an implicit fourth order compact finite difference scheme for one dimensional burgers equation abstract a fourth order compact finite difference method is proposed in this paper to solve one dimensional burgers equation the newly proposed method is based on the hopf cole transformation which transforms the original nonlinear burgers equation into a linear heat equation and transforms the dirichlet boundary condition into the robin boundary condition the linear heat equation is then solved by an implicit fourth order compact finite difference scheme a compact fourth order formula is also developed to approximate the robin boundary conditions while the initial condition for the heat equation is approximated using simpson s rule to maintain overall fourth order accuracy numerical experiments have been conducted to demonstrate the efficiency and high order accuracy of this method the numerical results also show that the method is unconditionally stable as there is no constraint on time step size

weak formulation based haar wavelet method for solving differential equations the haar wavelet based discretization method for solving differential equations is developed nonlinear burgers equation is considered as a test problem both strong and weak formulations based approaches are discussed the discretization scheme proposed is based on the weak formulation an attempt is made to combine the advantages of the fem and haar wavelets the obtained numerical results have been validated against a closed form analytical solution as well as fem results good agreement with the exact solution has been observed

least squares quadratic b spline finite element method for the regularised long wave equation abstract an approximate solution consisting of a combination of the quadratic b spline functions is incorporated into the least squares method an application of this method is presented for computing the solution of the regularised long wave rlw equation quadratic b spline solution of the rlw equation leads to tridiagonal matrix system which is solved easily by using the thomas algorithm performance of this scheme is tested by obtaining a solitary wave solution of the equation and by studying the development of the undular bore numerical results of the proposed algorithm is shown to have higher accuracy in terms of l 2 error norm compared studying the migration of the solitary wave a fourier stability analysis of the method is investigated

a modified reproducing kernel method for solving burgers equation arising from diffusive waves in fluid dynamics abstract as we known reproducing kernel method rkm has been presented for solving differential equations for initial and boundary value problems however the direct application of the rkm presented in the previous works cannot produce good numerical results for burgers equation to solve this problem this paper give a modified reproducing kernel method by piecewise technique the exact solution is given by reproducing kernel functions in a series expansion form the approximation solution is expressed by n term summation of reproducing kernel functions the three numerical experiments results show that the piecewise method is more easily implemented and effective some numerical results are also compared with the results obtained by other methods

a numerical solution of burgers equation by pseudospectral method and darvishi s preconditioning in this paper we solve the burgers equation by pseudospectral method this equation is one dimensional quasilinear partial differential equation to reduce round off error of pseudospectral method we use darvishi s preconditioning the numerical results obtained by this method for different values of viscosity have been compared with the exact solution it was seen that they were in good agreement with each other because norm infinity of error is very small

numerical approach for solving diffusion problems using cubic b spline collocation method abstract a numerical approach for modeling the removal of homogeneous solutes from the porous structure is presented pore diffusion equations controlling the overall process are expressed in terms of concentration of solute as a function of position in bed and washing period a computational scheme specifying knot vector s and control points de boor points to solve diffusion model using cubic b spline collocation method cscm is given the knot vector provides sufficient resolution and control points yield an accurate approximation of the desired curve convergence of numerical scheme is of o h 2 four problems have been worked out by the proposed scheme results are plotted graphically for fairly wide range of parameters a comparison with the previous techniques demonstrates the superiority of the proposed method on basis of computational time efficiency and accuracy

brief lq performance bounds for adaptive output feedback controllers for functionally uncertain nonlinear systems we consider functionally uncertain systems which can be written in an output feedback form where the nonlinearities are functions of the output only the uncertainty is described by a weighted l 2 norm about a nominal system and an approximate adaptive design is given which ensures output practical stability the main result requires knowledge of the weighted l 2 uncertainty level an upper bound on the lq performance of the output transient and the control input is derived where the cost penalises the output transient and the control effort on the time interval where the output lies outside the prescribed neighbourhood of zero to which we achieve convergence

a numerical solution of burgers equation by finite element method constructed on the method of discretization in time abstract in this paper the galerkin finite element method constructed on the method of discretization in time was applied to solve the one dimensional nonlinear burgers equation the system of nonlinear equations obtained for each time step was solved by using the newton method in order to show the efficiency of the presented method the numerical solutions obtained for various values of viscosity were compared with the exact solutions it was seen that they were in excellent agreement

efficient numerical techniques for burgers equation this paper presents new efficient numerical techniques for solving one dimensional quasi linear burgers equation burgers equation is used as a model problem in the study of turbulence boundary layer behavior shock waves convection dominated diffusion phenomena gas dynamics acoustic attenuation in fog and continuum traffic simulation using a non linear cole hopf transformation the burgers equation is reduced to one dimensional diffusion equation the linearized diffusion equation is semi discretized by using method of lines mol which leads to a system of ordinary differential equations in time resulting system of ordinary differential equations is solved by backward differentiation formulas bdf of order one two and three and the analysis of numerical errors are presented numerical results for modest values of kinematic viscosity are compared with the exact solution to demonstrate the efficiency of proposed numerical methods

a finite element approach for solution of burgers equation in this study for the burgers equation which is used a model problem in turbulence and shock wave theory was given a finite element solution considering the numerical results obtained for different values of viscosity compared with cole s analytical solution it was seen that they were in good agreement

