An unconditionally stable spatial sixth-order CCD-ADI method for the two-dimensional linear telegraph equation

The telegraph equation is one of the important models in many physics and engineering. In this work, we discuss the high-order compact finite difference method for solving the two-dimensional second-order linear hyperbolic equation. By using a combined compact finite difference method for the spatial discretization, a high-order alternating direction implicit method (ADI) is proposed. The method is O(τ² + h⁶) accurate, where τ, h are the temporal step size and spatial size, respectively. Von Neumann linear stability analysis shows that the method is unconditionally stable. Finally, numerical examples are used to illustrate the high accuracy of the new difference scheme.     

Efficient linear and unconditionally energy stable schemes for the modified phase field crystal equation

Science China Mathematics - In this paper, we construct efficient schemes based on the scalar auxiliary variable block-centered finite difference method for the modified phase field crystal...     

An unconditionally stable alternating direction implicit scheme for the two space dimensional linear hyperbolic equation

We report a new unconditionally stable implicit alternating direction implicit (ADI) scheme of O(k² + h²) for the difference solution of linear hyperbolic equation u_tt + 2αu_t + β²u = u_xx + u_yy + f(x, y, t), αβ ≥ 0, 0 < x, y < 1, t > 0 subject to appropriate initial and Dirichlet boundary conditions, where α > 0 and β ≥ 0 are real numbers. The resulting system of algebraic equations is solved by split method. Numerical results are provided to demonstrate the efficiency and accuracy of the method. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 684–688, 2001     

An unconditionally stable linearized difference scheme for the fractional Ginzburg-Landau equation

In this paper, we propose a linearized implicit finite difference scheme for solving the fractional Ginzburg-Landau equation. The scheme, which involves three time levels, is unconditionally stable and second-order accurate in both time and space variables. Moreover, the unique solvability, the unconditional stability, and the convergence of the method in the \(L^{\infty }\)-norm are proved by the energy method and mathematical induction. Compared with the implicit midpoint difference scheme (Wang and Huang J. Comput. Phys. 312, 31–49, 2016), current linearized method generally reduces the computational cost. Finally, numerical results are presented to confirm the theoretical results.     

An unconditionally energy stable finite difference scheme for a stochastic Cahn-Hilliard equation

In this work, the MMC-TDGL equation, a stochastic Cahn-Hilliard equation, is solved numerically by using the finite difference method in combination with a convex splitting technique of the energy functional. For the non-stochastic case, we develop an unconditionally energy stable difference scheme which is proved to be uniquely solvable. For the stochastic case, by adopting the same splitting of the energy functional, we construct a similar and uniquely solvable difference scheme with the discretized stochastic term. The resulted schemes are nonlinear and solved by Newton iteration. For the long time simulation, an adaptive time stepping strategy is developed based on both first- and second-order derivatives of the energy. Numerical experiments are carried out to verify the energy stability, the efficiency of the adaptive time stepping and the effect of the stochastic term.     

A second order splitting method for the Cahn-Hilliard equation

Summary A semi-discrete finite element method requiring only continuous element is presented for the approximation of the solution of the evolutionary, fourth order in space, Cahn-Hilliard equation. Optimal order error bounds are derived in various norms for an implementation which uses mass lumping. The continuous problem has an energy based Lyapunov functional. It is proved that this property holds for the discrete problem.Research partially supported by NSF Grant DMS-8896141     

Highly efficient and unconditionally energy stable semi-discrete time-marching numerical scheme for the two-phase incompressible flow phase-field system with variable-density and viscosity

For the variable-density/viscosity Cahn-Hilliard phase-field model of the binary-phase incompressible fluid flow system, the development of easy-to-implement numerical schemes has long been known as a challenging problem. We develop a novel fully-decoupled numerical technique in this article which can achieve unconditional energy stability while explicitly discretizing nonlinear coupling items. The idea is invented on the basis of combining the Strang operator splitting method and the novel deco...     

One projection method for linear third order equation

In this paper we study the Galyorkin method with a special basis for a linear operator-differential equation of the third order in a separable Hilbert space. The projection method is based on the eigenvectors of the operator similar to the leading operator of equation. We obtain estimates for the convergence rate of approximate solutions in uniform topology.     

New unconditionally stable scheme for telegraph equation based on weighted Laguerre polynomials

This article proposes a new unconditionally stable scheme to solve one‐dimensional telegraph equation using weighted Laguerre polynomials. Unlike other numerical schemes, the time derivatives in the equation can be expanded analytically based on the Laguerre polynomials and basis functions. By applying a Galerkin temporal testing procedure and using the orthogonal property of weighted Laguerre polynomials, the time variable can be eliminated from computations, which results in an implicit equation. After solving the equation recursively one can obtain the numerical results of telegraph equation by using the expanded coefficients. Some numerical examples are considered to validate the accuracy and stability of this proposed scheme, and the results are compared with some existing numerical schemes.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1603–1615, 2017     

Uniqueness theorem for linear elliptic equation of the second order with constant coefficients

The interior uniqueness theorem for analytic functions was generalized by M. B. Balk to the case of polyanalytic functions of order n. He proved that if the zeros of a polyanalytic function have an accumulation point of order n, then this function is identically zero. In this paper the interior uniqueness theorem is generalized to the solution to a linear homogeneous second order differential equation of elliptic type with constant coefficients.     

An unconditionally stable spline difference scheme of O(k2+h4) for solving the second-order 1D linear hyperbolic equation

In this paper, the second-order linear hyperbolic equation is solved by using a new three-level difference scheme based on quartic spline interpolation in space direction and finite difference discretization in time direction. Stability analysis of the scheme is carried out. The proposed scheme is second-order accurate in time direction and fourth-order accurate in space direction. Finally, numerical examples are tested and results are compared with other published numerical solutions.     

Analysis of second order and unconditionally stable BDF2‐AB2 method for the Navier‐Stokes equations with nonlinear time relaxation

In this study, we first consider a second order time stepping finite element BDF2‐AB2 method for the Navier‐Stokes equations (NSE). We prove that the method is unconditionally stable and accurate. Second, we consider a nonlinear time relaxation model which consists of adding a term “” to the Navier‐Stokes Equations with the algorithm depends on BDF2‐AB2 method. We prove that this method is unconditionally stable, too. We applied the BDF2‐AB2 method to several numeral experiments including flow around the cylinder. We have also applied BDF2‐AB2 method with nonlinear time relaxation to some problems. It is observed that when the equilibrium errors are high, applying BDF2‐AB2 with nonlinear time relaxation method to the problem yields lower equilibrium errors.     

An efficient method for quadratic Riccati differential equation

In this paper, a new form of homotopy perturbation method (NHPM) has been adopted for solving the quadratic Riccati differential equation. In this technique, the solution is considered as a Taylor series expansion converges rapidly to the exact solution of the nonlinear equation. Having found the exact solution of the Riccati equation, the capability and the simplicity of the proposed technique is clarified.     

On the Cauchy problem for a second order semilinear parabolic equation with factored linear part

Communicated by J. A. Goldstein