investigated detail. Moreover, we carry...

In this paper we propose and analyze an energy stable numerical scheme for the Cahn-Hilliard equation, with second order accuracy in time fourth finite difference approximation space. particular, truncation error long stencil approximation, over a uniform grid periodic boundary condition, is analyzed, via help of discrete Fourier analysis instead standard Taylor expansion. This turn results reduced regularity requirement test function. temporal apply BDF stencil, combined extrapolation...

In this paper we present a second order numerical scheme for the Cahn-Hilliard equation, with Fourier pseudo-spectral approximation in space. An additional Douglas-Dupont regularization term is introduced, which ensures energy stability. The bound of solution H2h and l? norms are obtained at theoretical level. Moreover, global nature method, propose linear iteration algorithm to solve non-linear system, due implicit treatment term. Some simulations verify efficiency algorithm.

We present a variant of second order accurate in time backward differentiation formula schemes for the Cahn-Hilliard equation, with Fourier collocation spectral approximation space. A three-point stencil is applied temporal discretization, and concave term diffusion treated explicitly. An addition-al Douglas-Dupont regularization introduced, which ensures energy stability mild requirement. Various numerical simulations including verification accuracy, coarsening process decay rate are...