k version of finite element method in gas dynamics higher order global differentiability numerical solutions in this paper we consider and examine alternate finite element computational strategies for time dependent navier stokes equations describing high speed compressible flows with shocks in a viscous and conducting medium with the ultimate objective of establishing the desired features of a general mathematical and computational framework for such initial value problems ivp in which a the numerically computed solutions are in agreement with the physics of evolution described by the governing differential equations gdes i e the ivp b the solutions are admissible in the non discretized form of the gdes in the pointwise sense i e anywhere and everywhere in the entire space time domain and hence in the integrated sense as well c the numerical approximations progressively approach the same global differentiability in space and time as the theoretical solutions d it is possible to time march the solutions this is essential for efficiency as well as ensuring desired accuracy of the computed solution for the current increment of time i e to minimize the error build up in the time marching process e the computational process is unconditionally stable and non degenerate regardless of the choice of discretization nature of approximations and their global differentiability and the dimensionless parameters influencing the physics of the process f there are no issues of stability cfl number limitations and g the mathematical and computational methodology is independent of the nature of the space time differential operators we consider one dimensional compressible flow in a viscous and conducting medium with shocks as model problems to illustrate various features of the general mathematical and computational framework used here and to demonstrate that the proposed framework is general and is applicable to all ivp the riemann shock tube with a single diaphragm serves as a model problem the specific details presented in the paper discuss 1 choice of the form of the gdes i e strong form or weak form 2 various choices of variables the paper establishes and considers density velocity and temperature as variables of choice 3 details of the space time least squares ls integral forms meritorious over all others in all aspects are presented and choice of approximation spaces are discussed 4 in all numerical studies we consider a viscous and conducting medium with ideal gas law however results are also presented for non conducting medium extension of this work to real gas models will be presented in a separate paper it is worth noting that when the medium is viscous and conducting the solutions of gas dynamics equations are analytic 5 it is also significant to note that upwinding methods based on addition of artificial diffusion such as supg supg dc supg dc ls and their many variations are neither needed nor used in this present work 6 numerical studies are aimed at resolving the localized details of the shock structure i e shock relations shock width shock speed etc as well as the over all global behaviour of the solution in the entire space time domain 7 numerical studies are presented for riemann shock tube for high mach number flows with special emphasis also on time accuracy of the evolution which is ensured by requiring that the approximations for each increment of time satisfy non discretized form of the gdes in the pointwise sense and hence in the integrated sense as well 8 comparisons are made with published results as well as theoretical solutions when possible it is established that space time least squares processes are the only processes that yield variationally consistent space time integral forms and hence unconditionally non degenerate space time computational processes which when considered in higher order scalar product spaces provide the desired mathematical framework in which progressively higher order global differentiability solutions in space and time yield the same characteristics as the theoretical solutions of the ivp in all aspects copyright 2006 john wiley sons ltd

p version least squares finite element formulation for convection diffusion problems this paper presents a p version least squares finite element formulation of the convection diffusion equation the second order differential equation describing convection diffusion is reduced to a series of equivalent first order differential equations for which the least squares formulation is constructed using the same order of approximation for each of the dependent variables the hierarchical approximation functions and the nodal variable operators are established by first constructing the one dimensional hierarchical approximation functions of orders p and p and the corresponding nodal variable operators in and direction and then taking their products numerical results are presented and compared with analytical and numerical solutions for a two dimensional test problem to demonstrate the accuracy and the convergence characteristics of the present formulation the gaussian quadrature rule used to calculate the numerical values of the element matrices vectors as well as the error functional i e is established based on the highest degree of the polynomial in the integrands it is demonstrated that this quadrature rule with the present p version formulation produces excellent results for very low as well as extremely high peclet numbers 10 106 and furthermore the error functional i sum of the integrals of e2 is a monotonically decreasing function of the number of degrees of freedom as the p levels are increased for a fixed mesh it is shown that exact integration with the h version linear and parabolic elements produces inaccurate solutions at high peclet numbers results are also presented using reduced integration it is shown that the reduced integration with p version produces accurate values of the primary variable even for relatively low p levels but the error functional i when calculated using the proper integration rule has a much higher value due to errors in the derivatives of the primary variable and is a non monotonic function of the degrees of freedom as p levels are increased for a fixed mesh similar behaviour of the error functional i is also observed for the h models using linear elements when reduced integration is used although the h models using parabolic elements produce monotonic error functional behaviour as the number of degrees of freedom are increased the error values are inferior to the p version results using exact integration

continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamic systems we consider a dynamic system containing uncertain elements only the set of possible values of these uncertainties is known based on this information a class of state feedback controls is proposed in order to guarantee uniform ultimate boundedness of every system response within an arbitrarily small neighborhood of the zero state these feedback controls are continuous functions of the state

quadratic b spline finite element method for numerical solution of the burgers equation abstract in this paper the one dimensional nonlinear burgers equation was converted to a set of nonlinear ordinary differential equations by the method of discretization in time and then each of them was solved by applying the quadratic b spline finite element method in order to show the efficiency of the presented method two problems were considered the numerical solutions obtained for various values of viscosity were compared with the exact solutions it is seen that the results obtained by this way are quite satisfactory

