Compact difference schemes for heat equation with Neumann boundary conditions (II)

This is the further work on compact finite difference schemes for heat equation with Neumann boundary conditions subsequent to the paper, [Sun, Numer Methods Partial Differential Equations (NMPDE) 25 (2009), 1320–1341]. A different compact difference scheme for the one‐dimensional linear heat equation is developed. Truncation errors of the proposed scheme are O(τ² + h⁴) for interior mesh point approximation and O(τ² + h³) for the boundary condition approximation with the uniform partition. The new obtained scheme is similar to the one given by Liao et al. (NMPDE 22 (2006), 600–616), while the major difference lies in no extension of source terms to outside the computational domain any longer. Compared with ones obtained by Zhao et al. (NMPDE 23 (2007), 949–959) and Dai (NMPDE 27 (2011), 436–446), numerical solutions at all mesh points including two boundary points are computed in our new scheme. The significant advantage of this work is to provide a rigorous analysis of convergence order for the obtained compact difference scheme using discrete energy method. The global accuracy is O(τ² + h⁴) in discrete maximum norm, although the spatial approximation order at the Neumann boundary is one lower than that for interior mesh points. The analytical techniques are important and can be successfully used to solve the open problem presented by Sun (NMPDE 25 (2009), 1320–1341), where analyzed theoretical convergence order of the scheme by Liao et al. (NMPDE 22 (2006), 600–616) is only O(τ² + h^3.5) while the numerical accuracy is O(τ² + h⁴), and convergence order of theoretical analysis for the scheme by Zhao et al. (NMPDE 23 (2007), 949–959) is O(τ² + h^2.5), while the actual numerical accuracy is O(τ² + h³). Following the procedure used for the new obtained difference scheme in this work, convergence orders of these two schemes can be proved rigorously to be O(τ² + h⁴) and O(τ² + h³), respectively. Meanwhile, extension to the case involving the nonlinear reaction term is also discussed, and the global convergence order O(τ² + h⁴) is proved. A compact ADI difference scheme for solving two‐dimensional case is derived. Finally, several examples are given to demonstrate the numerical accuracy of new obtained compact difference schemes. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

The numerical analysis of two linearized difference schemes for the Benjamin–Bona–Mahony–Burgers equation

In the article, two linearized finite difference schemes are proposed and analyzed for the Benjamin–Bona–Mahony–Burgers (BBMB) equation. For the construction of the two-level scheme, the nonlinear term is linearized via averaging k and k + 1 floor, we prove unique solvability and convergence of numerical solutions in detail with the convergence order O(τ² + h²) . For the three-level linearized scheme, the extrapolation technique is utilized to linearize the nonlinear term based on ψ function. We obtain the conservation, boundedness, unique solvability and convergence of numerical solutions with the convergence order O(τ² + h²) at length. Furthermore, extending our work to the BBMB equation with the nonlinear source term is considered and a Newton linearized method is inserted to deal with it. The applicability and accuracy of both schemes are demonstrated by numerical experiments.     

The stability and convergence of two linearized finite difference schemes for the nonlinear epitaxial growth model

The numerical simulation of the dynamics of the molecular beam epitaxy (MBE) growth is considered in this article. The governing equation is a nonlinear evolutionary equation that is of linear fourth order derivative term and nonlinear second order derivative term in space. The main purpose of this work is to construct and analyze two linearized finite difference schemes for solving the MBE model. The linearized backward Euler difference scheme and the linearized Crank‐Nicolson difference scheme are derived. The unique solvability, unconditional stability and convergence are proved. The linearized Euler scheme is convergent with the convergence order of O(τ + h²) and linearized Crank‐Nicolson scheme is convergent with the convergence order of O(τ² + h²) in discrete L²‐norm, respectively. Numerical stability with respect to the initial conditions is also obtained for both schemes. Numerical experiments are carried out to demonstrate the theoretical analysis. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

