On the Behavior of Combination High-Order Compact Approximations with Preconditioned Methods in the Diffusion-Convection Equation

In this paper, a family of high-order compact finite difference methods in combination preconditioned methods are used for solution of the Diffusion-Convection equation. We developed numerical methods by replacing the time and space derivatives by compact finite-difference approximations. The system of resulting nonlinear finite difference equations are solved by preconditioned Krylov subspace methods. Numerical results are given to verify the behavior of high-order compact approximations in combination preconditioned methods for stability, convergence. Also, the accuracy and efficiency of the proposed scheme are considered.

A truncated implicit high-order finite-difference scheme combined with boundary conditions

In this paper, first we calculate finite-difference coefficients of implicit finite- difference methods （IFDM） for the first and second-order derivatives on normal grids and first- order derivatives on staggered grids and find that small coefficients of high-order IFDMs exist. Dispersion analysis demonstrates that omitting these small coefficients can retain approximately the same order accuracy but greatly reduce computational costs. Then, we introduce a mirrorimage symmetric boundary condition to improve IFDMs accuracy and stability and adopt the hybrid absorbing boundary condition （ABC） to reduce unwanted reflections from the model boundary. Last, we give elastic wave modeling examples for homogeneous and heterogeneous models to demonstrate the advantages of the proposed scheme.

EVOLUTION ANALYSIS OF TS WAVE AND HIGH-ORDER HARMONIC WAVES IN BOUNDARY LAYERS

Linear and nonlinear evolutions of TS wave and high-order harmonic waves in boundary layers are studied based on the parabolic stability equation （PSE）. Initial conditions are derived by the local method with the Landau expansion. The evolution process and characteristics of the disturbance amplitude and the velocity profile, etc. , especially stronger nonlinear effects, are computed by an efficient numerical method. Effects and regulations of different initial amplitudes, frequencies and pressure gradients on the evolution of disturbances are explored, which are directly relative to the stability and the transition in boundary layers. Simulation results are in good agreement with the data of the accuracy direct numerical simulation （DNS） using full Navier-Stokes equations.

A STUDY ON THE WEIGHT FUNCTION OF THE MOVING LEAST SQUARE APPROXIMATION IN THE LOCAL BOUNDARY INTEGRAL EQUATION METHOD

The meshless method is a new numerical technique presented in recent years.It uses the moving least square(MLS)approximation as a shape function.The smoothness of the MLS approximation is determined by that of the basic function and of the weight function,and is mainly determined by that of the weight function.Therefore,the weight function greatly affects the accuracy of results obtained.Different kinds of weight functions,such as the spline function, the Gauss function and so on,are proposed recently by many researchers.In the present work,the features of various weight functions are illustrated through solving elasto-static problems using the local boundary integral equation method.The effect of various weight functions on the accuracy, convergence and stability of results obtained is also discussed.Examples show that the weight function proposed by Zhou Weiyuan and Gauss and the quartic spline weight function are better than the others if parameters c and α in Gauss and exponential weight functions are in the range of reasonable values,respectively,and the higher the smoothness of the weight function,the better the features of the solutions.

POINTWISE CONVERGENCE RATE OF VANISHING VISCOSITY APPROXIMATIONS FOR SCALAR CONSERVATION LAWS WITH BOUNDARY

This article is concerned with the pointwise error estimates for vanishing vis- cosity approximations to scalar convex conservation laws with boundary.By the weighted error function and a bootstrap extrapolation technique introduced by Tadmor-Tang,an optimal pointwise convergence rate is derived for the vanishing viscosity approximations to the initial-boundary value problem for scalar convex conservation laws,whose weak entropy solution is piecewise C 2 -smooth with interaction of elementary waves and the ...