dielectric function for a material containing hyperspherical inclusions to o c 2 i multipole expansions by averaging over pairs of hyperspheres we have obtained the dielectric function for a binary mixture containing hyperspherical inclusions up to order c 2 where c is the volume fraction of inclusions the method used is based on multipole expansions for the potential of two spheres in a uniform eld and is a generalization of the method of jerey to d dimensional space numerical results are presented for the second order coecient in the low c expansion of the dielectric constant for arbitrary d these verify earlier known results as well as showing the dependence of on dimensionality which is particularly simple as d 1 and as d 1 ever since the time of maxwell 1873 the study of the properties of random mix tures or suspensions containing a low volume fraction of inclusions has been of much interest the problems concerned are applicable to the study of electrical conduc tion thermal conduction electric permittivity magnetic permeability and others by virtue of the universality of laplace s equation in this paper we will concentrate on the case of dielectric inclusions other cases being easily obtainable by renaming the symbols maxwell 1873 provided the exact rst order coecient to o c for a system of spherical inclusions other similar problems too c not within the univer sal class have also attracted much interest for example the o c coecient for the viscosity of a suspension containing a system of hard spheres was found by einstein 1906 more recent exact results for the o c coecient of electrical conductivity for inclusions of other shapes in two dimensions were obtained by thorpe 1992 using conformal mapping a method favoured by maxwell it took nearly 100 years before any serious quantitative studies were made of the second order o c 2 coecient the original work being done by batchelor 1972 and batchelor green 1972 who unfortunately undertook initially to study the more complex fluid suspensions problem to this day few exact results are known batchelor 1974 1977 we remark here that the second order coecient we are interested in is related to the so called huggins coecient h for fluid suspensions i e the expansion of the viscosity is given by 0 1 c h c 2

a numerical solution of burgers equation based on least squares approximation abstract burgers equation which is one dimensional non linear partial differential equation was converted to p non linear ordinary differential equations by using the method of discretization in time each of them was solved by the least squares method for various values of viscosity at different time steps the numerical solutions obtained were compared with the exact solutions it was seen that both of them were in excellent agreement while the exact solution was not available for viscosity smaller than 0 01 it was shown that mathematical structure of the equation for the obtained numerical solutions did not decay

an introduction to finite element methods for transient advection problems abstract an introduction is presented to time accurate finite element methods recently developed for solving unsteady problems governed by linear and nonlinear advection equations this includes methods making an explicit use of the characteristic curves methods based on the taylor series in time and least squares methods

a qualocation method for burgers equation in this paper a qualocation method for the one dimensional burgers equation is proposed a semidiscrete scheme along with optimal error estimates is discussed results of a numerical experiment performed support the theoretical results

a predictor corrector compact finite difference scheme for burgers equation abstract in this paper a compact predictor corrector finite difference scheme is proposed to solve the burgers equation the scheme is based on compact derivatives approximation by which we get the spatial approximations of first order derivatives and second order derivatives with fourth order accuracy both for inner nodes and boundary nodes for the first time derivative item a two step predictor corrector method called maccormack method is used numerical experiments show the scheme is in good agreement with the exact solutions

biorthogonal multiwavelets on the interval for numerical solutions of burgers equation to numerically solve the burgers equation in this paper we propose a general method for constructing wavelet bases on the interval 0 1 derived from symmetric biorthogonal multiwavelets on the real line in particular we obtain wavelet bases with simple structures on the interval 0 1 from the hermite cubic splines in comparison with all other known constructed wavelets on the interval 0 1 our constructed wavelet bases on the interval 0 1 from the hermite cubic splines not only have good approximation and symmetry properties with extremely short supports but also employ a minimum number of boundary wavelets with a very simple structure these desirable properties make them to be of particular interest in numerical algorithms our constructed wavelet bases on the interval 0 1 are then used to solve the nonlinear burgers equation our method is based on the finite difference formula combined with the collocation method therefore our proposed numerical scheme in this paper is abbreviated as mfdcm mixed finite difference and collocation method some numerical examples are provided to demonstrate the validity and applicability of our proposed method which can be easily implemented to produce a desired accuracy

robustness of the tuning functions adaptive backstepping design for linear systems we study robustness of the adaptive backstepping design with tuning functions for linear systems under assumptions on unmodeled dynamics and disturbances equal to those for certainty equivalence schemes we address an adaptive scheme not based on the certainty equivalence principle in the process of redesign for robustness we employ only leakage in the estimator we do not employ normalization neither static nor dynamic a fundamental difference between the tuning functions design and the certainty equivalence designs is that the controller in the former is inherently nonlinear while in the latter it is nonlinear only in the parameter estimate as a result achievable robustness results for the tuning functions scheme are not global but regional with a region of attraction inversely proportional to the size of the unmodeled dynamics the tracking error is proportional to the size of the uncertainties

a weak galerkin finite element method for burgers equation abstract we propose a weak galerkin wg finite element method for solving the one dimensional burgers equation based on a new weak variational form both semi discrete and fully discrete wg finite element schemes are established and analyzed we prove the existence of the discrete solution and derive the optimal order error estimates in a discrete h 1 norm and the standard l 2 norm respectively numerical experiments are presented to illustrate our theoretical analysis

