Feb 01, 2006 read a numerical solution of burgers equation by pseudospectral method and darvishis preconditioning, applied mathematics and computation on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. Numerical methods for partial differential equations 33. In this paper, a general framework is presented for analyzing numerical methods for the evolutionary equations that admit semigroup formulations. Pdf a pseudospectral method for a nonlocal kdvburgers. The techniques have been extensively used to solve a wide range of. A meshfree interpolation method was employed by islam et al. A pseudospectral method of solution of fishers equation. The burgers equation is known to steepen negative gradients leading to the formation of socalled shocks.
Our numerical experiments show that the chebyshev collocation method is an efficient and reliable scheme for solving burgers equations with. Pseudospectral methods have become increasingly popular for solving differential equations also they are very useful in providing highly accurate solutions to differential equations. In this paper, we propose an efficient and accurate numerical method for the one and two dimensional nonlinear viscous burgers equations and coupled viscous burgers equations with various values of viscosity subject to suitable initial and boundary conditions. In section 6 we demonstrate the method on an example. Other pseudospectral optimal control techniques, such as the bellman pseudospectral method, rely on nodeclustering at the initial time to produce optimal controls. The numerical results show the advantage of such a method.
Abstract in this paper, we present a new pseudospectral method to solve the initial value problem associated to a nonlocal kdv burgers equation involving a caputotype fractional derivative. Computing nearly singular solutions using pseudospectral. Ch2and19 for a galerkin spectral method for navierstokes equations with t. The fourier pseudospectral method has been studied for a one dimensional coupled system of viscous burgers equations. Burgers equation by pseudospectral method and darvishis preconditioning. By analysis and calculation, the perturbation solution and some conservation relations of the ilw burgers equation are obtained. One of the methods to solve partial differential equations is the spectral collocation method or the pseudospectral method. An hpadaptive pseudospectral method for solving optimal control problems christopher l. The proposed exact solutions overcome the long existing problem of. Two identical solutions of the general burgers equation are separately derived by a direct integration method and the simplest equation method with the bernoulli equation being the simplest equation. Abstractspectral methods fourier galerkin, fourier pseudospectral, chebyshev tau, chebyshev collocation, spectral element and standard finite differences. Convergence of spectral methods for burgers equation. Forced ilwburgers equation as a model for rossby solitary. The finite difference method is used in time direction, while the pseudospectral method is used in xdirection.
Next, we solve fpdes, including the time and spacefractional advectiondiffusion equation, time and spacefractional multiterm fpdes, and finally the spacefractional burgers equation. An implicitexplicit spectral method for burgers equation springerlink. In this paper, we present a numerical solution of onedimensional kortewegde vries equation with variant boundary conditions by the fourier pseudospectral method. Convergence of spectral method in time for burgers equation. The basic idea is, using an algebraic map, to transform the whole real line into a bounded interval where we can apply a fourier expansion. Spectral and finite difference solutions of burgers equation citeseerx. Numerical implementation of bdf2 via method of lines for. Request pdf a numerical solution of burgers equation by pseudospectral method and darvishis preconditioning in this paper, we solve the burgers equation by pseudospectral method. In section 7 we provide further details about how our in. A numerical solution of the laxs 7 order kdv equation by. Our numerical results confirm the exponential convergence of the fractional collocation method. A fourier pseudospectral method for solving coupled viscous burgers equations. The viscous burgers equation was presented in 1940 and in 1950 hopf and in 1951 cole independently introduced the method that has come to be known as the colehopf transformation to solve the viscous burgers equation 3.
A modified leapfrog scheme is constructed in such a way. Abdou and soliman 3 used variational iteration method for solving burgers and. Quarteroni,spectral and pseudospectral methods for parabolic problems with nonperiodic boundary conditions, calcolo, 18, 1981, 197218. Oct 15, 2014 read numerical solution of the coupled viscous burgers equations by chebyshevlegendre pseudospectral method, applied mathematics and computation on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. After submitting, as a motivation, some applications of this paradigmatic equations, we continue with the mathematical analysis of them. Kdv equation by pseudospectral method and darvishis preconditioning m. A study of wave trapping between two obstacles in the forced kortewegde vries equation. The generalized stability and the convergence are proved. This paper presents a computational technique based on the pseudo. Pdf the burger s equation serves as a useful mathematical model to be applied in fluid dynamic problems.
