A reduced MFE formulation based on POD for the non-stationary conduction-convection problems

In this article, a reduced mixed finite element (MFE) formulation based on proper orthogonal decomposition (POD) for the non-stationary conduction-convection problems is presented. Also the error estimates between the reduced MFE solutions based on POD and usual MFE solutions are derived. It is shown by numerical examples that the results of numerical computation are consistent with theoretical conclusions. Moreover, it is shown that the reduced MFE formulation based on POD is feasible and efficient in finding numerical solutions for the non-stationary conduction-convection problems.     

A dual-level method of fundamental solutions for three-dimensional exterior high frequency acoustic problems

Physical essence of the fictitious boundary of the method of fundamental solutions has been a mystery for a long time. In this study, we attempt to explain the reason why fictitious boundary has such a dramatic effect on numerical results. The influence law of the fictitious boundary on numerical results is revealed. Based on this understanding, a dual-level method of fundamental solutions with self-adaptive adjustment coefficients is proposed. The competitive attributes of the method are that it inherits the high numerical efficiency of the method of fundamental solutions, and it improves numerical stability significantly. The effect of the fictitious boundary on numerical results is eliminated by introducing the concept of equivalent slope. It should be noted that the dual-level method of fundamental solutions can simulate exterior high frequency acoustic problems under the lowest sampling frequency allowed by the Shannon's sampling theorem. Numerical experiment with up to non-dimensional wavenumber of 600 has successfully been conducted on a single laptop when one uses 100,000 degrees of freedom.     

Numerical studies of time-independent and time-dependent scattering by several elliptical cylinders

A numerical solution to the problem of time-dependent scattering by an array of elliptical cylinders with parallel axes is presented. The solution is an exact one, based on the separation-of-variables technique in the elliptical coordinate system, the addition theorem for Mathieu functions, and numerical integration. Time-independent solutions are described by a system of linear equations of infinite order which are truncated for numerical computations. Time-dependent solutions are obtained by numerical integration involving a large number of these solutions. First results of a software package generating these solutions are presented: wave propagation around three impenetrable elliptical scatterers. As far as we know, this method described has never been used for time-dependent multiple scattering.     

Numerical analysis of stochastic oscillators on supercomputers

The problem of numerical analysis of stochastic differential equations (SDEs) with oscillating solutions is investigated. The expectation and variance of SDE numerical solutions are shown as functions of the mesh size of integrating the generalized Euler method. Results of some numerical experiments on the simulation of linear and nonlinear stochastic oscillators on the supercomputer of the Siberian Supercomputer Center are presented.     

Comparison of numerical schemes for solving the advection equation

We report on the dispersion and dissipation properties of numerical schemes aimed at solving the one-dimensional advection equation. The study is based on the consistency error, which is explicitly calculated for various standard finite-difference schemes. The oscillation and damping features of the numerical solutions are shown to be explained via a generalized Airy-like function. In the specific case of the advection of a step function, the solutions of the equivalent equations are systematically calculated and shown to recover the numerical solutions. A particular emphasis is put on one third-order accurate scheme, which involves a weak smearing of the step.     

The impact of finite precision arithmetic and sensitivity on the numerical solution of partial differential equations

In this paper, we address some fundamental issues concerning “time marching” numerical schemes for computing steady state solutions of boundary value problems for nonlinear partial differential equations. Simple examples are used to illustrate that even theoretically convergent schemes can produce numerical steady state solutions that do not correspond to steady state solutions of the boundary value problem. This phenomenon must be considered in any computational study of nonunique solutions to partial differential equations that govern physical systems such as fluid flows. In particular, numerical calculations have been used to “suggest” that certain Euler equations do not have a unique solution. For Burgers' equation on a finite spatial interval with Neumann boundary conditions the only steady state solutions are constant (in space) functions. Moreover, according to recent theoretical results, for any initial condition the corresponding solution to Burgers' equation must converge to a constant as t → ∞. However, we present a convergent finite difference scheme that produces false nonconstant numerical steady state “solutions.” These erroneous solutions arise out of the necessary finite floating point arithmetic inherent in every digital computer. We suggest the resulting numerical steady state solution may be viewed as a solution to a “nearby” boundary value problem with high sensitivity to changes in the boundary conditions. Finally, we close with some comments on the relevance of this paper to some recent “numerical based proofs” of the existence of nonunique solutions to Euler equations and to aerodynamic design.     

