ONE-DIMENSIONAL ADAPTIVE GRID GENERATION

In this work, we give an adaptive grid generation method which allows a single point to be added in the regions of large variation. This method uses a quadrature rule as a weight function. Our weight function measures the variation of the solution function on each subinterval of the solution domain. The method is applied to obtain the numerical solutions of some differential equations. A comparison of the numerical solution obtained by this method and other methods is given.


INTRODUCTION
It is well known that the choice ofthe mesh points plays an important role in the accuracy of the numerical solution ofdifferential equations.For example, when the solution has large variations like large gradients, peaks, or boundary layers in some parts ofthe solution domain, we need more grid points in such regions than in regions where the solution changes smoothly.This type omesh generation is known as adaptive grid generation.
During the last two decades, a significant interest in generating and applying adaptive grids to the numerical solutions of differential equations is surfaced (see for example Denny and Landis ]; Eiseman  [2]; Lentini and Pereyra [3]; Matsuno and Dwyer [4]; and Thompson [5]).The common property of most adaptive grid methods is that they divide the solution domain into subintervals such that some positive weight function has roughly equal value over each subinterval.Most adaptive grid generation methods differ from each other in the choice ofthe weight fimction and agree in calculating the value of the given weight function at a single point.Most forms ofthe weight functions used in literature depend on the first derivative, or the second derivative, or the tnmcation error ofthe solution.
Equidistdbution schemes in literature usually use a cm4tinear coordinate transformation to transform the physical domain into a computational domain where the mesh points are equally spaced and then construct the adaptive mesh by using the differential equation w x + x 0 w where w is the given weight function.In the nxt section, we show that the use of such techniques introduces a numerical diffusion to the solution ofthe problem under consideration.
In this paper, we give an adaptive grid generation method based on using a quadrature rule as a weight function If the variation ofthe solution is large on a subinterval, then the quadrature will be large.The advantage of our weight function is that it is calculated on a subinterval ofthe solution domain, not at a single point ofthat domain.To avoid the numerical diffusion introduced in most adaptive methods, we use finite differences on irregular grid to solve the given problems in the physical domain without the use of any transformations. 2.

FINITE DIFFERENCES AND TRUNCATION ERRORS
Adaptive methods in literature usually use a curvilinear coordinate transformation to transform the physical domain into a computational domain where the mesh points are equally spaced.A one dimensional transformation can be written as x(= p(- where N is the number of subintervals in the solution domain, i.e., h l/N, h is the step size in the uniform mesh.The first and second derivatives ofthe solution in physical domain take the form and u u,, (2.2) xg ugg ugxg (2.3) xx x x in the computational domain.Hoffman [6] compared the tnmcation errors ofthe first derivatives ofthe solution function in the physical and computational domains.Thompson et al.
[7] proved that if u is x approximated in the physical domain by using a central difference for u and the exact value of x, then the truncation error in equation (2.2) is given by T x Xn_l

Xn=l
The dominant part of T is x h2p u h2pu h2pUxxx (2.5) T1 =-6---x--xx -g The first term of T contains a numerical viscosity UxWhich usually causes troubles and may distroy the solution.This term can be eliminated ifwe use central difference approximation for x.In this case, we have U U('+I U('-I +T. (2.6) The dominant part of is In the physical domain, the first derivative u x as follows (see fig. 2.1) can be approximated by using the central difference x where the dominant part ofthe mmcation error is given by The dominant part of the tnmcation error in equation (2.9) is the same as that given in equation (2.7).Hence, the central difference approximation of the first derivative u has the same truncation error in x both physical and computational domain.
Now, we compare the truncation errors for the central difference approximation of the second derivative u in the physical and computational domains.In the physical domain, we have The main part ofthe mmcation error T is In the computational domain, we have u u(i+l)-2u(i)+u(_ 1) (u(i+l)-U(i_l)XX +l-2Xi +xt_ 1) where the dominant part ofthe truncation error T is given by /T (2.12) The first term in equation (2.13) is troublesome since it depends on the derivative u which is being approximated.Tiffs term vanishes only when h h 2 This leads to uniform mesh in the physical domain which is not our interest in this work.
Hence, the use of central differences for the derivatives ofthe transformation which eliminate the term u in the tnmcation error ofthe first derivative now introduces an error that depend on u This x xx error introduces a numerical diffusion.
From the above discussion, it is clear that the truncation errors ofthe first derivative u and the x second derivative u (as shown in equations (2.7) and (2.13)) contain a numerical diffusion term in the computational domain while in the physical domain, there is a numerical diffusion term only in the truncation error of the first derivative (see equation (2.9)).De Rivas [8] suggested the following approximation x hlh2 (h +h2) which has the main pan ofthe truncation error T 6 hlh2Uxx x Therefore, it is preferable t solve a problem in the physical d than t solve it in the cmpttil domain.A compariso f the maximum error between the exact sotuti and the uerical soluti btained in physical and compmatial dmains is given in exale 4.1.
Manteffel and WNte [9] showed that the difference scheme obtained by using equations (2.10) ad (2.14) yields a second order accate solutio deite the fct that the trcati error is flwer rder. 3.

