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