A dimensional split preconditioner for Stokes and linearized Navier-Stokes equations

In this paper we introduce a new preconditioner for linear systems of saddle point type arising from the numerical solution of the Navier-Stokes equations. Our approach is based on a dimensional splitting of the problem along the components of the velocity field, resulting in a convergent fixed-point iteration. The basic iteration is accelerated by a Krylov subspace method like restarted GMRES. The corresponding preconditioner requires at each iteration the solution of a set of discrete scalar elliptic equations, one for each component of the velocity field. Numerical experiments illustrating the convergence behavior for different finite element discretizations of Stokes and Oseen problems are included.     

椭圆型界面问题的破裂再生核方法

本文基于一维椭圆型界面问题提出了一种有效的数值方法.首先,根据模型构建一个崭新的破裂再生核空间.其次,应用破裂再生核方法给出了此类界面问题的近似解,并讨论该方法的收敛性.最后,通过几个有效的数值算例来说明该方法的精确性和稳定性.     

TWO-STAGE FOURTH-ORDER ACCURATE TIME DISCRETIZATIONS FOR 1D AND 2D SPECIAL RELATIVISTIC HYDRODYNAMICS

This paper studies the two-stage fourth-order accurate time discretization [J.Q. Li and Z.F. Du, SIAM J. Sci. Comput., 38 (2016)] and its application to the special relativistic hydrodynamical equations. Our analysis reveals that the new two-stage fourth-order accurate time discretizations can be proposed. With the aid of the direct Eulerian GRP (generalized Riemann problem) methods and the analytical resolution of the local "quasi 1D" GRP, the two-stage fourth-order accurate time discretizations are successfully implemented for the 1D and 2D special relativistic hydrodynamical equations. Several numerical experiments demonstrate the performance and accuracy as well as robustness of our schemes.     

Preliminary tests of a hybrid numerical-asymptotic method for solving nonlinear advection-diffusion equations in a domain limited by a self-adjusting boundary

An advection-diffusion equation is examined in which the diffusion coefficient is proportional to a positive power of the dependent variable, h. Because the diffusion coefficient is zero where h is zero, it is believed that the domain where h ≠ 0 expands at a finite velocity, u_η, which must be calculated in an appropriate way if an accurate numerical solution technique is to be implemented. An asymptotic study leads to a local approximation of u_η. The latter is then utilized in a finite volume solution of an axisymmetric problem where an exact solution can be obtained. The accuracy of the numerical results is excellent. Some peculiarities of the numerical solution are highlighted, and it is shown that they are due to the simplistic nature of the axisymmetric problem solved, which may be partly responsible for the high quality of the numerical results. Although the preliminary results are very encouraging, further testing of the method is needed, especially in fully two-dimensional cases.     

Splitting-based conjugate gradient method for a multi-dimensional linear inverse heat conduction problem

In this paper we consider a multi-dimensional inverse heat conduction problem with time-dependent coefficients in a box, which is well-known to be severely ill-posed, by a variational method. The gradient of the functional to be minimized is obtained by the aid of an adjoint problem, and the conjugate gradient method with a stopping rule is then applied to this ill-posed optimization problem. To enhance the stability and the accuracy of the numerical solution to the problem, we apply this scheme to the discretized inverse problem rather than to the continuous one. The difficulties with large dimensions of discretized problems are overcome by a splitting method which only requires the solution of easy-to-solve one-dimensional problems. The numerical results provided by our method are very good and the techniques seem to be very promising.     

一维Allen-Cahn方程紧差分格式的离散最大化原则和能量稳定性研究

本文主要研究相场模拟中的Allen-Cahn模型,考虑一维Allen-Cahn方程紧差分方法的数值逼近.建立具有O(τ2+h4)精度的全离散紧差分格式,证明在合理的步长比和时间步长的约束下,其数值解满足离散最大化原则,在此基础上,研究了全离散格式的能量稳定性.最后给出数值算例.     

A plane wave method combined with local spectral elements for nonhomogeneous Helmholtz equation and time-harmonic Maxwell equations

In this paper we are concerned with plane wave discretizations of nonhomogeneous Helmholtz equation and time-harmonic Maxwell equations. To this end, we design a plane wave method combined with local spectral elements for the discretization of such nonhomogeneous equations. This method contains two steps: we first solve a series of nonhomogeneous local problems on auxiliary smooth subdomains by the spectral element method, and then apply the plane wave method to the discretization of the resulting (locally homogeneous) residue problem on the global solution domain. We derive error estimates of the approximate solutions generated by this method. The numerical results show that the resulting approximate solutions possess high accuracy.     

