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 article, we study a new second‐order energy stable Backward Differentiation Formula (BDF) finite difference scheme for the epitaxial thin film equation with slope selection (SS). One major challenge higher‐order‐in‐time temporal discretizations is how to ensure an unconditional stability without compromising numerical efficiency or accuracy. We propose framework designing using BDF method constant coefficient stabilizing terms. Based on property that establish, derive solution and...

We present a second-order-in-time finite difference scheme for the Cahn-Hilliard-Hele-Shaw equations. This numerical method is uniquely solvable and unconditionally energy stable. At each time step, this leads to system of nonlinear equations that can be efficiently solved by multigrid solver. Owing stability, we derive an $\ell^2 (0, T; H_h^3)$ stability scheme. To overcome difficulty associated with convection term $\nabla · (\phi \mathit{\boldsymbol{u}})$, perform $\ell^∞ H_h^1)$ error...

Abstract This article is to discuss the bilinear and linear immersed finite element (IFE) solutions generated from algebraic multigrid solver for both stationary moving interface problems. For numerical methods based on difference formulation a structured mesh independent of interface, stiffness matrix system usually not symmetric positive-definite, which demands extra efforts design efficient methods. On other hand, arising IFE are naturally positive-definite. Hence IFE-AMG algorithm...

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...

We present a second-order-in-time finite difference scheme for the Cahn-Hilliard-Hele-Shaw equations. This numerical method is uniquely solvable and unconditionally energy stable. At each time step, this leads to system of nonlinear equations that can be efficiently solved by multigrid solver. Owing stability, we derive an $\ell^2 (0,T; H_h^3)$ stability scheme. To overcome difficulty associated with convection term $\nabla \cdot (\phi \boldsymbol{u})$, perform $\ell^\infty H_h^1)$ error...

A discontinuous Galerkin (DG) finite‐element interior calculus is used as a common framework to describe various DG approximation methods for second‐order elliptic problems. Using the framework, symmetric interior‐penalty methods, local and dual‐wind will be compared by expressing all of in primal form. The penalty‐free nature method both motivated better understand analytic properties methods. Consideration given Neumann boundary conditions with numerical experiments that support...

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,...

A linear discrete state-space model of a methanol/water binary batch distillation column is developed based on theoretical analysis dynamic mass balance and vapor-liquid phase balance, this used to design predictive control (MPC) strategy. The composition methanol inside the estimated using an empirical temperature-composition relationship model. state space MPC algorithm presented in detail, strategy implemented industrial computer directly column. Control experiments show that gives smooth...

We present and analyze an unconditionally energy stable convergent finite difference scheme for the Functionalized Cahn-Hilliard equation. One key difficulty associated with stability is based on fact that one nonlinear functional term in expansion appears as non-convex, non-concave. To overcome this subtle difficulty, we add two auxiliary terms to make combined convex, which turns yields a convex-concave decomposition of physical energy. As result, application convex splitting methodology...

The dynamic matrix control (DMC) strategy is extended to a nonlinear system using Hammerstein model that consists of static polynomial function followed in series by linear step response element. A new DMC algorithm (named NLH-DMC) presented detail. set point tracking and disturbance rejection simulation results strong PH neutralization process show NLH-DMC gives better performance than (LDMC) the PID (NL-PID) controller commonly used industry. Further experiment demonstrates not only good...

In this paper, we study a novel second-order energy stable Backward Differentiation Formula (BDF) finite difference scheme for the epitaxial thin film equation with slope selection (SS). One major challenge higher oder in time temporal discretization is how to ensure an unconditional stability and efficient numerical implementation. We propose general framework designing order by using BDF method constant coefficient stabilized terms. Based on property, derive $L^\infty_h (0,T; H_{h}^2)$...

In this paper we analyze and implement a second-order-in-time numerical scheme for the three-dimensional phase field crystal (PFC) equation. The was proposed in [46], with unique solvability unconditional energy stability established. However, its convergence analysis remains open. We present detailed article, which maximum norm estimate of solution over grid points plays an essential role. Moreover, outline multigrid method to solve highly nonlinear cubic domain, various results are...

Some new nonlinear weakly singular difference inequalities are discussed, which generalize some known and can be used in the analysis of Volterra-type equations with kernel. An application to upper bound solutions a equation is also presented.

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.

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...