A class of difference scheme for solving telegraph equation by new non-polynomial spline methods

In this paper, by using a new non-polynomial parameters cubic spline in space direction and compact finite difference in time direction, we get a class of new high accuracy scheme of O(τ⁴ + h²) and O(τ⁴ + h⁴) for solving telegraph equation if we suitably choose the cubic spline parameters. Meanwhile, stability condition of the difference scheme has been carried out. Finally, numerical examples are used to illustrate the efficiency of the new difference scheme.     

Further results on ultraconvergence derivative recovery for odd-order rectangular finite elements

For rectangular finite element, we give a superconvergence method by SPR technique based on the generalization of a new ultraconvergence record and the sharp Green function estimates, by which we prove that the derivative has ultra-convergence of order O(h ^k+3) (k ⩾ 3 being odd) and displacement has order of O(h ^k+4) (k ⩾ 4 being even) at the locally symmetry points.      

Non-polynomial spline method for the solution of two-dimensional linear wave equations with a nonlinear source term

In this paper, two classes of methods are developed for the solution of two space dimensional wave equations with a nonlinear source term. We have used non-polynomial cubic spline function approximations in both space directions. The methods involve some parameters, by suitable choices of the parameters, a new high accuracy three time level scheme of order O(h ⁴ + k ⁴ + τ ² + τ ² h ² + τ ² k ²) has been obtained. Stability analysis of the methods have been carried out. The results of some test problems are included to demonstrate the practical usefulness of the proposed methods. The numerical results for the solution of two dimensional sine-Gordon equation are compared with those already available in literature.     

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     

On the use of high order difference methods for the system of one space second order nonlinear hyperbolic equations with variable coefficients

Implicit difference schemes of O(k⁴ + k²h² + h⁴), where k0, h 0 are grid sizes in time and space coordinates respectively, are developed for the efficient numerical integration of the system of one space second order nonlinear hyperbolic equations with variable coefficients subject to appropriate initial and Dirichlet boundary conditions. The proposed difference method for a scalar equation is applied for the wave equation in cylindrical and spherical symmetry. The numerical examples are given to illustrate the fourth order convergence of the methods.     

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.     

Analysis of a new finite difference/local discontinuous Galerkin method for the fractional Cattaneo equation

In this paper, we first present a new finite difference scheme to approximate the time fractional derivatives, which is defined in the sense of Caputo, and give a semidiscrete scheme in time with the truncation error O((Δt)^3?α ), where Δt is the time step size. Then a fully discrete scheme based on the semidiscrete scheme for the fractional Cattaneo equation in which the space direction is approximated by a local discontinuous Galerkin method is presented and analyzed. We prove that the method is unconditionally stable and convergent with order O(h ^k+1 + (Δt)^3?α ), where k is the degree of piecewise polynomial. Numerical examples are also given to confirm the theoretical analysis.     

Implicit compact difference schemes for the fractional cable equation

In this paper, we propose two implicit compact difference schemes for the fractional cable equation. The first scheme is proved to be stable and convergent in l_∞-norm with the convergence order O(τ + h⁴) by the energy method, where new inner products defined in this paper gives great convenience for the theoretical analysis. Numerical experiments are presented to demonstrate the accuracy and effectiveness of the two compact schemes. The computational results show that the two new schemes proposed in this paper are more accurate and effective than the previous.     

A new difference scheme with high accuracy and absolute stability for solving convection–diffusion equations

In this paper, we use a semi-discrete and a padé approximation method to propose a new difference scheme for solving convection–diffusion problems. The truncation error of the difference scheme is O(h⁴+τ⁵). It is shown through analysis that the scheme is unconditionally stable. Numerical experiments are conducted to test its high accuracy and to compare it with Crank–Nicolson method.     

A new high accuracy locally one-dimensional scheme for the wave equation

In this paper, a new locally one-dimensional (LOD) scheme with error of O(Δt⁴+h⁴) for the two-dimensional wave equation is presented. The new scheme is four layer in time and three layer in space. One main advantage of the new method is that only tridiagonal systems of linear algebraic equations have to be solved at each time step. The stability and dispersion analysis of the new scheme are given. The computations of the initial and boundary conditions for the two intermediate time layers are explicitly constructed, which makes the scheme suitable for performing practical simulation in wave propagation modeling. Furthermore, a comparison of our new scheme and the traditional finite difference scheme is given, which shows the superiority of our new method.     

Finite element analysis for a nonlinear diffusion model in image processing

The optimal error estimate O(h^k+1) for a popular nonlinear diffusion model widely used in image processing is proved for the standard k^th-order (k ≥ 1) conforming tensor-product finite elements in the L²-norm. The optimal L²-estimate is obtained by the integral identity technique [1–3] without using the classic Nitsche duality argument [4].     

Multistep characteristic method for incompressible flow in porous media

In this paper we present a multistep difference scheme for the problem of miscible displacement of incompressible fluid flow in porous media. The discretization involves a three-level time scheme based on the characteristic method and a five-point finite difference scheme for space discretization. We prove that the convergence is of order O(h²+(Δt)²), which is in contrast to the convergence of order O(h+Δt) proved for a singlestep characteristic with the same space discretization. Numerical experiments demonstrate the stability and second-order convergence of the scheme.     

