COMPACT FOURTH-ORDER FINITE DIFFERENCE SCHEMES FOR HELMHOLTZ EQUATION WITH HIGH WAVE NUMBERS

In this paper, two fourth-order accurate compact difference schemes are presented for solving the Helmholtz equation in two space dimensions when the corresponding wave numbers are large. The main idea is to derive and to study a fourth-order accurate compact difference scheme whose leading truncation term, namely, the O（h^4） term, is independent of the wave number and the solution of the Helmholtz equation. The convergence property of the compact schemes are analyzed and the implementation of solving the resulting linear algebraic system based on a FFT approach is considered. Numerical results are presented, which support our theoretical predictions.     

A Trilayer Difference Scheme for One-Dimensional Parabolic Systems

In order to obtain the numerical solution for a one-dimensional parabolic system, an unconditionally stable difference method is investigated in [1]. If the number of unknown functions is M, for each time step only M times of calculation are needed. The rate of convergence is $O(\tau+h^2)$. On the basis of [1], an alternating calculation difference scheme is presented in [2]; the rate of the convergence is $O(\tau^2+h^2)$. The difference schemes in [1] and [2] are economic ones. For the $\alpha$-$th$ equation, only $U_{\alpha}$ is an unknown function; the others $U_{\beta}$ are given evaluated either in the last step or in the present step. So the practical calculation is quite convenient. The purpose of this paper is to derive a trilayer difference scheme for one-dimensional parabolic systems. It is known that the scheme is also unconditionally stable and the rate of convergence is $O(\tau^2+h^2)$.     

Delta-Wave in a Simple 1-D 2x2 Hyperbolic Problem

A simple one-dimensional $2\times 2$ hyperbolic system is considered in the paper. The model contains a linear hyperbolic equation, as well as a hyperbolic equation of which the coefficients are about the solution of the linear one. The exact solution is presented and discussed, then numerical experiments are given by TVD (or MmB) type schemes for Riemann problems. From the results, we know that the solutions do have $\delta-$waves for some suitable initial data.     

半导体瞬态问题的修正迎风有限体积格式

半导体器件的瞬时状态由包含三个拟线性偏微分方程所组成的方程组的初边值问题来描述.其中电子位势方程足椭圆型的,电子和空穴浓度方程是对流扩散型的.对电子位势方程采用一次元有限体积法米逼近,对电子浓度和空穴浓度方程采用修正的迎风有限体积方法来逼近,并进行详细的理论分析,关于位势得到O(h Δt)阶的H1模误差估计结果,关于浓度得到O(h2 Δt)阶的L2模误差估计结果.最后,给出数值例子.     

Numerical Solution of the Reaction-Diffusion Equation

In this paper, we consider the numerical solution for the reaction-diffusion equation. A finite difference scheme and the basic error equality are given. Then the error estimations are proved for the periodic problem with $v(x,t)\geq 0$, the first and second boundary value problems with $v(x,t)\geq v_0＞0$, and for $v(U)\geq v_0›0$. Under some conditions such estimations imply the stabilities and convergences of the schemes.     

Exponential Time Differencing-Padé Finite Element Method for Nonlinear Convection-Diffusion-Reaction Equations with Time Constant Delay

In this paper, ETD3-Padé and ETD4-Padé Galerkin finite element methods are proposed and analyzed for nonlinear delayed convection-diffusion-reaction equations with Dirichlet boundary conditions. An ETD-based RK is used for time integration of the corresponding equation. To overcome a well-known difficulty of numerical instability associated with the computation of the exponential operator, the Padé approach is used for such an exponential operator approximation, which in turn leads to the corresponding ETD-Padé schemes. An unconditional $L^2$ numerical stability is proved for the proposed numerical schemes, under a global Lipshitz continuity assumption. In addition, optimal rate error estimates are provided, which gives the convergence order of $O(k^{3}+h^{r})$ (ETD3-Padé) or $O(k^{4}+h^{r})$ (ETD4-Padé) in the $L^2$ norm, respectively. Numerical experiments are presented to demonstrate the robustness of the proposed numerical schemes.     