Variational inequalities of multilayer viscoelastic systems with interlayer Tresca friction: Existence and uniqueness of solution and convergence of numerical solution

The mathematical model of multilayer viscoelastic system based on the actual structure of asphalt pavement and the properties of its numerical solutions are studied. Firstly, based on the partial differential equations, the variational inequality model of the multilayer viscoelastic system is derived, and the existence and uniqueness of its solutions are further proved. Then, based on the finite element theory, the convergence property and error analysis of the semi-discrete numerical solutions of the variational inequalities are further studied. Finally, the convergence and error analysis of the fully discrete numerical solutions of the variational inequalities are derived by the forward difference method. The conclusion of the above error analysis also confirms the feasibility of studying the mechanical response of asphalt pavement based on variational inequality.     

Superconvergence of splitting positive definite mixed finite element for parabolic optimal control problems

In this paper, we investigate the superconvergence of fully discrete splitting positive definite mixed finite element (MFE) methods for parabolic optimal control problems. For the space discretization, the state and co-state are approximated by the lowest order Raviart–Thomas MFE spaces and the control variable is approximated by piecewise constant functions. The time discretization of the state and co-state are based on finite difference methods. We derive the superconvergence between the projections of exact solutions and numerical solutions or the exact solutions and postprocessing numerical solutions for the control, state and co-state. A numerical example is provided to validate the theoretical results.     

Numerical methods for ordinary differential equations on matrix manifolds

In recent years differential systems whose solutions evolve on manifolds of matrices have acquired a certain relevance in numerical analysis. A classical example of such a differential system is the well-known Toda flow. This paper is a partial survey of numerical methods recently proposed for approximating the solutions of ordinary differential systems evolving on matrix manifolds. In particular, some results recently obtained by the author jointly with his co-workers will be presented. We will discuss numerical techniques for isospectral and isodynamical flows where the eigenvalues of the solutions are preserved during the evolution and numerical methods for ODEs on the orthogonal group or evolving on a more general quadratic group, like the symplectic or Lorentz group. We mention some results for systems evolving on the Stiefel manifold and also review results for the numerical solution of ODEs evolving on the general linear group of matrices.     

求解非线性互补问题的微分方程方法

本文构造了一种求解非线性互补问题的微分方程方法.在一定条件下,证明了微分方程系统的平衡点是非线性互补问题的解并且基于一般微分方程系统的数值积分建立了一个数值算法.在适当的条件下,证明了此算法产生的序列解是收敛的.本文最后给出了数值结果,该结果表明了此微分方程方法的有效性.     

二重数值积分公式的高精度对偶公式及其应用

给出了计算二重积分的Simpson公式与两点高斯公式的对偶公式的构造过程,得到与之对应的高精度对偶修正解,提高了二重数值积分公式的计算精度,同时给出了二重积分的一种估值方法.最后,应用于几个典型的数值算例,计算结果表明：对偶修正解比对应的数值积分公式及其对偶公式的解有更高的计算精度和更快的收敛速度.     

A numerical approach for solving an extended Fisher-Kolomogrov-Petrovskii-Piskunov equation

In the present paper a numerical method, based on finite differences and spline collocation, is presented for the numerical solution of a generalized Fisher integro-differential equation. A composite weighted trapezoidal rule is manipulated to handle the numerical integrations which results in a closed-form difference scheme. A number of test examples are solved to assess the accuracy of the method. The numerical solutions obtained, indicate that the approach is reliable and yields results compatible with the exact solutions and consistent with other existing numerical methods. Convergence and stability of the scheme have also been discussed.     

