Advances in Applied Mathematics and Mechanics AAMM Volume 4, Number 1

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Volume 4, Number 1

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Convergence Analysis of the Spectral Methods for Weakly Singular Volterra Integro-Differential Equations with Smooth Solutions

Yunxia Wei & Yanping Chen

10.4208/aamm.10-m1055

Adv. Appl. Math. Mech., 4 (2012), pp. 1-20.

Preview Full PDF 2827 27478 Abstract

Abstract

The theory of a class of spectral methods is extended to Volterra integro-differential equations which contain a weakly singular kernel $(t-s)^{-\mu}$ with $0<\mu<1$. In this work, we consider the case when the underlying solutions of weakly singular Volterra integro-differential equations are sufficiently smooth. We provide a rigorous error analysis for the spectral methods, which shows that both the errors of approximate solutions and the errors of approximate derivatives of the solutions decay exponentially in $L^\infty$-norm and weighted $L^2$-norm. The numerical examples are given to illustrate the theoretical results.

The Sensitivity Analysis for the Flow Past Obstacles Problem with Respect to the Reynolds Number

Kazufumi Ito, Zhilin Li & Zhonghua Qiao

10.4208/aamm.11-m1110

Adv. Appl. Math. Mech., 4 (2012), pp. 21-35.

Preview Full PDF 2738 27231 Abstract

Abstract

In this paper, numerical sensitivity analysis with respect to the Reynolds number for the flow past obstacle problem is presented. To carry out such analysis, at each time step, we need to solve the incompressible Navier-Stokes equations on irregular domains twice, one for the primary variables; the other is for the sensitivity variables with homogeneous boundary conditions. The Navier-Stokes solver is the augmented immersed interface method for Navier-Stokes equations on irregular domains. One of the most important contributions of this paper is that our analysis can predict the critical Reynolds number at which the vortex shading begins to develop in the wake of the obstacle. Some interesting experiments are shown to illustrate how the critical Reynolds number varies with different geometric settings.

Application of Homotopy Analysis Method for Solving Systems of Volterra Integral Equations

M. Matinfar, M. Saeidy & J. Vahidi

10.4208/aamm.10-m1143

Adv. Appl. Math. Mech., 4 (2012), pp. 36-45.

Preview Full PDF 2844 26962 Abstract

Abstract

In this paper, we prove the convergence of homotopy analysis method (HAM). We also apply the homotopy analysis method to obtain approximate analytical solutions of systems of the second kind Volterra integral equations. The HAM solutions contain an auxiliary parameter which provides a convenient way of controlling the convergence region of series solutions. It is shown that the solutions obtained by the homotopy-perturbation method (HPM) are only special cases of the HAM solutions. Several examples are given to illustrate the efficiency and implementation of the method.

A Finite Volume Method Based on the Constrained Nonconforming Rotated Q₁-Constant Element for the Stokes Problem

Jing Qi, Wanfu Tian & Yonghai Li

10.4208/aamm.11-m1156

Adv. Appl. Math. Mech., 4 (2012), pp. 46-71.

Preview Full PDF 2824 27771 Abstract

Abstract

We construct a finite volume element method based on the constrained nonconforming rotated Q₁-constant element (CNRQ₁-P₀) for the Stokes problem. Two meshes are needed, which are the primal mesh and the dual mesh. We approximate the velocity by CNRQ₁ elements and the pressure by piecewise constants. The errors for the velocity in the H¹ norm and for the pressure in the L² norm are O(h) and the error for the velocity in the L² norm is O(h²). Numerical experiments are presented to support our theoretical results.

A Consistent Characteristic Boundary Condition for General Fluid Mixture and Its Implementation in a Preconditioning Scheme

Hua-Guang Li, Nan Zong, Xi-Yun Lu & Vigor Yang

10.4208/aamm.11-m1160

Adv. Appl. Math. Mech., 4 (2012), pp. 72-92.

Preview Full PDF 2992 27481 Abstract

Abstract

Characteristic boundary conditions that are capable of handling general fluid mixtures flow at all flow speeds are developed. The formulation is based on fundamental thermodynamics theories incorporated into an efficient preconditioning scheme in a unified manner. Local one-dimensional inviscid (LODI) relations compatible to the preconditioning system are proposed to obtain information carried by incoming characteristic waves at boundaries accurately. The approach has been validated against a variety of sample problems at a broad range of fluid states and flow speeds. Both acoustic waves and hydrodynamic flow features can pass through the boundaries of computational domain transparently without any unphysical reflection or spurious distortion. The approach can be reliably applied to fluid flows at extensive thermodynamic states and flow speeds in numerical simulations. Moreover, the use of the boundary condition shows to improve the computational efficiency.

Mathematical Simulation of Cloaking Metamaterial Structures

Jichun Li & Yunqing Huang

10.4208/aamm.10-m11109

Adv. Appl. Math. Mech., 4 (2012), pp. 93-101.

Preview Full PDF 2843 27455 Abstract

Abstract

In this paper we present a rigorous derivation of the material parameters for both the cylinder and rectangle cloaking structures. Numerical results using these material parameters are presented to demonstrate the cloaking effect.

Convergence of a Distributional Monte Carlo Method for the Boltzmann Equation

Christopher R. Schrock & Aihua W. Wood

10.4208/aamm.10-m11113

Adv. Appl. Math. Mech., 4 (2012), pp. 102-121.

Preview Full PDF 2769 27033 Abstract

Abstract

Direct Simulation Monte Carlo (DSMC) methods for the Boltzmann equation employ a point measure approximation to the distribution function, as simulated particles may possess only a single velocity. This representation limits the method to converge only weakly to the solution of the Boltzmann equation. Utilizing kernel density estimation we have developed a stochastic Boltzmann solver which possesses strong convergence for bounded and $L^\infty$ solutions of the Boltzmann equation. This is facilitated by distributing the velocity of each simulated particle instead of using the point measure approximation inherent to DSMC. We propose that the development of a distributional method which incorporates distributed velocities in collision selection and modeling should improve convergence and potentially result in a substantial reduction of the variance in comparison to DSMC methods. Toward this end, we also report initial findings of modeling collisions distributionally using the Bhatnagar-Gross-Krook collision operator.

Application of Improved (G'/G)–Expansion Method to Traveling Wave Solutions of Two Nonlinear Evolution Equations

Xiaohua Liu, Weiguo Zhang & Zhengming Li

10.4208/aamm.11-m11126

Adv. Appl. Math. Mech., 4 (2012), pp. 122-130.

Preview Full PDF 2898 27668 Abstract

Abstract

In this work, the improved (G'/G)-expansion method is proposed for constructing more general exact solutions of nonlinear evolution equation with the aid of symbolic computation. In order to illustrate the validity of the method we choose the RLW equation and SRLW equation. As a result, many new and more general exact solutions have been obtained for the equations. We will compare our solutions with those gained by the other authors.

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