An analysis for a high‐order difference scheme for numerical solution to utt = A(x,t)uxx + F(x,t, u,ut, ux)

This article is concerned with a high‐order implicit difference scheme presented by Mohanty, Jain, and George for the nonlinear hyperbolic equation u_tt = A(x, t)u_xx + F(x, t, u, u_t, u_x) with Dirichlet boundary conditions. Some prior estimates of the difference solution are obtained by the energy methods. The solvability of the difference scheme is proved by the energy method and Brower's fixed point theorem. Similarly, the uniqueness, the convergence in L_∞‐norm and the stability of the difference solution are obtained. A numerical example is provided to demonstrate the validity of the theoretical results. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 23: 484–498, 2007     

A new highly accurate discretization for three‐dimensional singularly perturbed nonlinear elliptic partial differential equations

We employ a new fourth‐order compact finite difference formula based on arithmetic average discretization to solve the three‐dimensional nonlinear singularly perturbed elliptic partial differential equation ε(u_xx + u_yy + u_zz) = f(x, y, z, u, u_x, u_y, u_z), 0 < x, y, z < 1, subject to appropriate Dirichlet boundary conditions prescribed on the boundary, where ε > 0 is a small parameter. We also describe new fourth‐order methods for the estimates of (?u/?x), (?u/?y), and (?u/?z), which are quite often of interest in many physical problems. In all cases, we require only a single computational cell with 19 grid points. The proposed methods are directly applicable to solve singular problems without any modification. We solve three test problems numerically to validate the proposed derived fourth‐order methods. We compare the advantages and implementation of the proposed methods with the standard central difference approximations in the context of basic iterative methods. Numerical examples are given to verify the fourth‐order convergence rate of the methods. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2006     

A second‐order accurate difference scheme for the two‐dimensional Burgers' system

In this article, a Crank‐Nicolson‐type finite difference scheme for the two‐dimensional Burgers' system is presented. The existence of the difference solution is shown by Brouwer fixed‐point theorem. The uniqueness of the difference solution and the stability and L₂ convergence of the difference scheme are proved by energy method. An iterative algorithm for the difference scheme is given in detail. Furthermore, a linear predictor–corrector method is presented. The numerical results show that the predictor–corrector method is also convergent with the convergence order of two in both time and space. At last, some comments are provided for the backward Euler scheme. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

An O(k2 + kh2 + h4) arithmetic average discretization for the solution of 1‐D nonlinear parabolic equations

This article develops a new two‐level three‐point implicit finite difference scheme of order 2 in time and 4 in space based on arithmetic average discretization for the solution of nonlinear parabolic equation ε u_xx = f(x, t, u, u_x, u_t), 0 < x < 1, t > 0 subject to appropriate initial and Dirichlet boundary conditions, where ε > 0 is a small positive constant. We also propose a new explicit difference scheme of order 2 in time and 4 in space for the estimates of (?u/?x). The main objective is the proposed formulas are directly applicable to both singular and nonsingular problems. We do not require any fictitious points outside the solution region and any special technique to handle the singular problems. Stability analysis of a model problem is discussed. Numerical results are provided to validate the usefulness of the proposed formulas. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

The Crank‐Nicolson/Adams‐Bashforth scheme for the time‐dependent Navier‐Stokes equations with nonsmooth initial data

In this article, we study the stability and convergence of the Crank‐Nicolson/Adams‐Bashforth scheme for the two‐dimensional nonstationary Navier‐Stokes equations with a nonsmooth initial data. A finite element method is applied for the spatial approximation of the velocity and pressure. The time discretization is based on the implicit Crank‐Nicolson scheme for the linear terms and the explicit Adams‐Bashforth scheme for the nonlinear term. Moreover, we prove that the scheme is almost unconditionally stable for a nonsmooth initial data u₀ with div u₀ = 0, i.e., the time step τ satisfies: τ ≤ C₀ if u₀ ∈ H¹ ∩ L^∞; τ |log h| ≤ C₀ if u₀ ∈ H¹ for the mesh size h and some positive constant C₀. Finally, we obtain some error estimates for the discrete velocity and pressure under the above stability condition. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 28: 155‐187, 2012     

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     

A fast second‐order difference scheme for the space–time fractional equation

In this paper, a fast second‐order accurate difference scheme is proposed for solving the space–time fractional equation. The temporal Caputo derivative is approximated by ?L2 ‐1_σ formula which employs the sum‐of‐exponential approximation to the kernel function appeared in Caputo derivative. The second‐order linear spline approximation is applied to the spatial Riemann–Liouville derivative. At each time step, a fast algorithm, the preconditioned conjugate gradient normal residual method with a circulant preconditioner (PCGNR), is used to solve the resulting system that reduces the storage and computational cost significantly. The unique solvability and unconditional convergence of the difference scheme are shown by the discrete energy method. Numerical examples are given to verify numerical accuracy and efficiency of the difference schemes.     