Balanced truncation model order reduction in limited time intervals for large systems

In this article we investigate model order reduction of large-scale systems using time-limited balanced truncation, which restricts the well known balanced truncation framework to prescribed finite time intervals. The main emphasis is on the efficient numerical realization of this model reduction approach in case of large system dimensions. We discuss numerical methods to deal with the resulting matrix exponential functions and Lyapunov equations which are solved for low-rank approximations. Our main tool for this purpose are rational Krylov subspace methods. We also discuss the eigenvalue decay and numerical rank of the solutions of the Lyapunov equations. These results, and also numerical experiments, will show that depending on the final time horizon, the numerical rank of the Lyapunov solutions in time-limited balanced truncation can be smaller compared to standard balanced truncation. In numerical experiments we test the approaches for computing low-rank factors of the involved Lyapunov solutions and illustrate that time-limited balanced truncation can generate reduced order models having a higher accuracy in the considered time region.     

刚性方程组M~.(t)+C~.X(t)+KX(t)=F(t)解的探讨

讨论了刚性常微分方程组(1)的解析解和数值解,给出了解的一般形式和应用该算法的数值例子.     

Delay-Dependent Stability of Linear Multistep Methods for Neutral Systems with Distributed Delays

This paper considers the asymptotic stability of linear multistep (LM) methods for neutral systems with distributed delays. In particular, several sufficient conditions for delay-dependent stability of numerical solutions are obtained based on the argument principle. Compound quadrature formulae are used to compute the integrals. An algorithm is proposed to examine the delay-dependent stability of numerical solutions. Several numerical examples are performed to verify the theoretical results.     

Exponential stability of numerical solutions for a class of stochastic age-dependent capital system with Poisson jumps

Recently, numerical solutions of stochastic differential equations have received a great deal of attention. Numerical approximation schemes are invaluable tools for exploring their properties. In this paper, we introduce a class of stochastic age-dependent (vintage) capital system with Poisson jumps. We also give the discrete approximate solution with an implicit Euler scheme in time discretization. Using Gronwall’s lemma and Barkholder-Davis-Gundy’s inequality, some criteria are obtained for the exponential stability of numerical solutions to the stochastic age-dependent capital system with Poisson jumps. It is proved that the numerical approximation solutions converge to the analytic solutions of the equations under the given conditions, where information on the order of approximation is provided. These error bounds imply strong convergence as the timestep tends to zero. A numerical example is used to illustrate the theoretical results.     

Convergence and Mean-Square Stability of Exponential Euler Method for Semi-Linear Stochastic Delay Integro-Differential Equations

In this paper, the numerical methods for semi-linear stochastic delay integro-differential equations are studied. The uniqueness, existence and stability of analytic solutions of semi-linear stochastic delay integro-differential equations are studied and some suitable conditions for the mean-square stability of the analytic solutions are also obtained. Then the numerical approximation of exponential Euler method for semi-linear stochastic delay integro-differential equations is constructed and the convergence and the stability of the numerical method are studied. It is proved that the exponential Euler method is convergent with strong order $\frac{1}{2}$ and can keep the mean-square exponential stability of the analytical solutions under some restrictions on the step size. In addition, numerical experiments are presented to confirm the theoretical results.     

带Markov跳时变随机种群收获系统数值解的收敛性

讨论了一类带Markov跳时变随机种群收获系统的数值解问题.利用EulerMaruyama方法给出了时变种群系统的数值解表达式,在局部Lipschitz条件下,证明了方程的数值解在均方意义下收敛于其解析解.最后,通过数值例子对所给出的结论进行了验证.     

New analytical and CWENO numerical solutions for axisymmetric horizontal flows with heat sources

Some new nonlinear analytical solutions are found for axisymmetric horizontal flows dominated by strong heat sources. These flows are common in multiscale atmospheric and oceanic flows such as hurricane embryos and ocean gyres. The analytical solutions are illustrated with several examples. The proposed exact solutions provide analytical support for previous numerical observations and can be also used as benchmark problems for validating numerical models. A central weighted essentially non-oscillatory (CWENO) reconstruction is also employed for numerical simulation of the corresponding integro-differential equations. Due to the use of the same polynomial reconstruction for all derivatives and integral terms, the balance between those terms is well preserved, and the method can precisely reproduce the exact solutions, which are hard to capture by traditional upwind schemes. The developed analytical solutions were employed to evaluate the performance of the numerical method, which showed an excellent performance of the numerical model in terms of numerical diffusion and oscillation.