p version least squares finite element formulation for axisymmetric incompressible non newtonian fluid flow abstract the paper presents a p version least squares finite element formulation lsfef for axisymmetric incompressible non newtonian fluid flow the dimensionless form of the equations describing the fluid motion are cast into a set of first order coupled partial differential equations involving non newtonian stresses as auxiliary variables the pressure velocities primary variables and the stresses auxiliary variables are interpolated using equal order c0 continuity p version hierarchical approximation functions the least squares functional or error functional is constructed using the system of coupled first order non linear partial differential equations integrated sum of squares of the errors resulting from the individual equations for the entire discretization without linearization approximations or assumptions the minimization of this least squares error functional results in finding a solution vector for which the partial derivatives of the error functional with respect to the nodal degrees of freedom becomes zero this is accomplished by using newton s method with a line search the paper presents an implementation of the power law model for non newtonian viscosity numerical examples are presented to demonstrate the convergence characteristics and the accuracy of the proposed formulation

fourth order finite difference method for solving burgers equation abstract in this paper we present fourth order finite difference method for solving nonlinear one dimensional burgers equation this method is unconditionally stable the convergence analysis of the present method is studied and an upper bound for the error is derived numerical comparisons are made with most of the existing numerical methods for solving this equation

a numerical solution of burgers equation in this paper the burgers equation which is one dimensional quasi linear parabolic partial differential equation was solved by a variational method constructed on the method of discretization in time the numerical results obtained by these ways for various values of viscosity have been compared with the exact solution it was seen that they were in good agreement with each other

a direct variational methods applied to burgers equation in this study a compact approximate method in limiting form for calculating the solution of burgers equation with appropriate boundary conditions is presented the results obtained by present method are found to be in good agreement with those due to earlier authors and offers appreciable advantages for burgers like nonlinear problems

a space time coupled p version least squares finite element formulation for unsteady two dimensional navier stokes equations this paper presents a p version least squares finite element formulation for two dimensional unsteady fluid flow described by navier stokes equations where the effects of space and time are coupled the dimensionless form of the navier stokes equations are first cast into a set of first order differential equations by introducing auxiliary variables this permits the use of c0 element approximation the element properties are derived by utilizing the p version approximation functions in both space and time and then minimizing the error functional given by the space time integral of the sum of squares of the errors resulting from the set of first order differential equations this results in a true space time coupled least squares minimization procedure the application of least squares minimization to the set of coupled first order partial differential equations results in finding a solution vector which makes gradient of error functional with respect to a null vector this is accomplished by using newton s method with a line search a time marching procedure is developed in which the solution for the current time step provides the initial conditions for the next time step equilibrium iterations are carried out for each time step until the error functional and each component of the gradient of the error functional with respect to nodal degrees of freedom are below a certain prespecified tolerance the space time coupled p version approximation functions provide the ability to control truncation error which in turn permits very large time steps what literally requires hundreds of time steps in uncoupled conventional time marching procedures can be accomplished in a single time step using the present space time coupled approach the generality success and superiority of the present formulation procedure is demonstrated by presenting specific numerical examples for transient couette flow and transient lid driven cavity the results are compared with the analytical solutions and those reported in the literature the formulation presented here is ideally suited for space time adaptive procedures the element error functional values provide a mechanism for adaptive h p or hp refinements

numerical solutions of nonlinear burgers equation with modified cubic b splines collocation method abstract in this paper a numerical method is proposed to approximate the solution of the nonlinear burgers equation the method is based on collocation of modified cubic b splines over finite elements so that we have continuity of the dependent variable and its first two derivatives throughout the solution range we apply modified cubic b splines for spatial variable and derivatives which produce a system of first order ordinary differential equations we solve this system by using ssp rk43 or ssp rk54 scheme this method needs less storage space that causes less accumulation of numerical errors the numerical approximate solutions to the burgers equation have been computed without transforming the equation and without using the linearization illustrative eleven examples are included to demonstrate the validity and applicability of the technique easy and economical implementation is the strength of this method

a space time coupled p version least squares finite element formulation for unsteady fluid dynamics problems this paper presents a p version least squares finite element formulation for unsteady fluid dynamics problems where the effects of space and time are coupled the dimensionless form of the differential equations describing the problem are first cast into a set of first order differential equations by introducing auxiliary variables this permits the use of c element approximation the element properties are derived by utilizing p version approximation functions in both space and time and then minimizing the error functional given by the space time integral of the sum of squares of the errors resulting from the set of first order differential equations this results in a true space time coupled least squares minimization procedure a time marching procedure is developed in which the solution for the current time step provides the initial conditions for the next time step the space time coupled p version approximation functions provide the ability to control truncation error which in turn permits very large time steps what literally requires hundreds of time steps in uncoupled conventional time marching procedures can be accomplished in a single time step using the present space time coupled approach for non linear problems the non linear algebraic equations resulting from the least squares process are solved using newton s method with a line search this procedure results in a symmetric hessian matrix equilibrium iterations are carried out for each time step until the error functional and each component of the gradient of the error functional with respect to nodal degrees of freedom are below a certain prespecified tolerance the generality success and superiority of the present formulation procedure is demonstrated by presenting specific formulations and examples for the advection diffusion and burgers equations the results are compared with the analytical solutions and those reported in the literature the formulation presented here is ideally suited for space time adaptive procedures the element error functional values provide a mechanism for adaptive h p or hp refinements the work presented in this paper provides the basis for the extension of the space time coupled least squares minimization concept to two and three dimensional unsteady fluid flow