Development of a numerical phase optimized upwinding combined compact difference scheme for solving the Camassa–Holm equation with different initial solitary waves

In this article, the solution of Camassa–Holm (CH) equation is solved by the proposed two‐step method. In the first step, the sixth‐order spatially accurate upwinding combined compact difference scheme with minimized phase error is developed in a stencil of four points to approximate the first‐order derivative term. For the purpose of retaining both of the long‐term accurate Hamiltonian property and the geometric structure inherited in the CH equation, the time integrator used in this study should be able to conserve symplecticity. In the second step, the Helmholtz equation governing the pressure‐like variable is approximated by the sixth‐order accurate three‐point centered compact difference scheme. Through the fundamental and numerical verification studies, the integrity of the proposed high‐order scheme is demonstrated. Another aim of this study is to reveal the wave propagation nature for the investigated shallow water equation subject to different initial wave profiles, whose peaks take the smooth, peakon, and cuspon forms. The transport phenomena for the cases with/without inclusion of the linear first‐order advection term κu_x in the CH equation will be addressed. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1645–1664, 2015     

Quintic trigonometric spline based numerical scheme for nonlinear modified Burgers' equation

This paper presents a numerical method based on quintic trigonometric B‐splines for solving modified Burgers' equation (MBE). Here, the MBE is first discretized in time by Crank–Nicolson scheme and the resulting scheme is solved by quintic trigonometric B‐splines. The proposed method tackles nonlinearity by using a linearization process known as quasilinearization. A rigorous analysis of the stability and convergence of the proposed method are carried out, which proves that the method is unconditionally stable and has order of convergence O(h⁴ + k²). Numerical results presented are very much in accordance with the exact solution, which is established by the negligible values of L₂ and L_∞ errors. Computational efficiency of the scheme is proved by small values of CPU time. The method furnishes results better than those obtained by using most of the existing methods for solving MBE.     

A fourth-order difference method for elliptic equations with nonlinear first derivative terms

We present a nine-point fourth-order finite difference method for the nonlinear second-order elliptic differential equation Au_xx + Bu_yy = f(x, y, u, u_x, u_y) on a rectangular region R subject to Dirichlet boundary conditions u(x, y) = g(x, y) on ?R. We establish, under appropriate conditions O(h⁴)-convergence of the finite difference scheme. Numerical examples are given to illustrate the method and its fourth-order convergence.     

Application of a Mickens finite‐difference scheme to the cylindrical Bratu‐Gelfand problem

The boundary value problem Δu + λe^u = 0 where u = 0 on the boundary is often referred to as “the Bratu problem.” The Bratu problem with cylindrical radial operators, also known as the cylindrical Bratu‐Gelfand problem, is considered here. It is a nonlinear eigenvalue problem with two known bifurcated solutions for λ < λ_c, no solutions for λ > λ_c and a unique solution when λ = λ_c. Numerical solutions to the Bratu‐Gelfand problem at the critical value of λ_c = 2 are computed using nonstandard finite‐difference schemes known as Mickens finite differences. Comparison of numerical results obtained by solving the Bratu‐Gelfand problem using a Mickens discretization with results obtained using standard finite differences for λ < 2 are given, which illustrate the superiority of the nonstandard scheme. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 327–337, 2004     

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     

Existence and convergence for quasi‐static evolution in brittle fracture

This paper investigates the mathematical well‐posedness of the variational model of quasi‐static growth for a brittle crack proposed by Francfort and Marigo in [15]. The starting point is a time discretized version of that evolution which results in a sequence of minimization problems of Mumford and Shah type functionals. The natural weak setting is that of special functions of bounded variation, and the main difficulty in showing existence of the time‐continuous quasi‐static growth is to pass to the limit as the time‐discretization step tends to 0. This is performed with the help of a jump transfer theorem which permits, under weak convergence assumptions for a sequence {u_n} of SBV‐functions to its BV‐limit u, to transfer the part of the jump set of any test field that lies in the jump set of u onto that of the converging sequence {u_n}. In particular, it is shown that the notion of minimizer of a Mumford and Shah type functional for its own jump set is stable under weak convergence assumptions. Furthermore, our analysis justifies numerical methods used for computing the time‐continuous quasi‐static evolution. © 2003 Wiley Periodicals, Inc.     

Weighted estimate for the convergence rate of a projection difference scheme for a quasilinear parabolic equation

A novel technique is proposed for analyzing the convergence of a projection difference scheme as applied to the initial value problem for a quasilinear parabolic operator-differential equation with initial data u ₀ ∈ H. The technique is based on the smoothing property of solutions to the differential problem for t > 0. Under certain conditions on the nonlinear term, a new estimate of order \(O(\sqrt \tau + h)\) for the convergence rate in a weighted energy norm is obtained without using a priori assumptions on the additional smoothness of weak solutions.     

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.     

