We have seen that in this case spectral methods yield a highly accurate and simple way to calculate derivatives. The techniques have been extensively used to solve a wide range of. 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. Quarteroni,spectral and pseudospectral methods for parabolic problems with nonperiodic boundary conditions, calcolo, 18, 1981, 197218. Next, we solve fpdes, including the time and spacefractional advectiondiffusion equation, time and spacefractional multiterm fpdes, and finally the spacefractional burgers equation. 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. An hpadaptive pseudospectral method for solving optimal.
In a more abstract way, the pseudospectral method deals with the multiplication of two functions and as part of a partial differential equation. 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. Burgers equation by pseudospectral method and darvishis preconditioning. In this paper, we present a numerical solution of onedimensional kortewegde vries equation with variant boundary conditions by the fourier pseudospectral method. Convergence analysis of threelevel fourier pseudospectral. Mapped chebyshev pseudospectral method for unsteady flow. Abdou and soliman 3 used variational iteration method for solving burgers and. Rao university of florida gainesville, fl 32611 abstract an hpadaptive pseudospectral method is presented for numerically solving optimal control. Hermite pseudospectral method for nonlinear partial differential equations. Kdv equation by pseudospectral method and darvishis preconditioning m. In section 6 we demonstrate the method on an example. Dispersion and stability of fourier solutions the goal of this lecture is to shed light at one end of the axis of fd.
Pseudospectral method 4, 11 for the following nonlinear wave equations. Other pseudospectral optimal control techniques, such as the bellman pseudospectral method, rely on nodeclustering at the initial time to produce optimal controls. A comparison of fourier pseudospectral method and finite volume method used to solve. Numerical implementation of bdf2 via method of lines for time. By analysis and calculation, the perturbation solution and some conservation relations of the ilwburgers equation are obtained. Ps optimal control theory has been used in ground and flight systems in military and industrial applications. Numerical methods for partial differential equations 33. This framework is then applied to spectral and pseudospectral methods for the burgers equation, using trigonometric, chebyshev, and legendre polynomials. View enhanced pdf access article on wiley online library. Numerical implementation of bdf2 via method of lines for. Convergence of spectral method in time for burgers equation. Fourier pseudospectral method for twodimensional vorticity. Fractional spectral collocation method siam journal on. Rbfps method and fourier pseudospectral method for solving.
A fourier pseudospectral method for solving coupled. Efficient chebyshev pseudospectral methods for viscous. After submitting, as a motivation, some applications of this paradigmatic equations, we continue with the mathematical analysis of them. 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. We establish some approximation results in the next section. Firstly, we discretize the burgers equation in one dimensional space with chebyshev pseudospectral method.
Numerical methods for partial differential equations 31. 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. Pdf a comparison of fourier pseudospectral method and finite. Ch2and19 for a galerkin spectral method for navierstokes equations with t. Hopf barycentric gegenbauer integral pseudospectral method. 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. Pseudospectral methods have become increasingly popular for solving differential equations also they are very useful in providing highly accurate solutions to differential equations. Spectral and pseudospectral methods for the linearized burgers equation were proposed by gottlieb and orszag 8. 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. One of the methods to solve partial differential equations is the spectral collocation method or the pseudospectral method. 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.
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. A basic pseudospectral method for optimal control is based on the covector mapping principle. 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. It combines pseudospectral ps theory with optimal control theory to produce ps optimal control theory. In this paper, we generalize this method to twodimensional vorticity equations. The acoustic wave equation with the fourier method. An implicitexplicit spectral method for burgers equation springerlink. The numerical results are compared with the exact solutions. On the other hand, the authors 7,10 developed a pseudospectral method by using riesz spherical means to get better results. Spectral and finite difference solutions of burgers equation citeseerx. Finally, with the help of pseudospectral method, the numerical solutions of the forced ilw burgers equation are given. Convergence of spectral methods for burgers equation siam.
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. Numerical solution of the coupled viscous burgers equation. A practical guide to pseudospectral methods by bengt fornberg. A numerical solution of the laxs 7 order kdv equation by. To simplify the notation, the timedependence is dropped. Pdf a pseudospectral method for a nonlocal kdvburgers. So the hermite pseudospectral method is more preferable in actual calculations. Numerical solution of kortewegde vries equation by. Our numerical results confirm the exponential convergence of the fractional collocation method. Onedimensional coupled burgers equation and its numerical solution by an implicit logarithmic finitedifference method. 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.
Khanib a department of mathematics, razi university kermanshah 67149, iran b department of mathematics, ilam university p. 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. Pdf a comparison of fourier pseudospectral method and. Optimal order of convergence is obtained, which implies the spectral accuracy of these methods. Then as an example, we provide a hermite pseudospectral scheme for the burgers equation on the. Spectral methods for differential problems tiberiu popoviciu. Four test problem with known exact solutions were studied to demonstrate the accuracy of the present method. A pseudospectral method for a nonlocal kdvburgers equation posed on r.