Ps optimal control theory has been used in ground and flight systems in military and industrial applications. Hermite pseudospectral method for nonlinear partial. It combines pseudospectral ps theory with optimal control theory to produce ps optimal control theory. Finally, with the help of pseudospectral method, the numerical solutions of the forced ilw burgers equation are given. Pdf a comparison of fourier pseudospectral method and finite. This paper considers a general burgers equation with the nonlinear term coefficient being an arbitrary constant. A pseudospectral method for the onedimensional fractional laplacian on r jorge cayama 1carlota m.
The nonlinear term in the fvm, is discretion by using upwind and centraldifferencing schemes, showing a rate of convergence for the secondorder accurate. Finally, numerical results obtained by this way are compared with the exact solution to show the efficiency of the method. We have seen that in this case spectral methods yield a highly accurate and simple way to calculate derivatives. Development of the tau method for the numerical solution of twodimensional linear volterra integrodifferential equations. Rbfps method and fourier pseudospectral method for solving. The fourier method can be considered as the limit of the finitedifference method as the length of the operator tends to the number of points along a particular dimension. Fractional spectral collocation method siam journal on. Pdf the burgers equation serves as a useful mathematical model to be applied in fluid dynamic problems. A basic pseudospectral method for optimal control is based on the covector mapping principle. For solving burgers equation with periodic boundary conditions, this paper presents a fully spectral discretization method. A numerical solution of the kdvburgers equation by spectral hikari.
We establish some approximation results in the next section. A fourier pseudospectral method for the good boussinesq equation with secondorder temporal accuracy kelong cheng,1 wenqiang feng,2 sigal gottlieb,3 cheng wang3 1department of mathematics, southwest university of science and technology, mianyang, sichuan 621010, peoples republic of china. In a more abstract way, the pseudospectral method deals with the multiplication of two functions and as part of a partial differential equation. Spectral methods for differential problems tiberiu popoviciu. Mapped chebyshev pseudospectral method for unsteady flow. Basic implementation of multipleinterval pseudospectral.
In this paper, quasi linear onedimensional burgers equation is solved by method of lines mol in which the spatial derivatives are approximated by finite differences. Numerical solution of kortewegde vries equation by. Pseudospectral method 4, 11 for the following nonlinear wave equations. This allows us to use the newton iterative method to obtain a very accurate approximation up to digits of accuracy to the exact solution of the 1d burgers equation arbitrarily close to the singularity time.
A modified pseudospectral method for solving trajectory optimization problems with singular arc. So the hermite pseudospectral method is more preferable in actual calculations. To simplify the notation, the timedependence is dropped. Pdf a comparison of fourier pseudospectral method and. Khanib a department of mathematics, razi university kermanshah 67149, iran b department of mathematics, ilam university p. Abstractin this paper, a fourier pseudospectral method for numerical approximation of a periodic initial boundary value problem for the kortewegde vries burgers equation is developed. In this paper, generalized burgersfisher equation was solved by combination of pseudospectral collocation with a new preconditioning scheme and forth order rungekutta method. Read a numerical solution of burgers equation by pseudospectral method and darvishis preconditioning, applied mathematics and computation on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. They are closely related to spectral methods, but complement the basis by an additional pseudospectral basis, which allows representation of functions on a quadrature grid. An hpadaptive pseudospectral method for solving optimal. A comparison of fourier pseudospectral method and finite volume method used to solve. We obtain accurate stable solutions of fe for relatively large values of, with the appropriate division of the domain xl,xr. Distributed optimal control of the viscous burgers. Convergence analysis of threelevel fourier pseudospectral.
View enhanced pdf access article on wiley online library. Four test problem with known exact solutions were studied to demonstrate the accuracy of the present method. Optimal order of convergence is obtained, which implies the spectral accuracy of these methods. Fourier pseudospectral method for twodimensional vorticity. Rbfps method and fourier pseudospectral method for. The space derivatives are calculated in the wavenumber domain by multiplication of the spectrum with. Stability and convergence analysis of fully discrete. Numerical solution of the coupled viscous burgers equation. Efficient chebyshev pseudospectral methods for viscous. This framework is then applied to spectral and pseudospectral methods for the burgers equation, using trigonometric, chebyshev, and legendre polynomials. Abstract in this paper, we present a new pseudospectral method to solve the initial value problem associated to a nonlocal kdvburgers equation involving a caputotype fractional derivative. Spectral methods are powerful numerical methods used for the solution of ordinary and partial differential equations. Fourier galerkin approximation in the spatial direction and chebyshev pseudospectral approximation in the time direction.