Two-Step Scheme for Backward Stochastic Differential Equations

In this paper, a stochastic linear two-step scheme has been presented to approximate backward stochastic differential equations (BSDEs). A necessary and sufficient condition is given to judge the $\mathbb{L}_2$-stability of our numerical schemes. This stochastic linear two-step method possesses a family of $3$-order convergence schemes in the sense of strong stability. The coefficients in the numerical methods are inferred based on the constraints of strong stability and $n$-order accuracy ($n\in\mathbb{N}^+$). Numerical experiments illustrate that the scheme is an efficient probabilistic numerical method.     

Several Abstract Iterative Schemes for Solving the Bifurcation at Simple Eigenvalues

In this paper we consider the nonlinear operator equation $\lambda x=Lx+G(\lambda,x)$ where $L$ is a closed linear operator of $X-›X, X$ is a real Banach Space, with a simple eigenvalue $\lambda_0\neq 0$. We discretize its Liapunov-Schmidt bifurcation equation instead of the original nonlinear operator equation and estimate the approximating order of our approximate solution to the genuine solution. Our method is more convenient and more accurate. Meanwhile we put forward several abstract Newton-type iterative schemes, which are more efficient for practical computation, and get the result of their super-linear convergence.     

Convergence of Extrapolated BDF2 Finite Element Schemes for Unsteady Penetrative Convection Model

Extrapolated two-step backward difference (BDF2) in time and finite element in space discretization for the unsteady penetrative convection model is analyzed. Penetrative convection model employs a nonlinear equation of state making the problem more nonlinear. Optimal order error estimates are derived for the semi-discrete finite element spatial discretization. Two time discretization schemes based on linear extrapolation are proposed and analyzed, namely a coupled and a decoupled scheme. In particular, we show that although both schemes are unconditionally nonlinearly stable, the decoupled scheme converges unconditionally whereas coupled scheme requires that the time step be sufficiently small for convergence. These time discretization schemes can be implemented efficiently in practice, saving computational memory. Numerical computations and numerical convergence checks are presented to demonstrate the efficiency and the accuracy of the schemes.     

Square root 3 -Subdivision Schemes: Maximal Sum Rule Orders

Abstract. Subdivision with finitely supported masks is an efficient method to create discrete multiscale representations of smooth surfaces for CAGD applications. Recently a new subdivision scheme for triangular meshes, called $\sqrt 3$ -subdivision , has been studied. In comparison to dyadic subdivision, which is based on the dilation matrix 2I , $\sqrt 3$ -subdivision is based on a dilation M with det M=3 . This has certain advantages, for example, a slower growth for the number of control points. This paper concerns the problem of achieving maximal sum rule orders for stationary $\sqrt 3$ -subdivision schemes with given mask support, which is important because the sum rule order characterizes the order of the polynomial reproduction, and provides an upper bound on the Sobolev smoothness of the surface. We study both interpolating and approximating schemes for a natural family of symmetric mask support sets related to squares of sidelength 2n in Z ² , and obtain exact formulas for the maximal sum rule order for arbitrary n . For approximating schemes, the solution is simple, and schemes with maximal sum rule order are realized by an explicit family of schemes based on repeated averaging [15]. In the interpolating case, we use properties of multivariate Lagrange polynomial interpolation to prove the existence of interpolating schemes with maximal sum rule orders. These can be found by solving a linear system which can be reduced in size by using symmetries. From this, we construct some new examples of smooth (C ² ,C ³ ) interpolating $\sqrt 3$ -subdivision schemes with maximal sum rule order and symmetric masks. The construction of associated dual schemes is also discussed.     