Compact difference schemes for heat equation with Neumann boundary conditions

In this article, two recent proposed compact schemes for the heat conduction problem with Neumann boundary conditions are analyzed. The first difference scheme was proposed by Zhao, Dai, and Niu (Numer Methods Partial Differential Eq 23, (2007), 949–959). The unconditional stability and convergence are proved by the energy methods. The convergence order is O(τ² + h^2.5) in a discrete maximum norm. Numerical examples demonstrate that the convergence order of the scheme can not exceeds O(τ² + h³). An improved compact scheme is presented, by which the approximate values at the boundary points can be obtained directly. The second scheme was given by Liao, Zhu, and Khaliq (Methods Partial Differential Eq 22, (2006), 600–616). The unconditional stability and convergence are also shown. By the way, it is reported how to avoid computing the values at the fictitious points. Some numerical examples are presented to show the theoretical results. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Unconditional superconvergent analysis of a new mixed finite element method for Ginzburg–Landau equation

In this article, unconditional superconvergent analysis of a linearized fully discrete mixed finite element method is presented for a class of Ginzburg–Landau equation based on the bilinear element and zero‐order Nédélec's element pair (Q₁₁/Q₀₁ × Q₁₀). First, a time‐discrete system is introduced to split the error into temporal error and spatial error, and the corresponding error estimates are deduced rigorously. Second, the unconditional superclose and optimal estimate of order O(h² + τ) for u in H¹‐norm and p = ?u in L²‐norm are derived respectively without the restrictions on the ratio between h and τ, where h is the subdivision parameter and τ, the time step. Third, the global superconvergent results are obtained by interpolated postprocessing technique. Finally, some numerical results are carried out to confirm the theoretical analysis.     

Two energy-conserving and compact finite difference schemes for two-dimensional Schrödinger-Boussinesq equations

In this paper, we study two compact finite difference schemes for the Schrödinger-Boussinesq (SBq) equations in two dimensions. The proposed schemes are proved to preserve the total mass and energy in the discrete sense. In our numerical analysis, besides the standard energy method, a “cut-off” function technique and a “lifting” technique are introduced to establish the optimal H¹ error estimates without any restriction on the grid ratios. The convergence rate is proved to be of O(τ² + h⁴) with the time step τ and mesh size h. In addition, a fast finite difference solver is designed to speed up the numerical computation of the proposed schemes. The numerical results are reported to verify the error estimates and conservation laws.

     

Numerical analysis of semi-linear parabolic systems in diffusion–reaction problems

In this paper, we investigate problems of approximation for the solution of a system of coupled semi-linear parabolic partial differential equations that model diffusion-reaction problems in chemical engineering. Given that the solutions belong to H^s (0, ∞), we consider finite-element approximations on bounded domains (0, R(h)) such that lim_h→0[R(h)] = ∞. Optimal convergence estimates are found to depend on the asymptotic behaviour of the solution and its regularity near t = 0. In the L²-norm, they cannot exceed an order of O((;h²/t^3/4) + h²[In h]²). For that purpose, a Wheeler-type argument is also generalized to non-coercive elliptic forms. Fully discrete schemes that preserve the positivity of the solutions are also considered. Due to the singularity at t = 0, they lead to estimates of the order O(τ^1/4 + h²/t^3/4).     

Recursive polynomial interpolation algorithm (RPIA)

Two difference schemes are derived for numerically solving the one-dimensional time distributed-order fractional wave equations. It is proved that the schemes are unconditionally stable and convergent in the \(L^{\infty }\) norm with the convergence orders O(τ ² + h ²+Δγ ²) and O(τ ² + h ⁴+Δγ ⁴), respectively, where τ,h, and Δγ are the step sizes in time, space, and distributed order. A numerical example is implemented to confirm the theoretical results.     

Local discontinuous Galerkin methods with explicit Runge‐Kutta time marching for nonlinear carburizing model

A fully discrete local discontinuous Galerkin (LDG) method coupled with 3 total variation diminishing Runge‐Kutta time‐marching schemes, for solving a nonlinear carburizing model, will be analyzed and implemented in this paper. On the basis of a suitable numerical flux setting in the LDG method, we obtain the optimal error estimate for the Runge‐Kutta–LDG schemes by energy analysis, under the condition τ ≤ λh², where h and τ are mesh size and time step, respectively, λ is a positive constant independent of h. Numerical experiments are presented to verify the accuracy and capability of the proposed schemes. For the carburizing diffusion processes of steel and the diffusion simulation for Cu‐Ni system, the numerical results show good agreement with the experimental results.     