Numerical Solution of the Two-Sided Space–Time Fractional Telegraph Equation Via Chebyshev Tau Approximation

The operational matrices of left Caputo fractional derivative, right Caputo fractional derivative, and Riemann–Liouville fractional integral, for shifted Chebyshev polynomials, are presented and derived. We propose an accurate and efficient spectral algorithm for the numerical solution of the two-sided space–time Caputo fractional-order telegraph equation with three types of non-homogeneous boundary conditions, namely, Dirichlet, Robin, and non-local conditions. The proposed algorithm is based on shifted Chebyshev tau technique combined with the derived shifted Chebyshev operational matrices. We focus primarily on implementing the novel algorithm both in temporal and spatial discretizations. This algorithm reduces the problem to a system of algebraic equations greatly simplifying the problem. This system can be solved by any standard iteration method. For confirming the efficiency and accuracy of the proposed scheme, we introduce some numerical examples with their approximate solutions and compare our results with those achieved using other methods.     

Quasi-one-dimensional method for the two-dimensional inverse problem of magnetotelluric sounding

The article presents a quasi-one-dimensional method for solving the inverse problem of electromagnetic sounding. The quasi-one-dimensional method is an iteration process that in each iteration solves a parametric one-dimensional inverse problem and a two-dimensional direct problem. The solution results of these problems are applied to update the input values for the parametric one-dimensional inverse problem in the next iteration. The method has been implemented for a two-dimensional inverse problem of magnetotelluric sounding in a quasi-layered medium.     

A posteriori error estimation for arbitrary order FEM applied to singularly perturbed one-dimensional reaction-diffusion problems

FEM discretizations of arbitrary order r are considered for a singularly perturbed one-dimensional reaction-diffusion problem whose solution exhibits strong layers. A posteriori error bounds of interpolation type are derived in the maximum norm. An adaptive algorithm is devised to resolve the boundary layers. Numerical experiments complement our theoretical results.     

A Hybrid Immersed Interface Method for Driven Stokes Flow in an Elastic Tube

We present a hybrid numerical method for simulating fluid flow through a compliant, closed tube, driven by an internal source and sink. Fluid is assumed to be highly viscous with its motion described by Stokes flow. Model geometry is assumed to be axisymmetric, and the governing equations are implemented in axisymmetric cylindrical coordinates, which capture 3D flow dynamics with only 2D computations. We solve the model equations using a hybrid approach: we decompose the pressure and velocity fields into parts due to the surface forcings and due to the source and sink, with each part handled separately by means of an appropriate method. Because the singularly-supported surface forcings yield an unsmooth solution, that part of the solution is computed using the immersed interface method. Jump conditions are derived for the axisymmetric cylindrical coordinates. The velocity due to the source and sink is calculated along the tubular surface using boundary integrals. Numerical results are presented that indicate second-order accuracy of the method.     

Abstract theory of hybridizable discontinuous Galerkin methods for second-order quasilinear elliptic problems

An abstract theory for discretizations of second-order quasilinear elliptic problems based on the mixed-hybrid discontinuous Galerkin method. Discrete schemes are formulated in terms of approximations of the solution to the problem, its gradient, flux, and the trace of the solution on the interelement boundaries. Stability and optimal error estimates are obtained under minimal assumptions on the approximating space. It is shown that the schemes admit an efficient numerical implementation.     

An efficient cubic B-spline and bicubic B-spline collocation method for numerical solutions of multidimensional nonlinear stochastic quadratic integral equations

This paper aims to develop a novel numerical approach on the basis of B-spline collocation method to approximate the solution of one-dimensional and two-dimensional nonlinear stochastic quadratic integral equations. The proposed approach is based on the hybrid of collocation method, cubic B-spline, and bi-cubic B-spline interpolation and Itô approximation. Using this method, the problem solving turns into a nonlinear system solution of equations that is solved by a suitable numerical method. Also, the convergence analysis of this numerical approach has been discussed. In the end, examples are given to test the accuracy and the implementation of the method. The results are compared with the results obtained by other methods to verify that this method is accurate and efficient.     