On the global weak solutions for a modified two‐component Camassa‐Holm equation

In this paper, we investigate the existence of global weak solutions to the Cauchy problem of a modified two‐component Camassa‐Holm equation with the initial data satisfying lim_{x → ±∞}u₀(x) = u_±. By perturbing the Cauchy problem around a rarefaction wave, we obtain a global weak solution for the system under the assumption u_? ≤ u₊. The global weak solution is obtained as a limit of approximation solutions. The key elements in our analysis are the Helly theorem and the estimation of energy for approximation solutions in $H^1(\mathbb {R})\times H^1(\mathbb {R})In this paper, we investigate the existence of global weak solutions to the Cauchy problem of a modified two‐component Camassa‐Holm equation with the initial data satisfying lim_{x → ±∞}u₀(x) = u_±. By perturbing the Cauchy problem around a rarefaction wave, we obtain a global weak solution for the system under the assumption u_? ≤ u₊. The global weak solution is obtained as a limit of approximation solutions. The key elements in our analysis are the Helly theorem and the estimation of energy for approximation solutions in $H^1(\mathbb {R})\times H^1(\mathbb {R})$ and some a priori estimates on the first‐order derivatives of approximation solutions.     

Optimal error estimates of the local discontinuous Galerkin method for nonlinear second‐order elliptic problems on Cartesian grids

In this paper, we study the local discontinuous Galerkin (LDG) methods for two‐dimensional nonlinear second‐order elliptic problems of the type u_xx + u_yy = f(x, y, u, u_x, u_y) , in a rectangular region Ω with classical boundary conditions on the boundary of Ω . Convergence properties for the solution and for the auxiliary variable that approximates its gradient are established. More specifically, we use the duality argument to prove that the errors between the LDG solutions and the exact solutions in the L² norm achieve optimal (p + 1)th order convergence, when tensor product polynomials of degree at most p are used. Moreover, we prove that the gradient of the LDG solution is superclose with order p + 1 toward the gradient of Gauss–Radau projection of the exact solution. The results are valid in two space dimensions on Cartesian meshes using tensor product polynomials of degree p ≥ 1 , and for both mixed Dirichlet–Neumann and periodic boundary conditions. Preliminary numerical experiments indicate that our theoretical findings are optimal.     

Kernel regression estimation for continuous spatial processes

We investigate here a kernel estimate of the spatial regression function r(x) = E(Y _u | X _u = x), x ∈ ℝ^d, of a stationary multidimensional spatial process { Z _u = (X _u, Y _u), u ∈ ℝ ^N }. The weak and strong consistency of the estimate is shown under sufficient conditions on the mixing coefficients and the bandwidth, when the process is observed over a rectangular domain of ℝ^N. Special attention is paid to achieve optimal and suroptimal strong rates of convergence. It is also shown that this suroptimal rate is preserved by using a suitable spatial sampling scheme.      

Numerical analysis of the Allen-Cahn equation and approximation for mean curvature flows

Summary.  We propose and analyze a semi-discrete (in time) scheme and a fully discrete scheme for the Allen-Cahn equation u _t −Δu+ɛ⁻² f(u)=0 arising from phase transition in materials science, where ɛ is a small parameter known as an ``interaction length'. The primary goal of this paper is to establish some useful a priori error estimates for the proposed numerical methods, in particular, by focusing on the dependence of the error bounds on ɛ. Optimal order and quasi-optimal order error bounds are shown for the semi-discrete and fully discrete schemes under different constraints on the mesh size h and the time step size k and different regularity assumptions on the initial datum function u ₀ . In particular, all our error bounds depend on only in some lower polynomial order for small ɛ. The cruxes of the analysis are to establish stability estimates for the discrete solutions, to use a spectrum estimate result of de Mottoni and Schatzman [18, 19] and Chen [12] and to establish a discrete counterpart of it for a linearized Allen-Cahn operator to handle the nonlinear term. Finally, as a nontrivial byproduct, the error estimates are used to establish convergence and rate of convergence of the zero level set of the fully discrete solution to the motion by mean curvature flow and to the generalized motion by mean curvature flow. Received April 30, 2001 / Revised version received March 20, 2002 / Published online July 18, 2002 Mathematics Subject Classification (1991): 65M60, 65M12, 65M15, 35B25, 35K57, 35Q99, 53A10 Correspondence to: A. Prohl     

Analysis and computation of least‐squares methods for a compressible Stokes problem

We develop and analyze a negative norm least‐squares method for the compressible Stokes equations with an inflow boundary condition. Least‐squares principles are derived for a first‐order form of the equations obtained by using ω = ?×u and φ = ? · u as new dependent variables. The resulting problem is incompletely elliptic, i.e., it combines features of elliptic and hyperbolic equations. As a result, well‐posedness of least‐squares functionals cannot be established using the ADN elliptic theory and so we use direct approaches to prove their norm‐equivalence. The article concludes with numerical examples that illustrate the theoretical convergence rates. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006