numerical solution of nonlinear burgers equation using high accuracy multi quadric quasi interpolation in this paper a numerical method is presented to approximate the solution to nonlinear burgers equation which is related to many scientific research topics a numerical scheme by using high accuracy multi quadric quasi interpolation is presented in which a kind of multi quadric quasi interpolant is used to approximate the derivatives of the solution in spatial domain and finite difference is used to approximate the derivatives of the solution in temporal domain the advantage of the scheme is that it is mesh free and in each time step only a multi quadric quasi interpolant is employed so that the algorithm is easy to implement the numerical results of this scheme are also shown and compared with other numerical schemes

second order interactions in a medium containing perfect conducting hyperspherical inclusions we consider the interaction between two hyperspherical inclusions surrounded by a medium experiencing an e field in the perfect conducting limit by the use of d dimensional dipole moments we show that the dielectric function of the medium can be calculated to any order n in particular d n the coefficient of o c 2 in the series expansion for the dielectric function is determined in terms of a dimensional dependence which even though it is mathematically complex proves to be superior in convergence to other methods we calculate the potential difference between the two hyperspheres for various limits including the all important closely packed limit using the theory of continued fractions we investigate the convergence of the interaction terms between the two inclusions and obtain results that reduce the enormous number of calculations that need to be computed as n the latter may be useful in the pursuit of a theory that resums the complicated interaction terms present in the two body medium problem with a view towards an improved effective medium theory

a least squares finite element scheme for the ew equation abstract the equal width ew equation is solved by a least squares technique using linear space time finite elements in simulations of the migration of a single solitary wave this algorithm is shown to have good accuracy and conservation the development of solitary waves from an arbitrary initial condition within an ew environment is examined

generalized multiscale finite element methods for space time heterogeneous parabolic equations abstract in this paper we consider local multiscale model reduction for problems with multiple scales in space and time we developed our approaches within the framework of the generalized multiscale finite element method gmsfem using space time coarse cells the main idea of gmsfem is to construct a local snapshot space and a local spectral decomposition in the snapshot space previous research in developing multiscale spaces within gmsfem focused on constructing multiscale spaces and relevant ingredients in space only in this paper our main objective is to develop a multiscale model reduction framework within gmsfem that uses space time coarse cells we construct space time snapshot and offline spaces we compute these snapshot solutions by solving local problems a complete snapshot space will use all possible boundary conditions however this can be very expensive we propose using randomized boundary conditions and oversampling cf calo et al 2016 we construct the local spectral decomposition based on our analysis as presented in the paper we present numerical results to confirm our theoretical findings and to show that using our proposed approaches we can obtain an accurate solution with low dimensional coarse spaces we discuss using online basis functions constructed in the online stage and using the residual information online basis functions use global information via the residual and provide fast convergence to the exact solution provided a sufficient number of offline basis functions we present numerical studies for our proposed online procedures we remark that the proposed method is a significant extension compared to existing methods which use coarse cells in space only because of 1 the parabolic nature of cell solutions 2 extra degrees of freedom associated with space time cells and 3 local boundary conditions in space time cells

numerical solutions of the burgers equation by the least squares quadratic b spline finite element method in this study a least squares quadratic b spline finite element method for calculating the numerical solutions of the one dimensional burgers like equations with appropriate boundary and initial conditions is presented three test problems have been studied to demonstrate the accuracy of the present method results obtained by the method have been compared with the exact solution of each problem and are found to be in good agreement with each other a fourier stability analysis of the method is also investigated

finite element methods for nonlinear advection abstract the taylor galerkin method for the temporal and spatial discretization of mixed initial boundary value problems is applied to derive numerical schemes for solving a single scalar conservation law equation the discretization in time is performed before the spatial approximation by introducing second order and third order accurate generalizations of the standard two level euler scheme with the help of taylor series expansions in the time step the equations in weak form are then spatially discretized by means of the conventional galerkin finite element method to obtain a new class of parameter free one step linearly implicit schemes for the solution of nonlinear hyperbolic problems numerical results for the so called inviscid burgers equation in one and two dimensions are presented to illustrate the properties of the proposed taylor galerkin schemes

a space time least square finite element scheme for advection diffusion equations abstract a space time least square finite element scheme is presented for the advection diffusion problems at moderate to high peclet numbers this scheme is designed to eliminate spurious oscillations and can be used to define the steady state solution as the asymptotic transient solution for large time numerical results using linear elements in a 1d space and bilinear elements in a 2d space demonstrate the accuracy and the stability of the new scheme

numerical solution of one dimesional burgers equation explicit and exact explicit finite difference methods this paper presents finite difference solution and analytical solution of the finite difference approximations based on the standard explicit method to the one dimensional burgers equation which arises frequently in the mathematical modelling used to solve problems in fluid dynamics results obtained by these ways for some modest values of viscosity have been compared with the exact fourier one it is shown that they are in good agreement with each other

the generalized boundary element approach to burgers equation the generalized boundary element method is presented for the numerical solution of burgers equation the new method is based on the set of boundary integral equations derived for each subdomain by using the fundamental solution for the linearized differential operator of the equation the resulting system of quasi non linear equations is solved implicitly with use of a simple iterative procedure the adaptability and the accuracy of the proposed method are demonstrated by three examples and a comparison of the numerical results with the exact solution or other existing solutions is shown for the first example