A second order numerical scheme for the solution of the one-dimensional Boussinesq equation

A predictor–corrector (P-C) scheme is applied successfully to a nonlinear method arising from the use of rational approximants to the matrix-exponential term in a three-time level recurrence relation. The resulting nonlinear finite-difference scheme, which is analyzed for local truncation error and stability, is solved using a P-C scheme, in which the predictor and the corrector are explicit schemes of order 2. This scheme is accelerated by using a modification (MPC) in which the already evaluated values are used for the corrector. The behaviour of the P-C/MPC schemes is tested numerically on the Boussinesq equation already known from the bibliography free of boundary conditions. The numerical results are derived for both the bad and the good Boussinesq equation and conclusions from the relevant known results are derived.      

Finite Difference/Collocation Method for Two-Dimensional Sub-Diffusion Equation with Generalized Time Fractional Derivative

In this paper, we propose a finite difference/collocation method for two-dimensional time fractional diffusion equation with generalized fractional operator. The main purpose of this paper is to design a high order numerical scheme for the new generalized time fractional diffusion equation. First, a finite difference approximation formula is derived for the generalized time fractional derivative, which is verified with order $2-\alpha$ $(0<\alpha<1)$. Then, collocation method is introduced for the two-dimensional space approximation. Unconditional stability of the scheme is proved. To make the method more efficient, the alternating direction implicit method is introduced to reduce the computational cost. At last, numerical experiments are carried out to verify the effectiveness of the scheme.     

Error Estimate of a New Conservative Finite Difference Scheme for the Klein-Gordon-Dirac System

In this paper, we derive and analyze a conservative Crank-Nicolson-type finite difference scheme for the Klein-Gordon-Dirac (KGD) system. Differing from the derivation of the existing numerical methods given in literature where the numerical schemes are proposed by directly discretizing the KGD system, we translate the KGD equations into an equivalent system by introducing an auxiliary function, then derive a nonlinear Crank-Nicolson-type finite difference scheme for solving the equivalent system. The scheme perfectly inherits the mass and energy conservative properties possessed by the KGD, while the energy preserved by the existing conservative numerical schemes expressed by two-level's solution at each time step. By using energy method together with the 'cut-off' function technique, we establish the optimal error estimate of the numerical solution, and the convergence rate is $\mathcal{O}(τ^2 + h^2)$ in $l^∞$-norm with time step $τ$ and mesh size $h.$ Numerical experiments are carried out to support our theoretical conclusions.     

Generalized and Unified Families of Interpolating Subdivision Schemes

We present generalized and unified families of $(2n)$-point and $(2n-1)$-point $p$-ary interpolating subdivision schemes originated from Lagrange polynomial for any integers $n ≥ 2$ and $p ≥ 3$. Almost all existing even-point and odd-point interpolating schemes of lower and higher arity belong to this family of schemes. We also present tensor product version of generalized and unified families of schemes. Moreover, error bounds between limit curves and control polygons of schemes are also calculated. It has been observed that error bounds decrease when complexity of the scheme decrease and vice versa. Furthermore, error bounds decrease with the increase of arity of the schemes. We also observe that in general the continuity of interpolating scheme do not increase by increasing complexity and arity of the scheme.     

General Full Implicit Strong Taylor Approximations for Stiff Stochastic Differential Equations

In this paper, we present the backward stochastic Taylor expansions for a Ito process, including backward Ito-Taylor expansions and backward Stratonovich-Taylor expansions. We construct the general full implicit strong Taylor approximations (including Ito-Taylor and Stratonovich-Taylor schemes) with implicitness in both the deterministic and the stochastic terms for the stiff stochastic differential equations (SSDE) by employing truncations of backward stochastic Taylor expansions. We demonstrate that these schemes will converge strongly with corresponding order $1,2,3,\ldots$ Mean-square stability has been investigated for full implicit strong Stratonovich-Taylor scheme with order $2$, and it has larger mean-square stability region than the explicit and the semi-implicit strong Stratonovich-Taylor schemes with order $2$. We can improve the stability of simulations considerably without too much additional computational effort by using our full implicit schemes. The full implicit strong Taylor schemes allow a larger range of time step sizes than other schemes and are suitable for SSDE with stiffness on both the drift and the diffusion terms. Our numerical experiment shows these points.     