Wind Structure in an Intermediate Boundary Layer Model Based on Ekman Momentum Approximation

A quasi three–dimensional, intermediate planetary boundary layer (PBL) model is developed by including inertial acceleration with the Ekman momentum approximation, but a nonlinear eddy viscosity based on Blackadar’s scheme was included to improve the theoretical model proposed by Tan and Wu (1993). The model could keep the same complexity as the classical Ekman model in numerical, but extends the conventional Ekman model to include the horizontal accelerated flow with the Ekman momentum approximation. A comparison between this modified Ekman model and other simplified accelerating PBL models is made. Results show that the Ekman model overestimates (underestimates) the wind speed and pumping velocity in the cyclonic (anticyclonic) shear flow due to the neglect of the acceleration flow, however, the semi–geostrophic Ekman model overestimates the acceleration effects resulting from the underestimating (overestimating) of the wind speed and pumping velocity in the cyclonic (anticyclonic) shear flow. The Ekman momentum approximation boundary layer model could be applied to the baroclinic atmosphere. The baroclinic Ekman momentum approximation boundary layer solution has both features of classical baroclinic Ekman layer and the Ekman momentum approximate boundary lager.

Exponential Convergence of Finite-dimensional Approximations to Linear Bond-based Peridynamic Boundary Value Problems

In this paper, we demonstrate that the finite-dimensional approximations to the solutions of a linear bond-based peridynamic boundary value problem converge to the exact solution exponentially with the analyticity assumption of the forcing term, therefore greatly improve the convergence rate derived in literature.

RESTRICTED NONLINEAR APPROXIMATION AND SINGULAR SOLUTIONS OF BOUNDARY INTEGRAL EQUATIONS

This paper studies several problems , which are potentially relevant for the construction of adaptive numerical schemes. First, biorthogonal spline wavelets on [0,1] are chosen as a starting point for characterizations of functions in Besom spaces B(?)(0,1) with 0<σ<∞ and (1+σ)-1<γ<∞. Such function spaces are known to be related to nonlinear approximation. Then so called restricted nonlinear approximation procedures with respect to Sobolev space norms are considered. Besides characterization results Jackson type estimates for various tree-type and tresholding algorithms are investigated. Finally known approximation results for geometry induced singularity functions of boundary integeral equations are combined with the characterization results for restricted nonlinear approximation to show Besov space regularity results.

Real scalar field scattering with polynomial approximation around Schwarzschild–de Sitter black-hole

As one of the fitting methods, the polynomial approximation is effective to process sophisticated problem. In this paper, we employ this approach to handle the scattering of scalar field around the Schwarzschild-de Sitter blackhole. The complicated relationship between tortoise coordinate and radial coordinate is replaced by the approximate polynomial. The Schroedinger-like equation, the real boundary conditions and the polynomial approximation construct a full Sturm Liouville type problem. Then this boundary value problem can be solved numerically for two limiting cases： the first one is the Nariai black-hole whose horizons are close to each other, the second one is the black-hole with the horizons widely separated. Compared with previous results （Brevik and Tian）, the field near the event horizon and cosmological horizon can have a better description.

Solution of the Euler Equations with Approximate Boundary Conditions for Thin Airfoils

This paper presents an efficient numerical method for solving the Euler equations on rectilinear grids. Wall boundary conditions on the surface of an airfoil are implemented by using their first order expansions on the airfoil chord line, which is placed along a grid line. However, the method is not restricted to flows with small disturbances since there are no restrictions on the magnitude of the velocity or pressure perturbations. The mathematical formulation and the numerical implementation of the wall boundary conditions in a finite volume Euler code are described. Steady transonic flows are calculated about the NACA 0006, NACA 0012 and NACA 0015 airfoils, corresponding to thickness ratios of 6%, 12%, and 15%, respectively. The computed results, including surface pressure distributions, the lift coefficient, the wave drag coefficient, and the pitching moment coefficient, at angles of attack from 0° to 8° are compared with solutions at the same conditions by FLO52, a well established Euler code using body fitted curvilinear grids. Results demonstrate that the method yields acceptable accuracies even for the relatively thick NACA 0015 airfoil and at high angles of attack. This study establishes the potential of extending the method to computing unsteady fluid structure interaction problems, where the use of a stationary rectilinear grid offers substantial advantages in both computer time and human work since it would not require the generation of time dependent body fitted grids.