CONSTRUCTION OF ADAPTIVE MESH
In this section, we describe our procedure for constructing the adaptive mesh.To illustrate this, let us consider the differential equation where a, b, c, g are, in general, functions ofx and the boundary values u(0) and u(1) are given.
Let u(x) eC4[0,1] be an approximate solution of equation (3.1) obtained by some interpolation of the discrete numerical solution obtained on crude uniform mesh.For example, the function u(x) can be approximated by using spline or any piecewise polynomial approximations.In this work, we use quadratic spline polynomial to approximate the solution.Then by considering the midpoint Xm=(Xi + xi +1) 2.
. . . .u(x)dx---( x , + , l x,)[u(x,)+4u(xt)+2u(x,,)+4u(x,.)+u(x,/i)ge vue on a btaL we have a ge v ofe or tion (3.4).h s ca, we add e dpot x to e m pots d reat e procede d me goppg cdte is m fisfied.R w o at e nmedcal rets dd on e propeRies ofe d. ffe me ratio is ge, e converg ofe kae lufion mod y be log.h work, we ratio Mn to be 4 S Mn 4. s worL e procede of cong adape m was to op wh e n of adapte me pots is e e as e o m. procede c be ehed e foflog algo ALGOM approte fion u of equation ( 3 5. geat gs 2-4 tfl e oppg cdt tfied. 4.

NUMERICAL RESULTS
In this section, we give the numerical solution of some examples on adaptive mesh which is generated by the above algorithm.A comparison ofthe numerical solution obtained on the adaptive mesh and on a uniform mesh is given.d2u EXAMPLE 4.1.
--+4u 20x +4x 0_< x < dx In this example, we compare the numerical results obtained in the physical and computational domain.The boundary conditions are obtained fi'om the exact solution u(x)= x5.The results are given in table 4.1.Also, a comparison of the numerical results obtained by our method and some other methods is given in table 4.2.The adaptive mesh is shown in fig. 4.1.d2u du 0 0_<x_<l EXAMPLE 4.2.dx q dx with the boundary conditions u(0) 0 and u(1)= 1.This example is considered by Lick and Gaskins   [10].It has a boundary layer ofthickness 1/q near the fight boundary.Numerical results for q=20 are given in table 4.3, and the adaptive mesh is shown in figure 4.2.rx d2u du EXAMPLE 4.3.+--0 0 _< x _< r-dx dx where the boundary conditions are u(0) 0 and u(1) 1.The first and higher order derivatives ofthe solution function have singuhdfies at x=0 for r > 1.The numerical readts obtained for 1.25 are given in table 4.4, and the adaptive mesh distribution is shown in figure 4 dx dx with boundary conditions u(0) 0 and u(1) sm(x/-1).Numerical results given in table 4.5 are due to t=l 0. The adaptive mesh is shown in figure 4.4.
REMARK.In all tables, adaptive , adaptive2, and adaptive represent the numerical results on adaptive grid obtained by using the first derivative, second derivative, and our quadrature rule (equation (3.4)) as a weight function.

CONCLUSIONS
In this work, we have analyzed the numerical solution of differential equations in the physical and computational domains.The difference approximations ofthe first derivatives have the same truncation errors in both domains while the difference approximations ofthe second derivatives introduce artificial viscosity in the computational domain.The artificial viscosity can be avoided by solving the differential equations in the physical domain.
The adaptive grid generation methods generates a well suited mesh for the problems under consideration specially ifthe solution ofthese problems has a large variation in some parts ofthe solution domain.Also the use of adaptive methods does not require any priori knowledge about the solution or even the locations ofks large variations.
The numerical results presented in this paper show that the error obtained by using our adaptive method is ofalmost 50% ofthe error obtained by using other methods. []] [2] [31 [8] [9] [10]

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
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