Conservative numerical schemes for the Ostrovsky equation

The Ostrovsky equation describes gravity waves under the influence of Coriolis force. It is known that solutions of this equation conserve the L² norm and an energy function that is determined non-locally. In this paper we propose four conservative numerical schemes for this equation: a finite difference scheme and a pseudospectral scheme that conserve the norm, and the same types of schemes that conserve the energy. A numerical comparison of these schemes is also provided, which indicates that the energy conservative schemes perform better than the norm conservative schemes.     

Well‐posedness and asymptotic analysis for a Penrose–Fife type phase field system

In this paper, an asymptotic analysis of the (non‐conserved) Penrose–Fife phase field system for two vanishing time relaxation parameters ε and δ is developed, in analogy with the similar analyses for the phase field model proposed by G. Caginalp (Arch. Rational Mech. Anal. 1986; 92 :205–245), which were carried out by Rossi and Stoth (Adv. Math. Sci. Appl. 2003; 13 :249–271; Quart. Appl. Math. 1995; 53 :695–700). Although formally the singular limits for ε ↓ 0 and for ε and δ ↓ 0 are, respectively, the viscous Cahn–Hilliard equation and the Cahn–Hilliard equation, it turns out that the Penrose–Fife system is indeed a bad approximation for these equations. Therefore, we consider an alternative approximating phase field system, which could be viewed as a generalization of the classical Penrose–Fife phase field system, featuring a double non‐linearity given by two maximal monotone graphs. A well‐posedness result is proved for such a system, and it is shown that the solutions converge to the unique solution of the viscous Cahn–Hilliard equation as ε ↓ 0, and of the Cahn–Hilliard equation as ε ↓ 0 and δ ↓ 0. Copyright © 2004 John Wiley & Sons, Ltd.     

On exact dimensional splitting for a multidimensional scalar quasilinear hyperbolic conservation law

A dimensional splitting scheme is applied to a multidimensional scalar homogeneous quasilinear hyperbolic equation (conservation law). It is proved that the splitting error is zero. The proof is presented for the above partial differential equation in an arbitrary number of dimensions. A numerical example is given that illustrates the proved accuracy of the splitting scheme. In the example, the grid convergence of split (locally one-dimensional) compact and bicompact difference schemes and unsplit bicompact schemes combined with high-order accurate time-stepping schemes (namely, Runge–Kutta methods of order 3, 4, and 5) is analyzed. The errors of the numerical solutions produced by these schemes are compared. It is shown that the orders of convergence of the split schemes remain high, which agrees with the conclusion that the splitting error is zero.     

Splitting schemes for pseudoparabolic equations

We construct and analyze splitting schemes for a pseudoparabolic equation. We suggest a three-level vector additive scheme in which part of the time derivatives is taken from the previous time level. Unconditional stability estimates for the splitting schemes are proved.     

First-Order and Second-Order,Chaos-Free,Finite Difference Schemes for Fisher Equation

A new class of finite difference schemes is constructed for Fisher partial differential equation i.e. the reaction-diffusion equation with stiff source term: $au(1-u)$. These schemes have the properties that they reduce to high fidelity algorithms in the diffusion-free case namely in which the numerical solutions preserve the properties in the exact solutions for arbitrary time step-size and reaction coefficient α＞0 and all nonphysical spurious solutions including bifurcations and chaos that normally appear in the standard discrete models of Fisher partial differential equation will not occur. The implicit schemes so developed obtain the numerical solutions by solving a single linear algebraic system at each step. The boundness and asymptotic behaviour of numerical solutions obtained by all these schemes are given. The approach constructing the above schemes can be extended to reaction-diffusion equations with other stiff source terms.