Entropy generation analysis of tangent hyperbolic fluid in quadratic Boussinesq approximation using spectral quasi-linearization method

In many industrial applications,heat transfer and tangent hyperbolic fluid flow processes have been garnering increasing attention,owing to their immense importance in technology,engineering,and science.These processes are relevant for polymer solutions,porous industrial materials,ceramic processing,oil recovery,and fluid beds.The present tangent hyperbolic fluid flow and heat transfer model accurately predicts the shear-thinning phenomenon and describes the blood flow characteristics.Therefore,the entropy production analysis of a non-Newtonian tangent hyperbolic material flow through a vertical microchannel with a quadratic density temperature fluctuation(quadratic/nonlinear Boussinesq approximation)is performed in the present study.The impacts of the hydrodynamic flow and Newton’s thermal conditions on the flow,heat transfer,and entropy generation are analyzed.The governing nonlinear equations are solved with the spectral quasi-linearization method(SQLM).The obtained results are compared with those calculated with a finite element method and the bvp4c routine.In addition,the effects of key parameters on the velocity of the hyperbolic tangent material,the entropy generation,the temperature,and the Nusselt number are discussed.The entropy generation increases with the buoyancy force,the pressure gradient factor,the non-linear convection,and the Eckert number.The non-Newtonian fluid factor improves the magnitude of the velocity field.The power-law index of the hyperbolic fluid and the Weissenberg number are found to be favorable for increasing the temperature field.The buoyancy force caused by the nonlinear change in the fluid density versus temperature improves the thermal energy of the system.

Approximation Theorem of K.Fan for Positive Mapping and Applications

In this paper, we investigate the validity of approximation theorm of K. Fan for a demicompact 1-set-contraction map defined on a closed ball,an annulus and a sphere in cones. From this, we improve all recent results of Lin [2]. As applications of our theorems, we discuss the existence of positive solutions to twopoint boundary value-problems of differential equations in Banach space. At the same time, the recent main results of (3) established by Guo Dajun are Generalized and supplemented.

On the approximate solution of synoptic flow equation with discontinuous boundary conditions

In the present paper we investigate existence and uniqueness generalized solution for initial boundary value problem of synoptic flow equation with discontinuous boundary conditions. We consider Rothe-Galerkin method for given problem and reduce numerical calculations.

DIFFERENCE SCHEMES BASING ON COEFFICIENT APPROXIMATION

In respect of variable coefficient differential equations, the equations of coefficient function approximation were more accurate than the coefficient to be frozen as a constant in every discrete subinterval. Usually, the difference schemes constructed based on Taylor expansion approximation of the solution do not suit the solution with sharp function. Introducing into local bases to be combined with coefficient function approximation, the difference can well depict more complex physical phenomena, for example, boundary layer as well as high oscillatory,with sharp behavior. The numerical test shows the method is more effective than the traditional one.

Approximate Formulation and Numerical Solution for Hypersingular Boundary Integral Equations in Plane Elasticity

Based on the fact that the singular boundary integrals in the sense of Cauchy principal value can be represented approximately by the mean values of two companion nearly singular boundary integrals, a vary general approach was developed in the paper. In the approach, the approximate formulation before discretization was constructed to cope with the difficulties encountered in the corner treatment in the formulations of hypersingular boundary integral equations. This makes it possible to solve the hypersingular boundary integral equation numerically in a non regularized form and in a local manner by using conforming C 0 quadratic boundary elements and standard Gaussian quadratures similar to those employed in the conventional displacement BIE formulations. The approximate formulation is very convenient to use because the corner information is comprised naturally in the representations of those approximate integrals. Numerical examples in plane elasticity show that with the present approach, the compatible or better results can be achieved in comparison with those of the conventional BIE formulations.

