Local error control in codes for ordinary differential equations

There are several schemes for the control of local error which are seen in differential equation solvers. The analysis attempts to explain how the selection of a scheme influences the behavior of global error seen in high quality production codes. Two rules of thumb for estimating global errors are given theoretical support when used in conjunction with suitable codes. Substantial numerical experiments support the analysis and conclusions.     

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.     

PARALLEL COMPUTING METHOD OF PURE ALTERNATIVE SEGMENT EXPLICIT-IMPLICIT DIFFERENCE SCHEME FOR NONLINEAR LELAND EQUATION

The research on the numerical solution of the nonlinear Leland equation has important theoretical significance and practical value. To solve nonlinear Leland equation, this paper offers a class of difference schemes with parallel nature which are pure alternative segment explicit-implicit(PASE-I) and implicit-explicit(PASI-E) schemes. It also gives the existence and uniqueness,the stability and the error estimate of numerical solutions for the parallel difference schemes. Theoretical analysis demonstrates that PASE-I and PASI-E schemes have obvious parallelism, unconditionally stability and second-order convergence in both space and time. The numerical experiments verify that the calculation accuracy of PASE-I and PASI-E schemes are better than that of the existing alternating segment Crank-Nicolson scheme, alternating segment explicit-implicit and implicit-explicit schemes. The speedup of PASE-I scheme is 9.89, compared to classical Crank-Nicolson scheme. Thus the schemes given by this paper are high efficient and practical for solving the nonlinear Leland equation.     

Weak convergence of finite element approximations of linear stochastic evolution equations with additive noise II. Fully discrete schemes

We present an abstract framework for analyzing the weak error of fully discrete approximation schemes for linear evolution equations driven by additive Gaussian noise. First, an abstract representation formula is derived for sufficiently smooth test functions. The formula is then applied to the wave equation, where the spatial approximation is done via the standard continuous finite element method and the time discretization via an I-stable rational approximation to the exponential function. It is found that the rate of weak convergence is twice that of strong convergence. Furthermore, in contrast to the parabolic case, higher order schemes in time, such as the Crank-Nicolson scheme, are worthwhile to use if the solution is not very regular. Finally we apply the theory to parabolic equations and detail a weak error estimate for the linearized Cahn-Hilliard-Cook equation as well as comment on the stochastic heat equation.     

Error Estimates for a Stochastic Impulse Control Problem

We obtain error bounds for monotone approximation schemes of a stochastic impulse control problem. This is an extension of the theory for error estimates for the Hamilton-Jacobi-Bellman equation. We obtain almost the same estimate on the rate of convergence as in the equation without impulsions [2], [3].     

Analysis of stability and error estimates for three methods approximating a nonlinear reaction-diffusion equation

We present the error analysis of three time-stepping schemes used in the discretization of a nonlinear reaction-diffusion equation with Neumann boundary conditions, relevant in phase transition. We prove $L^\infty$ stability by maximum principle arguments, and derive error estimates using energy methods for the implicit Euler, and two implicit-explicit approaches, a linearized scheme and a fractional step method. A numerical experiment validates the theoretical results, comparing the accuracy of the methods.     

New predictor-corrector approach for nonlinear fractional differential equations: error analysis and stability

In this paper, the predictor-corrector approach is used to propose two algorithms for the numerical solution of linear and non-linear fractional differential equations (FDE). The fractional order derivative is taken to be in the sense of Caputo and its properties are used to transform FDE into a Volterra-type integral equation. Simpson''s 3/8 rule is used to develop new numerical schemes to obtain the approximate solution of the integral equation associated with the given FDE. The error and stability analysis for the two methods are presented. The proposed methods are compared with the ones available in the literature. Numerical simulation is performed to demonstrate the validity and applicability of both the proposed techniques. As an application, the problem of dynamics of the new fractional order non-linear chaotic system introduced by Bhalekar and Daftardar-Gejji is investigated by means of the obtained numerical algorithms.     

Difference schemes for solving the Cauchy problem for a second-order operator differential equation

A class of finite-difference schemes for solving an ill-posed Cauchy problem for a second-order linear differential equation with a sectorial operator in a Banach space is studied. Time-uniform estimates of the convergence rate and the error of such schemes are obtained. Previously known estimates are improved due to an optimal choice of initial data for a difference scheme.     