moving least squares particle hydrodynamics method for burgers equation abstract in this paper a meshless method named moving least squares particle hydrodynamics mlsph method is applied to obtain numerical solutions of the burgers equation with a set of initial and boundary conditions the influence of kernel function and smoothing length on the accuracy of mlsph method is investigated several various forms of one and two dimensional burgers equations are successfully calculated it is concluded that the proposed method can obtain second order accuracy the way of non uniform distribution is applied to present the desired performance near the shocks the numerical results obtained by the proposed method are compared with the analytical solutions and the results obtained by some other methods the satisfactorily small errors indicate the high precision of the proposed method to solve this kind of partial differential equation

least squares finite element method for the advection diffusion equation the space time least squares finite element methods are constructed for the advection diffusion equation by using both linear shape functions and quadratic b spline shape functions two test problems are studied to demonstrate the accuracy of the present methods results of the two schemes have been compared

a difference scheme for burgers equation in an unbounded domain the numerical solution of the one dimensional burgers equation in an unbounded domain is considered two artificial boundaries are introduced to make the computational domain finite on both artificial boundaries two exact boundary conditions are proposed respectively to reduce the original problem to an initial boundary value problem in a finite computational domain a difference scheme is constructed by the method of reduction of order to solve the problem in the finite computational domain at each time level only a strictly diagonal dominated tridiagonal system of linear algebraic equations needs to be solved it is proved that the difference scheme is uniquely solvable and unconditional convergent with the convergence order 3 2 in time and order 2 in space in an energy norm a numerical example demonstrates the theoretical results

dielectric function for a material containing hyperspherical inclusions to o c2 ii method of images we have developed the method of images for the solution of two hyperspheres in d dimensions in a uniform field in order to study the o c 2 coefficient in the dielectric constant of the medium the method employs extensions of image theory for classical charges and dipoles for dielectric spheres which produce novel line charge and line dipole densities numerical results for with arbitrary integer d are consistent with our previous paper i this issue and demonstrate that the image method shows superior convergence properties although computationally less convenient the procedure is in principle calculable term by term and leads to useful analytic results in the weak scattering limit this work also produces a simple algebraic approximation that is useful everywhere

least squares finite elements for first order hyperbolic systems application de la methode des moindres carres a elements finis a l approximation des systemes hyperboliques du premier ordre analyse de la stabilite exemples

numerical solution of burgers equation by the sobolev gradient method abstract burgers equation is solved numerically with sobolev gradient methods a comparison is shown with other numerical schemes presented in this journal such as modified adomian method mam 1 and by a variational method vm which is based on the method of discretization in time 2 it is shown that the sobolev gradient methods are highly efficient while at the same time retaining the simplicity of steepest descent algorithms

gaussian networks for direct adaptive control a direct adaptive tracking control architecture is proposed and evaluated for a class of continuous time nonlinear dynamic systems for which an explicit linear parameterization of the uncertainty in the dynamics is either unknown or impossible the architecture uses a network of gaussian radial basis functions to adaptively compensate for the plant nonlinearities under mild assumptions about the degree of smoothness exhibit by the nonlinear functions the algorithm is proven to be globally stable with tracking errors converging to a neighborhood of zero a constructive procedure is detailed which directly translates the assumed smoothness properties of the nonlinearities involved into a specification of the network required to represent the plant to a chosen degree of accuracy a stable weight adjustment mechanism is determined using lyapunov theory the network construction and performance of the resulting controller are illustrated through simulations with example systems

group theoretic methods applied to burgers equation abstract in this study the group theoretic methods for calculating the solution of burgers equation with appropriate boundary and initial conditions is presented the application of a one parameter group reduces the number of independent variables by one and consequently the governing partial differential equation with the boundary and initial conditions to an ordinary differential equation with the appropriate corresponding conditions the obtained differential equation is solved analytically and the solution obtained in closed form for a specific choice of boundary condition

a class of high order compact difference schemes for solving the burgers equations abstract in this paper a class of high order compact difference method is introduced for solving the burgers equations firstly a linear high order compact difference scheme is proposed to solve the one dimensional burgers equation the scheme is fourth order accurate in space and second order accurate in time linear stability analysis is conducted to show the scheme is conditionally stable because only three grid points are involved in each time level thomas algorithm can be directly used to solve the tridiagonal linear system then this method is extended to solve the two dimensional and three dimensional coupled burgers equations numerical experiments are carried out to demonstrate the accuracy and dependability of the present method

local rbf method for multi dimensional partial differential equations abstract in this paper a local meshless differential quadrature collocation method is utilized to solve multi dimensional reaction convection diffusion pdes numerically in some cases global version of the meshless method is considered as well the meshless methods approximate solution on scattered and uniform nodes in both local and global sense in the case of convection dominated pdes the local meshless method is coupled with an upwind technique to avoid spurious oscillations for this purpose a physically motivated local domain is utilized in the flow direction both regular and irregular geometries are taken into consideration numerical experiments are performed to demonstrate effective applications and accuracy of the meshless method on regular and irregular domains