An iterative multi level algorithm for solving nonlinear ill–posed problems

Summary. The convergence analysis of Landweber's iteration for the solution of nonlinear ill–posed problem has been developed recently by Hanke, Neubauer and Scherzer. In concrete applications, sufficient conditions for convergence of the Landweber iterates developed there (although quite natural) turned out to be complicated to verify analytically. However, in numerical realizations, when discretizations are considered, sufficient conditions for local convergence can usually easily be stated. This paper is motivated by these observations: Initially a discretization is fixed and a discrete Landweber iteration is implemented until an appropriate stopping criterion becomes active. The output is used as an initial guess for a finer discretization. An advantage of this method is that the convergence analysis can be considered in a family of finite dimensional spaces. The numerical performance of this multi level algorithm is compared with Landweber's iteration. Received October 21, 1996 / Revised version received July 28, 1997     

Analytical and numerical investigations of a batch crystallization model

This article is concerned with the analytical and numerical investigations of a one-dimensional population balance model for batch crystallization processes. We start with a one-dimensional batch crystallization model and prove the local existence and uniqueness of the solution of this model. For this purpose Laplace transformation is used as a basic tool. A semi-discrete high resolution finite volume scheme is proposed for the numerical solution of the current model. The issues of positivity (monotonicity), consistency, stability and convergence of the proposed scheme for the current model are analyzed and proved. Finally, we give a numerical test problem. The numerical results of the proposed high resolution scheme are compared with the solution of the reduced four-moments model and the first-order upwind scheme.     

An iterative approach to solve a nonlinear moving boundary problem describing the solvent diffusion within glassy polymers

This paper consists in studying a mathematical model of solvent diffusion through the glassy polymers as a one-dimensional moving boundary problem with kinetic undercooling. We establish an iterative variable time-step method based on a nonstandard finite difference (NSFD) scheme to solve the considered moving boundary problem. The monotonicity and positivity of the numerical solution are proved. The numerical approach is investigated for three test problems composed of constant and inconstant diffusion coefficients for different values of parameters to demonstrate the validity and ability of the method.     

Theoretical and numerical analysis for the quasi-continuum approximation of a material particle model

In many applications materials are modeled by a large number of particles (or atoms) where any one of particles interacts with all others. Near or nearest neighbor interaction is expected to be a good simplification of the full interaction in the engineering community. In this paper we shall analyze the approximate error between the solution of the simplified problem and that of the full-interaction problem so as to answer the question mathematically for a one-dimensional model. A few numerical methods have been designed in the engineering literature for the simplified model. Recently much attention has been paid to a finite-element-like quasicontinuum (QC) method which utilizes a mixed atomistic/continuum approximation model. No numerical analysis has been done yet. In the paper we shall estimate the error of the QC method for this one-dimensional model. Possible ill-posedness of the method and its modification are discussed as well.

     

炸药爆炸作用下飞片的运动

刚性飞片在炸药爆炸作用下的一维抛掷问题,仅当爆炸气体多方指数等于三时可以求得解析解;一般情况下须利用计算机求出数值解本文利用了爆炸气体中反射冲击波的“弱”击波特性,使飞片运动和飞片后方流场之间相互耦合的复杂问题解耦而归结为求解常微分方程问题;然后用小参数摄动法求出多方指数接近于三的各种炸药驱动飞片问题的近似解析解,所得飞片终速和数值解符合很好;从而给出了用爆速和多方指数等两个炸药示性参量表出的估算飞片运动的良好近似公式.     

Fundamental mode exact schemes for unsteady problems

The problem of increasing the accuracy of an approximate solution is considered for boundary value problems for parabolic equations. For ordinary differential equations (ODEs), nonstandard finite difference schemes are in common use for this problem. They are based on a modification of standard discretizations of time derivatives and, in some cases, allow to obtain the exact solution of problems. For multidimensional problems, we can consider the problem of increasing the accuracy only for the most important components of the approximate solution. In the present work, new unconditionally stable schemes for parabolic problems are constructed, which are exact for the fundamental mode. Such two‐level schemes are designed via a modification of standard schemes with weights using Padé approximations. Numerical results obtained for a model problem demonstrate advantages of the proposed fundamental mode exact schemes.     

An accurate numerical solution for the transient heating of an evaporating spherical droplet

A recently derived numerical algorithm for one-dimensional time-dependent Stefan problems is extended for the purposes of solving a moving boundary problem for the transient heating of an evaporating spherical droplet. The Keller box finite-difference scheme is used, in tandem with the so-called boundary immobilization method. An important component of the work is the careful use of variable transformations that must be built into the numerical algorithm in order to preserve second-order accuracy in both time and space - an issue not previously discussed in relation to this widely-used scheme. In addition, we demonstrate that our solution is in close agreement with the solution obtained using an alternative numerical scheme that employs an analytic solution of the heat conduction equation inside the droplet, for which the droplet radius was assumed to be a piecewise linear function of time. The advantages of the new method are discussed.