On the perturbation algorithm for the semidiscrete scheme for the evolution equation and estimation of the approximate solution error using semigroups

In a Banach space, for the approximate solution of the Cauchy problem for the evolution equation with an operator generating an analytic semigroup, a purely implicit three-level semidiscrete scheme that can be reduced to two-level schemes is considered. Using these schemes, an approximate solution to the original problem is constructed. Explicit bounds on the approximate solution error are proved using properties of semigroups under minimal assumptions about the smoothness of the data of the problem. An intermediate step in this proof is the derivation of an explicit estimate for the semidiscrete Crank–Nicolson scheme. To demonstrate the generality of the perturbation algorithm as applied to difference schemes, a four-level scheme that is also reduced to two-level schemes is considered.     

The completely conservative difference schemes for the nonlinear Landau-Fokker-Planck equation

Conservativity and complete conservativity of finite difference schemes are considered in connection with the nonlinear kinetic Landau-Fokker-Planck equation. The characteristic feature of this equation is the presence of several conservation laws. Finite difference schemes, preserving density and energy are constructed for the equation in one- and two-dimensional velocity spaces. Some general methods of constructing such schemes are formulated. The constructed difference schemes allow us to carry out the numerical solution of the relaxation problem in a large time interval without error accumulation. An illustrative example is given.     

To the theory of asymptotically stable second-order accurate two-stage scheme for an inhomogeneous parabolic initial-boundary value problem

An asymptotically stable two-stage difference scheme applied previously to a homogeneous parabolic equation with a homogeneous Dirichlet boundary condition and an inhomogeneous initial condition is extended to the case of an inhomogeneous parabolic equation with an inhomogeneous Dirichlet boundary condition. It is shown that, in the class of schemes with two stages (at every time step), this difference scheme is uniquely determined by ensuring that high-frequency spatial perturbations are fast damped with time and the scheme is second-order accurate and has a minimal error. Comparisons reveal that the two-stage scheme provides certain advantages over some widely used difference schemes. In the case of an inhomogeneous equation and a homogeneous boundary condition, it is shown that the extended scheme is second-order accurate in time (for individual harmonics). The possibility of achieving second-order accuracy in the case of an inhomogeneous Dirichlet condition is explored, specifically, by varying the boundary values at time grid nodes by O(τ ²), where τ is the time step. A somewhat worse error estimate is obtained for the one-dimensional heat equation with arbitrary sufficiently smooth boundary data, namely, $O\left( {\tau ^2 \ln \frac{T} {\tau }} \right) $ , where T is the length of the time interval.     

Characterizing and minimizing the operator split error for Fisher’s equation

Operator splitting (OS) is a popular and convenient technique used to numerically solve reactive transport problems such as Fisher’s equation. Although OS has been widely used to solve Fisher’s equation, no characterization of the innate OS error has been presented. Here the exact characteristics of the OS error for travelling wave solutions of Fisher’s equation are revealed and explored. The analysis shows that the OS error behaves differently to previously studied linear problems by smoothing or steepening the wave front depending on the sequential order of splitting. Further analysis confirms that the OS error is reduced by implementing an alternating OS scheme.     

Symbolic computation of high order compact difference schemes for three dimensional linear elliptic partial differential equations with variable coefficients

We present a symbolic computation procedure for deriving various high order compact difference approximation schemes for certain three dimensional linear elliptic partial differential equations with variable coefficients. Based on the Maple software package, we approximate the leading terms in the truncation error of the Taylor series expansion of the governing equation and obtain a 19 point fourth order compact difference scheme for a general linear elliptic partial differential equation. A test problem is solved numerically to validate the derived fourth order compact difference scheme. This symbolic derivation method is simple and can be easily used to derive high order difference approximation schemes for other similar linear elliptic partial differential equations.     

Evolution Galerkin methods for hyperbolic systems in two space dimensions

The subject of the paper is the analysis of three new evolution Galerkin schemes for a system of hyperbolic equations, and particularly for the wave equation system. The aim is to construct methods which take into account all of the infinitely many directions of propagation of bicharacteristics. The main idea of the evolution Galerkin methods is the following: the initial function is evolved using the characteristic cone and then projected onto a finite element space. A numerical comparison is given of the new methods with already existing methods, both those based on the use of bicharacteristics as well as commonly used finite difference and finite volume methods. We discuss the stability properties of the schemes and derive error estimates.