Superconvergence analysis of a linearized three‐step backward differential formula finite element method for nonlinear Sobolev equation

A linearized three‐step backward differential formula (BDF) Galerkin finite element method (FEM) is developed for nonlinear Sobolev equation with bilinear element. Temporal error and spatial error are discussed through introducing a time‐discrete system. Solutions of the time‐discrete system are bounded in H²‐norm by the temporal error. Superconvergence results of order O(h² + τ³) in H¹‐norm for the original variable are deduced based on the spatial error. Some new tricks are utilized to get higher order of the temporal error and the spatial error. At last, two numerical examples are provided to support the theoretical analysis. Here, h is the subdivision parameter, and τ is the time step.     

Optimal l~∞ error estimates of finite difference methods for the coupled Gross-Pitaevskii equations in high dimensions

Due to the difficulty in obtaining the a priori estimate,it is very hard to establish the optimal point-wise error bound of a finite difference scheme for solving a nonlinear partial differential equation in high dimensions(2D or 3D).We here propose and analyze finite difference methods for solving the coupled GrossPitaevskii equations in two dimensions,which models the two-component Bose-Einstein condensates with an internal atomic Josephson junction.The methods which we considered include two conservative type schemes and two non-conservative type schemes.Discrete conservation laws and solvability of the schemes are analyzed.For the four proposed finite difference methods,we establish the optimal convergence rates for the error at the order of O(h~2+τ~2)in the l~∞-norm(i.e.,the point-wise error estimates)with the time stepτand the mesh size h.Besides the standard techniques of the energy method,the key techniques in the analysis is to use the cut-off function technique,transformation between the time and space direction and the method of order reduction.All the methods and results here are also valid and can be easily extended to the three-dimensional case.Finally,numerical results are reported to confirm our theoretical error estimates for the numerical methods.     

Finite difference schemes for the fourth-order parabolic equations with different boundary value conditions

In this paper, the fourth-order parabolic equations with different boundary value conditions are studied. Six kinds of boundary value conditions are proposed. Several numerical differential formulae for the fourth-order derivative are established by the quartic interpolation polynomials and their truncation errors are given with the aid of the Taylor expansion with the integral remainders. Effective difference schemes are presented for the third Dirichlet boundary value problem, the first Neumann boundary value problem and the third Neumann boundary value problem, respectively. Some new embedding inequalities on the discrete function spaces are presented and proved. With the method of energy analysis, the unique solvability, unconditional stability and unconditional convergence of the difference schemes are proved. The convergence orders of derived difference schemes are all O(τ² + h²) in appropriate norms. Finally, some numerical examples are provided to confirm the theoretical results.     

Finite element method for space-time fractional diffusion equation

In this paper, we consider two types of space-time fractional diffusion equations(STFDE) on a finite domain. The equation can be obtained from the standard diffusion equation by replacing the second order space derivative by a Riemann-Liouville fractional derivative of order β (1 < β ≤ 2), and the first order time derivative by a Caputo fractional derivative of order γ (0 < γ ≤ 1). For the 0 < γ < 1 case, we present two schemes to approximate the time derivative and finite element methods for the space derivative, the optimal convergence rate can be reached O(τ^2?γ + h²) and O(τ² + h²), respectively, in which τ is the time step size and h is the space step size. And for the case γ = 1, we use the Crank-Nicolson scheme to approximate the time derivative and obtain the optimal convergence rate O(τ² + h²) as well. Some numerical examples are given and the numerical results are in good agreement with the theoretical analysis.     