Hermite pseudospectral method for nonlinear partial differential equations. In this paper, we generalize this method to twodimensional vorticity equations. Pseudospectral optimal control is a joint theoreticalcomputational method for solving optimal control problems. Dispersion and stability of fourier solutions the goal of this lecture is to shed light at one end of the axis of fd. Read numerical solution of the coupled viscous burgers equations by chebyshevlegendre pseudospectral method, applied mathematics and computation on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. A pseudospectral method for the onedimensional fractional.
A short course in pseudospectral collocation methods for wave equations, with implementations in python. The acoustic wave equation with the fourier method. A numerical solution of burgers equation by pseudospectral. Box 69315516, ilam, iran abstract in this paper, we solve the laxs seventhorder kortewegde vires. Optimal order of convergence is obtained, which implies the spectral accuracy of these. The numerical results are compared with the exact solutions. Spacetime chebyshev pseudospectral method for burgers. In this work we provide a novel stability and convergence analysis for the fourier collocation pseudospectral method, coupled with a number of carefully tailored time discretizations for the three dimensional viscous burgers equation. A mapped chebyshev pseudospectral method is developed as an accurate and yet efficient approach to solve unsteady flows. Comparisons with finite differences for the elastic wave equation bengt fornberg abstract the pseudospectral or fourier method has been used recently by several investigators for forward seis mic modeling.
A fourier pseudospectral method for solving coupled viscous. Hopf barycentric gegenbauer integral pseudospectral method. A numerical solution of burgers equation by pseudospectral method and darvishis preconditioning. A fourier pseudospectral method for solving coupled. A pseudospectral method for a nonlocal kdvburgers equation posed on r. Rao university of florida gainesville, fl 32611 abstract an hpadaptive pseudospectral method is presented for numerically solving optimal control.
Convergence of spectral methods for burgers equation siam. A fourier pseudospectral method for the good boussinesq. Finally, with the help of pseudospectral method, the numerical solutions of the forced ilwburgers equation are given. Numerical implementation of bdf2 via method of lines for time. A numerical solution of burgers equation based on modi. In this paper, we present a new method for solving of the one dimensional burgers equation, that is the spacetime chebyshev pseudospectral method. Introduction the pseudospectral method in a nutshell the pseudospectral method in a nutshell principle of the pseudospectral method based on the fourier series use of sine and cosine functions for the expansions implies periodicity using chebyshev polynomials similar accuracy of common boundary conditions free surface, absorbing can be achieved. In this paper, a fourier pseudospectral method for numerical approximation of a periodic initial boundary value problem for the kortewegde vries burgers equation is developed. Stability and convergence analysis of fully discrete fourier. Preserving the conservation laws, the method discretizes a spatialderivative term implicitly, whereas a timederivative term is treated explicitly using the mapped chebyshev collocation operator. Pseudospectral methods, also known as discrete variable representation dvr methods, are a class of numerical methods used in applied mathematics and scientific computing for the solution of partial differential equations. Firstly, we discretize the burgers equation in one dimensional space with chebyshev pseudospectral method. Pdf efficient chebyshev pseudospectral methods for viscous.
Due requirement of the domains in fpm to be periodic, in the present work, the immersed boundary methodology is used, to solve the equation at nonperiodic domains. By analysis and calculation, the perturbation solution and some conservation relations of the ilwburgers equation are obtained. A new exact solution of burgers equation with linearized. Spectral and pseudospectral methods for the linearized burgers equation were proposed by gottlieb and orszag 8. On the other hand, the authors 7,10 developed a pseudospectral method by using riesz spherical means to get better results. Then as an example, we provide a hermite pseudospectral scheme for the burgers equation on the. Approximation of burgers equation by pseudospectral methods. Computing nearly singular solutions using pseudospectral methods. Onedimensional coupled burgers equation and its numerical solution by an implicit logarithmic finitedifference method. The aim of this paper is to develop the hermite pseudospectral method. This will lead us to confront one of the main problems. Up to now we have considered linear problems, which may be treated ex clusively in fourier space.
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