     

On some finite difference schemes for solution of hyperbolic heat conduction problems

We consider the accuracy of two finite difference schemes proposed recently in [Roy S., Vasudeva Murthy A.S., Kudenatti R.B., A numerical method for the hyperbolic-heat conduction equation based on multiple scale technique, Appl. Numer. Math., 2009, 59(6), 1419–1430], and [Mickens R.E., Jordan P.M., A positivity-preserving nonstandard finite difference scheme for the damped wave equation, Numer. Methods Partial Differential Equations, 2004, 20(5), 639–649] to solve an initial-boundary value problem for hyperbolic heat transfer equation. New stability and approximation error estimates are proved and it is noted that some statements given in the above papers should be modified and improved. Finally, two robust finite difference schemes are proposed, that can be used for both, the hyperbolic and parabolic heat transfer equations. Results of numerical experiments are presented.     

Fully discrete finite element approximations of the forced Fisher equation

We consider fully discrete finite element approximations of the forced Fisher equation that models the dynamics of gene selection/migration for a diploid population with two available alleles in a multidimensional habitat and in the presence of an artificially introduced genotype. Finite element methods are used to effect spatial discretization and a nonstandard backward Euler method is used for the time discretization. Error estimates for the fully discrete approximations are derived by applying the Brezzi-Rappaz-Raviart theory for the approximation of a class of nonlinear problems. The approximation schemes and error estimates are applicable under weaker regularity hypotheses than those that are typically assumed in the literature. The algorithms and analyses, although presented in the concrete setting of the forced Fisher equation, also apply to a wide class of semilinear parabolic partial differential equations.     

Error estimates of H^1-Galerkin mixed finite element method for Schrodinger equation

An H1-Galerkin mixed finite element method is discussed for a class of second order SchrSdinger equation. Optimal error estimates of semidiscrete schemes are derived for problems in one space dimension. At the same time, optimal error estimates are derived for fully discrete schemes. And it is showed that the H1-Galerkin mixed finite element approximations have the same rate of convergence as in the classical mixed finite element methods without requiring the LBB consistency condition.     

高阶Schrdinger方程的一族恒稳差分格式

1 引言 高阶Schrdinger方程在量子力学、非线性光学及流体力学中有着广泛的应用,其最简单的模型方程为,即 (其中,i=(-1)~(1/2),m为正整数) (1) 文[2-3]研究了方程(1)的辛算法及差分解法。文[2]先将方程(1)写成Hamilton形     

Construction of splitting schemes based on transition operator approximation

The stability analysis of approximate solutions to unsteady problems for partial differential equations is usually based on the use of the canonical form of operator-difference schemes. Another possibility widely used in the analysis of methods for solving Cauchy problems for systems of ordinary differential equations is associated with the estimation of the norm of the transition operator from the current time level to a new one. The stability of operator-difference schemes for a first-order model operator-differential equation is discussed. Primary attention is given to the construction of additive schemes (splitting schemes) based on approximations of the transition operator. Specifically, classical factorized schemes, componentwise splitting schemes, and regularized operator-difference schemes are related to the use of a certain multiplicative transition operator. Additive averaged operator-difference schemes are based on an additive representation of the transition operator. The construction of second-order splitting schemes in time is discussed. Inhomogeneous additive operator-difference schemes are constructed in which various types of transition operators are used for individual splitting operators.     

Discrete time high-order schemes for viscosity solutions of Hamilton-Jacobi-Bellman equations

Summary. A general method for constructing high-order approximation schemes for Hamilton-Jacobi-Bellman equations is given. The method is based on a discrete version of the Dynamic Programming Principle. We prove a general convergence result for this class of approximation schemes also obtaining, under more restrictive assumptions, an estimate in of the order of convergence and of the local truncation error. The schemes can be applied, in particular, to the stationary linear first order equation in . We present several examples of schemes belonging to this class and with fast convergence to the solution. Received July 4, 1992 / Revised version received July 7, 1993