uncertainty performance and model dependency in approximate adaptive nonlinear control we consider systems satisfying a matching condition that are functionally known up to weighted l sup 2 and l sup spl infin measures of uncertainty a modified lq measure of control and state transient performance is given and the performance of a class of approximate model based adaptive controllers is studied an upper performance bound is derived in terms of the uncertainty models the stability and state transient bounds require only the la uncertainty model control effort bounds require both l sup 2 and l sup spl infin uncertainty models and various structural properties of the model basis sufficient conditions are given to ensure that the performance is bounded independently of the model basis size

time accurate solution of advection diffusion problems by finite elements abstract the taylor galerkin method recently proposed for the spatial and temporal discretization of hyperbolic equations is employed to derive accurate and efficient numerical schemes for the solution of time dependent advection diffusion problems two distinct numerical strategies are discussed the first is suitable for evolutionary problems while the second is appropriate for situations in which a steady state is eventually reached for purely evolutionary problems the taylor galerkin method is applied to the complete advection diffusion equation and through modifications of the standard mass matrix is shown to generate an incremental form of the crank nicolson time stepping method such an incremental form also lends itself to the derivation of computationally simple but accurate explicit algorithms to deal with transient situations which evolve towards a highly convective steady state the global taylor galerkin method is found to be ineffective since it reduces to the standard galerkin formulation as the temporal term vanishes for such cases we suggest the use of a splitting up method in which advection and diffusion are treated separately by appropriate taylor galerkin methods the proposed methods are illustrated first for linear constant coefficient equations they are successively extended to deal with nonlinear and multi dimensional problems numerical applications indicate their effectiveness

least squares finite element formulation for hydrodynamic modeling of semiconductor devices a least squares finite element formulation is presented for the hydrodynamic modeling of semiconductor devices the formulation results in a symmetric and positive definite algebraic system its capability of adopting equal order interpolations for all unknown variables makes it simple to formulate and easy to program the semiconductor hydrodynamic equations coupled with the poisson equation are formulated as one unified equation system in this least squares finite element scheme the developed method is examined on 1 d and 2 d deep submicron semiconductor device structures to demonstrate its capability of handling the large gradients of variables and highly nonlinear source terms in the semiconductor hydrodynamic equations

numerical solution of the burgers equation by automatic differentiation abstract we compute the solution of the one dimensional burgers equation by marching the solution in time using a taylor series expansion our approach does not require symbolic manipulation and does not involve the solution of a system of linear or non linear algebraic equations instead we use recursive formulas obtained from the differential equation to calculate exact values of the derivatives needed in the taylor series we illustrate the effectiveness of our method by solving four test problems with known exact solutions the numerical solutions we obtain are in excellent agreement with the exact solutions while being superior to other previously reported numerical solutions

an exponentially fitted method for solving burgers equation in this paper an exponentially fitted method is used to numerically solve the one dimensional burgers equation the performance of the method is tested on the model involving moderately large reynolds numbers the obtained numerical results show that the method is efficient stable and reliable for solving burgers equation accurately even involving high reynolds numbers for which the exact solution fails copyright 2009 john wiley sons ltd

solutions for linear conservation laws with velocity fields in l rm infty a space time integrated least squares stils method is derived for solving the linear conservation law with a velocity field in l rm infty an existence and uniqueness result is given for the solution of this equation a maximum principle is established and finally a comparison with a renormalized solution is presented

a space time fully decoupled wavelet galerkin method for solving two dimensional burgers equations abstract a space time fully decoupled formulation for solving two dimensional burgers equations is proposed based on the coiflet type wavelet sampling approximation for a function defined on a bounded interval by applying a wavelet galerkin approach for spatial discretization nonlinear partial differential equations are first transformed into a system of ordinary differential equations in which all matrices are completely independent of time and never need to be updated in the time integration finally the mixed explicit implicit scheme is employed to solve the resulting semi discretization system by numerically studying three widely considered test problems results demonstrate that the proposed method has a much better accuracy and a faster convergence rate than many existing numerical methods most importantly the study also indicates that the present wavelet method is capable of solving the two dimensional burgers equation at high reynolds numbers

peridynamic least squares minimization this chapter presents the pd least square minimization lsm to construct the analytical expressions in integral form for pd approximation of a field variable and its derivatives on the basis of tse and the moving lsm of error similar to the pddo it is also based on the concept of pd interactions unlike the pddo it does not require the construction of pd functions at each point

the conductivity of a sheet containing a few polygonal holes and or superconducting inclusions i use conformal mapping techniques to determine the change in the conductivity of a sheet containing a few well separated holes the hole shapes studied are the equilateral triangle square pentagon and regular n gons i show that the conductivity can be written as sigma sigma 0 1 alpha n f o f 2 where f is the area fraction of the inclusions and the coefficient alpha n frac tan pi n 2 pi n gamma 4 left frac 1 n right gamma 2 left frac 2 n right which is 2 5811 2 1884 2 0878 for triangles squares and pentagons and tends to the circle limit of 2 as n rightarrow infty the coefficient alpha n is proportional to the induced dipole moment around the polygonal hole which can be found using an appropriate conformal mapping i have also examined and compared the results for long thin needle like holes in the shape of diamonds rectangles and ellipses in all cases the conductivity parallel to the needles has the limiting form sigma sigma 0 1 f while for the perpendicular conductivity i find that sigma sigma 0 1 n pi a 2 where 2a is the length of the needle and n is the number of needles per unit area for thicker needles the shape becomes important and i compare the results with recent analog experiments and computer simulations because of the reciprocity theorem all the results found here apply equally well to superconducting inclusions