Stability and convergence of a fully discrete local discontinuous Galerkin method for multi-term time fractional diffusion equations

In this paper, a fully discrete local discontinuous Galerkin method for a class of multi-term time fractional diffusion equations is proposed and analyzed. Using local discontinuous Galerkin method in spatial direction and classical L1 approximation in temporal direction, a fully discrete scheme is established. By choosing the numerical flux carefully, we prove that the method is unconditionally stable and convergent with order O(h ^k+1 + (Δt)^2?α ), where k, h, and Δt are the degree of piecewise polynomial, the space, and time step sizes, respectively. Numerical examples are carried out to illustrate the effectiveness of the numerical scheme.     

A new high accuracy non-polynomial tension spline method for the solution of one dimensional wave equation in polar co-ordinates

In this paper, we propose a new three-level implicit nine point compact finite difference formulation of O(k² + h⁴) based on non-polynomial tension spline approximation in r-direction and finite difference approximation in t-direction for the numerical solution of one dimensional wave equation in polar co-ordinates. We describe the mathematical formulation procedure in details and also discuss the stability of the method. Numerical results are provided to justify the usefulness of the proposed method.     

A three level linearized compact difference scheme for the Cahn-Hilliard equation

This article is devoted to the study of high order accuracy difference methods for the Cahn-Hilliard equation.A three level linearized compact difference scheme is derived.The unique solvability and unconditional convergence of the difference solution are proved.The convergence order is O(τ 2 + h 4 ) in the maximum norm.The mass conservation and the non-increase of the total energy are also verified.Some numerical examples are given to demonstrate the theoretical results.     

EQ 1 rot nonconforming finite element method for nonlinear dual phase lagging heat conduction equations

EQ rot 1 nonconforming finite element approximation to a class of nonlinear dual phase lagging heat conduction equations is discussed for semi-discrete and fully-discrete schemes. By use of a special property, that is, the consistency error of this element is of order O(h2 ) one order higher than its interpolation error O(h), the superclose results of order O(h2 ) in broken H1 -norm are obtained. At the same time, the global superconvergence in broken H1 -norm is deduced by interpolation postprocessing technique. Moreover, the extrapolation result with order O(h4 ) is derived by constructing a new interpolation postprocessing operator and extrapolation scheme based on the known asymptotic expansion formulas of EQ rot 1 element. Finally, optimal error estimate is gained for a proposed fully-discrete scheme by different approaches from the previous literature.     

Compact finite difference scheme for the solution of time fractional advection-dispersion equation

In this paper, a compact finite difference method is proposed for the solution of time fractional advection-dispersion equation which appears extensively in fluid dynamics. In this approach the time fractional derivative of mentioned equation is approximated by a scheme of order O(τ ^2???α ), 0?<?α?<?1, and spatial derivatives are replaced with a fourth order compact finite difference scheme. We will prove the unconditional stability and solvability of proposed scheme. Also we show that the method is convergence with convergence order O(τ ^2???α ?+?h ⁴). Numerical examples confirm the theoretical results and high accuracy of proposed scheme.     

Order of convergence for polynomial and analytical nodal methods

In [6] we analyzed the direct analytical nodal methods (ANM) of indexl and show that the corresponding mathematical methods are equivalent to the physical ones when the components of the matrices are calculated by generalized Radau reduced integration. In this article we extend the theorem 8 of [7] to the polynomial nodal methods (PNM) (exact calculation of moments) which are thus the order ofO(h ^l+3?δ _l0. We also show that the analytical nodal methods are only the order ofO(h ^l+2). Forl = 0 our numerical results confirm our theoretical results.     

Superconvergence analysis and extrapolation of quasi-Wilson nonconforming finite element method for nonlinear Sobolev equations

Quasi-Wilson nonconforming finite element approximation for a class of nonlinear Sobolev equa-tions is discussed on rectangular meshes. We first prove that this element has two special characters by novel approaches. One is that (▽h ( u-Ihu )1, ▽hvh) h may be estimated as order O ( h2 ) when u ∈ H3 (Ω), where Ihu denotes the bilinear interpolation of u , vh is a polynomial belongs to quasi-Wilson finite element space and ▽h denotes the piecewise defined gradient operator, h is the mesh size tending to zero. The other is that the consistency error of this element is of order O ( h2 ) /O ( h3 ) in broken H 1-norm, which is one/two order higher than its interpolation error when u ∈ H3 (Ω) /H4 (Ω). Then we derive the optimal order error estimate and su- perclose property via mean-value method and the known high accuracy result of bilinear element. Furthermore, we deduce the global superconvergence through interpolation post processing technique. At last, an extrapola- tion result of order O ( h3 ), two order higher than traditional error estimate, is obtained by constructing a new suitable extrapolation scheme.     

Correction of finite element estimates for Sturm-Liouville eigenvalues

Summary It is shown that a simple asymptotic correction technique of Paine, de Hoog and Anderssen reduces the error in the estimate of thekth eigenvalue of a regular Sturm-Liouville problem obtained by the finite element method, with linear hat functions and mesh lengthh, fromO(k ⁴ h ²) toO(k h ²). The result still holds when the matrix elements are evaluated by Simpson's rule, but if the trapezoidal rule is used the error isO(k ² h ²). Numerical results demonstrate the usefulness of the correction even for low values ofk.