Numerical solution to a one‐dimensional thermoplastic problem with unilateral constraint

In this article, we present a numerical simulation of one‐dimensional problem of quasi‐static contact with an elastic obstacle. A finite difference scheme is derived by the method of reduction of order on uniform meshes. The stability and convergence are proved. The convergence order is of O(τ² + h²), where τ and h are the time step size and the space step size, respectively. Some numerical examples demonstrate the theoretical results. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Analysis of a new class of forward semi-Lagrangian schemes for the 1D Vlasov Poisson equations

The Vlasov equation is a kinetic model describing the evolution of a plasma which is a globally neutral gas of charged particles. It is self-consistently coupled with Poisson’s equation, which rules the evolution of the electric field. In this paper, we introduce a new class of forward semi-Lagrangian schemes for the Vlasov–Poisson system based on a Cauchy Kovalevsky (CK) procedure for the numerical solution of the characteristic curves. Exact conservation properties of the first moments of the distribution function for the schemes are derived and a convergence study is performed that applies as well for the CK scheme as for a more classical Verlet scheme. A L ¹ convergence of the schemes will be proved. Error estimates [in O(Dt²+h² + \frach²Dt){O\left(\Delta{t}^2+h^2 + \frac{h^2}{\Delta{t}}\right)} for Verlet] are obtained, where Δt and h = max(Δx, Δv) are the discretization parameters.     

High accuracy analysis of the lowest order <Emphasis Type="Italic">H</Emphasis><Superscript>1</Superscript>-Galerkin mixed finite element method for nonlinear sine-Gordon equations

The lowest order H¹-Galerkin mixed finite element method (for short MFEM) is proposed for a class of nonlinear sine-Gordon equations with the simplest bilinear rectangular element and zero order Raviart-Thomas element. Base on the interpolation operator instead of the traditional Ritz projection operator which is an indispensable tool in the traditional FEM analysis, together with mean-value technique and high accuracy analysis, the superclose properties of order O(h²)/O(h² + τ²) in H¹-norm and H(div;Ω)-norm are deduced for the semi-discrete and the fully-discrete schemes, where h, τ denote the mesh size and the time step, respectively, which improve the results in the previous literature.     

A Priori Error Estimates for Finite Element Methods for H (2,1)-Elliptic Equations

The convergence of finite element methods for linear elliptic boundary value problems of second and forth order is well understood. In this article, we introduce finite element approximations of some linear semi-elliptic boundary value problem of mixed order on a two-dimensional rectangular domain Q. The equation is of second order in one direction and forth order in the other and appears in the optimal control of parabolic partial differential equations if one eliminates the control and the state (or the adjoint state) in the first order optimality conditions. We establish a regularity result and estimate for the finite element error of conforming approximations of this equation. The finite elements in use have a tensor product structure, in one dimension we use linear, quadratic or cubic Lagrange elements in the other dimension cubic Hermite elements. For these elements, we prove the error bound O(h ² + τ ^k ) in the energy norm and O((h ² + τ ^k )(h ² + τ)) in the L ²(Q)-norm.     

Maximum error estimates for a compact difference scheme of the coupled nonlinear Schrödinger–Boussinesq equations

The coupled nonlinear Schrödinger–Boussinesq (SBq) equations describe the nonlinear development of modulational instabilities associated with Langmuir field amplitude coupled to intense electromagnetic wave in dispersive media such as plasmas. In this paper, we present a conservative compact difference scheme for the coupled SBq equations and analyze the conservative property and the existence of the scheme. Then we prove that the scheme is convergent with convergence order O(τ² + h⁴) in L_∞‐norm and is stable in L_∞‐norm. Numerical results verify the theoretical analysis.     

Tension spline solution of nonlinear sine-Gordon equation

The sine-Gordon equation plays an important role in modern physics. By using spline function approximation, two implicit finite difference schemes are developed for the numerical solution of one-dimensional sine-Gordon equation. Stability analysis of the method has been given. It has been shown that by choosing the parameters suitably, we can obtain two schemes of orders O(k²+k²h²+h²)\mathcal{O}(k^{2}+k^{2}h^{2}+h^{2}) and O(k²+k²h²+h⁴)\mathcal{O}(k^{2}+k^{2}h^{2}+h^{4}). At the end, some numerical examples are provided to demonstrate the effectiveness of the proposed schemes.     

Finite element method for Grwünwald–Letnikov time-fractional partial differential equation

In this article, we consider the finite element methods (FEM) for Grwünwald–Letnikov time-fractional diffusion equation, which is obtained from the standard two-dimensional diffusion equation by replacing the first-order time derivative with a fractional derivative (of order α, with 0?h ^r+1?+?τ^2-α), where h, τ and r are the space step size, time step size and polynomial degree, respectively. A numerical example is presented to verify the order of convergence.