Fourier galerkin approximation in the spatial direction and chebyshev pseudospectral approximation in the time direction. 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. Box 69315516, ilam, iran abstract in this paper, we solve the laxs seventhorder kortewegde vires. For solving burgers equation with periodic boundary conditions, this paper presents a fully spectral discretization method. 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. The fourier pseudospectral method has been studied for a one dimensional coupled system of viscous burgers equations.
A numerical solution of the kdvburgers equation by spectral hikari. A modified pseudospectral method for solving trajectory optimization problems with singular arc. In this paper, a general framework is presented for analyzing numerical methods for the evolutionary equations that admit semigroup formulations. Pseudospectral optimal control is a joint theoreticalcomputational method for solving optimal control problems. Pdf the burger s equation serves as a useful mathematical model to be applied in fluid dynamic problems.
An hpadaptive pseudospectral method for solving optimal control problems christopher l. Convergence of spectral methods for burgers equation. Basic implementation of multipleinterval pseudospectral. Preserving the conservation laws, the method discretizes a spatialderivative term implicitly, whereas a timederivative term is treated explicitly using the mapped chebyshev collocation operator. A numerical solution of burgers equation by pseudospectral.
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 short course in pseudospectral collocation methods for wave equations, with implementations in python. 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. In this paper, generalized burgersfisher equation was solved by combination of pseudospectral collocation with a new preconditioning scheme and forth order rungekutta method. Stability and convergence analysis of fully discrete. Development of the tau method for the numerical solution of twodimensional linear volterra integrodifferential equations. A fourier pseudospectral method for the good boussinesq. Hermite pseudospectral method for nonlinear partial.
The generalized stability and the convergence are proved. A fourier pseudospectral method for solving coupled viscous burgers equations. The numerical results show the advantage of such a method. The aim of this paper is to develop the hermite pseudospectral method. The finite difference method is used in time direction, while the pseudospectral method is used in xdirection. 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. The nonlinear term in the fvm, is discretion by using upwind and centraldifferencing schemes, showing a rate of convergence for the secondorder accurate. Pdf efficient chebyshev pseudospectral methods for viscous. Abstractspectral methods fourier galerkin, fourier pseudospectral, chebyshev tau, chebyshev collocation, spectral element and standard finite differences. The expansion coefficients are determined by means of minimizing an object functional, and rapid convergence of the method is proved. A mapped chebyshev pseudospectral method is developed as an accurate and yet efficient approach to solve unsteady flows. Finally, numerical results obtained by this way are compared with the exact solution to show the efficiency of the method.
Distributed optimal control of the viscous burgers. Computing nearly singular solutions using pseudospectral methods. A pseudospectral method of solution of fishers equation. Spacetime chebyshev pseudospectral method for burgers. A numerical solution of burgers equation by pseudospectral method and darvishis preconditioning. The proposed exact solutions overcome the long existing problem of. Approximation of burgers equation by pseudospectral methods. 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. Rbfps method and fourier pseudospectral method for. Stability and convergence analysis of fully discrete fourier. This will lead us to confront one of the main problems. 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. We obtain accurate stable solutions of fe for relatively large values of, with the appropriate division of the domain xl,xr.
The stability and the convergence of proposed hermite pseudospectral scheme are proved strictly. In section 7 we provide further details about how our in. A pseudospectral method for the onedimensional fractional laplacian on r jorge cayama 1carlota m. The two numerical schemes discussed are the legendre pseudospectral method with lgl nodes and the chebyshev pseudospectral method with cgl nodes. 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. A modified leapfrog scheme is constructed in such a way. Our numerical experiments show that the chebyshev collocation method is an efficient and reliable scheme for solving burgers equations with. Forced ilwburgers equation as a model for rossby solitary. The burgers equation is known to steepen negative gradients leading to the formation of socalled shocks.
A study of wave trapping between two obstacles in the forced kortewegde vries equation. Up to now we have considered linear problems, which may be treated ex clusively in fourier space. 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. This paper presents a computational technique based on the pseudo. Finally, with the help of pseudospectral method, the numerical solutions of the forced ilwburgers equation are given. The coupled viscous burgers equation which is a nonlinear partial differential equation of the form. Spectral methods are powerful numerical methods used for the solution of ordinary and partial differential equations. Because standard chebyshev points make the corresponding spectral derivative. Pdf the burgers equation serves as a useful mathematical model to be applied in fluid dynamic problems. In this paper, we present a new method for solving of the one dimensional burgers equation, that is the spacetime chebyshev pseudospectral method. The space derivatives are calculated in the wavenumber domain by multiplication of the spectrum with.
A meshfree interpolation method was employed by islam et al. A new exact solution of burgers equation with linearized. A pseudospectral method for the onedimensional fractional. 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. This paper considers a general burgers equation with the nonlinear term coefficient being an arbitrary constant. A numerical solution of burgers equation based on modi. By analysis and calculation, the perturbation solution and some conservation relations of the ilw burgers equation are obtained.