Stabilized seventh-order dissipative compact scheme using simultaneous approximation terms

To ensure time stability of a seventh-order dissipative compact finite difference scheme, fourth-order boundary closures are used near domain boundaries previously. However, this would reduce the global convergence rate to fifth-order only. In this paper, we elevate the boundary closures to sixth-order to achieve seventh-order global accuracy. To keep the improved scheme time stable, the simultaneous approximation terms (SATs) are used to impose boundary conditions weakly. Eigenvalue analysis shows that the improved scheme is time stable. Numerical experiments for linear advection equations and one-dimensional Euler equations are implemented to validate the new scheme.

A staggered-grid high-order finite-difference modeling for elastic wave field in arbitrary tilt anisotropic media

The paper presents a staggered-grid any even-order accurate finite-difference scheme for two-dimensional (2D), three-component (3C), first-order stress-velocity elastic wave equation and its stability condition in the arbitrary tilt anisotropic media; and derives a perfectly matched absorbing layer (PML) boundary condition and its stag- gered-grid any even-order accurate difference scheme in the 2D arbitrary tilt anisotropic media. The results of nu- merical modeling indicate that the modeling precision is high, the calculation efficiency is satisfactory and the absorbing boundary condition is better. The wave-front shapes of elastic waves are complex in the anisotropic media, and the velocity of qP wave is not always faster than that of qS wave. The wave-front triplication of qS wave and its events in both reflected domain and propagated domain, which are not commonly hyperbola, is a common phenomenon. When the symmetry axis is tilted in the TI media, the phenomenon of S-wave splitting is clearly observed in the snaps of three components and synthetic seismograms, and the events of all kinds of waves are asymmetric.

The Exact and Approximate Solutions of Some Boundary Value Problems in Domains with Angular Points of the Boundary

On the basis of the exact solution of biharmonic problems of elasticity theory in a half-strip one possible reason is shown of those problems that arise when an approximate or numerical approaches leading the solution of boundary value problems to infinite systems of linear algebraic equations. Construction of exact solutions of some boundary value problems for differential equations in partial derivatives is not possible without their extensions to Riemann surfaces. Moreover, each of the boundary value problem corresponds to its Riemann surface. This fact is important to consider when developing an effective approximate and numerical methods of solving boundary value problems.

Numerical Studies on the Generation and Propagation of Tsunami Waves Based on the High-Order Spectral Method

An effective numerical model for wave propagation over three-dimensional(3D)bathymetry was developed based on the High-Order Spectral(HOS)method and combined with a moving bottom boundary.Based on this model,tsunami waves caused by various mechanisms were simulated and analyzed.Two-dimensional bed upthrust and the effect of the uplift velocity of the bathymetry on the wave profiles of tsunami waves were studied.Next,tsunami waves caused by 3D submarine slides were generated and the effects of the slide velocity,slide dimension and water depth on the tsunami waves were analyzed.Based on wavelet analysis,the properties of the tsunami wave propagation were investigated.The results show that the bottom movement can significantly affect the generation and propagation of tsunami waves and the studies could help understand the mechanisms of tsunamis caused by a moving bottom boundary.

MULTIPLE RECIPROCITY METHOD WITH TWO SERIES OF SEQUENCES OF HIGH-ORDER FUNDAMENTAL SOLUTION FOR THIN PLATE BENDING

The boundary value problem of plate bending problem on two_parameter foundation was discussed.Using two series of the high_order fundamental solution sequences, namely, the fundamental solution sequences for the multi_harmonic operator and Laplace operator, applying the multiple reciprocity method(MRM), the MRM boundary integral equation for plate bending problem was constructed. It proves that the boundary integral equation derived from MRM is essentially identical to the conventional boundary integral equation. Hence the convergence analysis of MRM for plate bending problem can be obtained by the error estimation for the conventional boundary integral equation. In addition, this method can extend to the case of more series of the high_order fundamental solution sequences.