numerical solution of burgers equation by cubic hermite collocation method abstract in this paper numerical solution of the non linear burgers equation are obtained by using cubic hermite collocation method chcm the advantage of the method is continuity of the dependent variable and its derivative throughout the solution range a linear stability analysis shows that the numerical scheme based on crank nicolson approximation in time is unconditionally stable this method is applied on some test problems with different choice of collocation points to validate the accuracy of the method the obtained numerical results show that the method is efficient robust and reliable even for high reynolds numbers for which the exact solution fails moreover the method can be applied to a wide class of nonlinear partial differential equations

numerical methods for nonlinear second order hyperbolic partial differential equations ii rothe s techniques for 1 d problems abstract two families of rothe s methods that are based on the discretization of the time variable and keeping the spatial one continuous for the solution of second order hyperbolic equations with damping and nonlinear source terms are presented the first family is based on time integration and results in a volterra integro differential equation which upon approximating certain time integrals can be written as an ordinary differential equation in space upon discretizing the spatial derivatives by means of finite difference formula this family results in nonlinear algebraic equations which contain exponential terms that depend on the time step and its accuracy is shown to be lower than that of an explicit second order accurate discretization in both space and time of second order hyperbolic equations the second family of methods is based on the discretization of the time derivatives and time linearization and results in linear ordinary differential equations in space upon freezing the coefficients of these equations one can integrate analytically the resulting linear ordinary differential equations to obtain piecewise exponential solutions which are continuous in space and three point finite difference equations which depend exponentially on the time step spatial grid size and the diffusion relaxation damping and reaction times the finite difference equations are shown to result in non diagonally dominant matrices unless the time step is smaller than the characteristic relaxation diffusion damping and reaction times to avoid this problem two rothe s techniques that do not account for the jacobian of the reaction terms in the ordinary differential operator are developed and it is shown that the rothe s techniques belonging to the second family are as accurate as the linearly implicit methods presented in part i provided that the relaxation time is smaller than the critical time and the solution does not oscillate in space when the relaxation time is greater than the critical one rothe s methods have been found to be less accurate than the finite difference techniques of part i the rothe s techniques belonging to the second family are generalized to systems of nonlinear second order hyperbolic equations and mixed systems of parabolic and second order hyperbolic partial differential equations

a sixth order compact finite difference scheme to the numerical solutions of burgers equation abstract a numerical solution of the one dimensional burgers equation is obtained using a sixth order compact finite difference method to achieve this a tridiagonal sixth order compact finite difference scheme in space and a low storage third order total variation diminishing runge kutta scheme in time have been combined the scheme is implemented to solve two test problems with known exact solutions comparisons of the computed results with exact solutions showed that the method is capable of achieving high accuracy and efficiency with minimal computational effort the present results are also seen to be more accurate than some available results given in the literature

linearly implicit approximate factorization exponential methods for multi dimensional reaction diffusion equations abstract linearly implicit approximate factorization methods are used to reduce the solution of multi dimensional reaction diffusion equations to a sequence of two point boundary value problems in ordinary differential equations which are solved analytically in a piecewise manner it is shown that depending on the allocation of the transient diffusion and reaction terms to the second order ordinary differential operators several families of piecewise exponential methods can be obtained when only diffusion is included a quadratic analytical solution is obtained whereas if transient and diffusion diffusion and reaction and transient diffusion and reaction terms are included piecewise exponential methods result it is shown that exponential methods that only account for reaction and diffusion in the differential operator may result in ill conditioned systems of algebraic equation whereas if diffusion transient and reaction are included ill conditioning is avoided if the time step is sufficiently small it is also shown that the most robust exponential method is that that accounts for transient and diffusion terms in the differential operator and this method provides solutions in good accord with those of a linearly implicit approximate factorization technique which employs a three point second order accurate discretization of the diffusion terms it is also shown that a hybrid exponential method that solves the transient reaction and diffusion terms at odd time levels and the transient and diffusion terms at even time levels also provides very accurate solutions

local variably scaled newton basis functions collocation method for solving burgers equation abstract in the present paper a meshless method which constructs derivatives discretizations based on variably scaled newton basis functions vsnbfs interpolants for localized sets of nodes is provided to solve burgers equation in the native hilbert space of the reproducing kernel using such formulas for discretizations of pdes when each internal node considered as a center leads to a sparse global of system of ordinary differential equations odes to be formed by a simple assembly of function values at the local system centrepoints for efficiency and stability reasons we use the interpolation with variably scaled kernels which are introduced recently by bozzini et al the vsnbfs can be constructed from a symmetric positive definite variably scaled kernel the main advantage of using the local method is that the overlapping domains of influence result in many small matrices with the dimension of the number of nodes included in the domain of influence for each center node instead of a large collocation matrix and hence sparse global derivative matrices from local contributions therefore the method needs less computer storage and flops the method is tested on six benchmark problems on burgers equation with mixed boundary conditions five number of nodes are used in the domain of influence of the local support the numerical results for different values of reynolds numbers re are compared with analytical solutions as well as some other numerical methods it is shown that the proposed method is efficient accurate and stable for flow with reasonably high values of re numerical results also show that using variably scaled kernels leads to results that are better than using constant scaled kernels with respect to both stability and error