Spectral Methods for Time-Dependent Problems: Analysis and Applications
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On the other hand, many problems of scientific interest, such as the interaction of solitary waves and homogeneous turbulence, can be modeled by PDEs with periodic boundary conditions.
Furthermore, even if an original problem is not periodic, the periodicity may be induced by using coordinate transforms, such as polar, spherical and cylindrical coordinates. Indeed, there are numerous circumstances where the problems are periodic in one or two directions, and non-periodic in other directions. In such cases, it is natural to use Fourier series in the periodic directions and other types of spectral expansions, such as Legendre or Chebyshev polynomials, in the non-periodic directions cf. The Fourier spectral method is only appropriate for problems with periodic boundary conditions.
If a Fourier method is applied to a non-periodic problem, it inevitably induces the so-called Gibbs phenomenon, and reduces the global convergence rate to O N -1 cf. Gottlieb and Orszag Consequently, one should not apply a Fourier method to problems with non-periodic boundary conditions. Instead, one should use orthogonal polynomials which are eigenfunctions of some singular Sturm-Liouville problems. The commonly used orthogonal polynomials include the Legendre, Chebyshev, Hermite and Laguerre polynomials.
We consider in this chapter spectral algorithms for solving the two-point boundary value problem.
High-order differential equations often arise from mathematical modeling of a variety of physical phenomena. For example, higher even-order differential equations may appear in astrophysics, structural mechanics and geophysics, and higher odd-order differential equations, such as the Korteweg-de Vries KdV equation, are routinely used in modeling nonlinear waves and nonlinear optics. Generalized submersiveness of second-order ordinary differential equations. Journal of Geometric Mechanics , , 1 2 : Jacobi fields for second-order differential equations on Lie algebroids.
Conference Publications , , special : Jaume Llibre , Amar Makhlouf.
Periodic solutions of some classes of continuous second-order differential equations. A spectral collocation method for solving initial value problems of first order ordinary differential equations. Qiong Meng , X. Multiple solutions of second-order ordinary differential equation via Morse theory. Raegan Higgins. Asymptotic behavior of second-order nonlinear dynamic equations on time scales. Superposition rules and second-order Riccati equations. Journal of Geometric Mechanics , , 3 1 : Subharmonic oscillations for some second-order differential equations without Landesman-Lazer conditions.
Conference Publications , , Special : Alessandro Fonda , Fabio Zanolin. Bounded solutions of nonlinear second order ordinary differential equations. Bernd Kawohl , Vasilii Kurta. A Liouville comparison principle for solutions of singular quasilinear elliptic second-order partial differential inequalities. Kyeong-Hun Kim , Kijung Lee.
Euler , N. Euler , M. On nonlocal symmetries generated by recursion operators: Second-order evolution equations. Decay rates for second order evolution equations in Hilbert spaces with nonlinear time-dependent damping.
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Paola Buttazzoni , Alessandro Fonda. Periodic perturbations of scalar second order differential equations. Kunquan Lan. Eigenvalues of second order differential equations with singularities. A Legendre-Gauss collocation method for nonlinear delay differential equations.
Can Huang , Zhimin Zhang. The spectral collocation method for stochastic differential equations. Ugur G. On the optimal control of the free boundary problems for the second order parabolic equations. Convergence of the method of finite differences. Oscillations in a second-order discontinuous system with delay. Constructing continuous stationary covariances as limits of the second-order stochastic difference equations.
Spectral method
American Institute of Mathematical Sciences. We propose an efficient Legendre-Gauss collocation algorithm for second-order nonlinear ordinary differential equations ODEs. We also design a Legendre-Gauss-type collocation algorithm for time-dependent second-order nonlinear partial differential equations PDEs , which can be implemented in a synchronous parallel fashion.
Numerical results indicate the high accuracy and effectiveness of the suggested algorithms.
Keywords: second-order ordinary differential equations , Legendre spectral collocation method , time-dependent second-order partial differential equations. Citation: Lijun Yi, Zhongqing Wang. References: [1] I. Google Scholar [2] I. Google Scholar [3] P. Google Scholar [4] C. Google Scholar [5] J. In other words, spectral methods take on a global approach while finite element methods use a local approach.
Partially for this reason, spectral methods have excellent error properties, with the so-called "exponential convergence" being the fastest possible, when the solution is smooth. However, there are no known three-dimensional single domain spectral shock capturing results shock waves are not smooth.
Spectral methods can be used to solve ordinary differential equations ODEs , partial differential equations PDEs and eigenvalue problems involving differential equations. When applying spectral methods to time-dependent PDEs, the solution is typically written as a sum of basis functions with time-dependent coefficients; substituting this in the PDE yields a system of ODEs in the coefficients which can be solved using any numerical method for ODEs. Eigenvalue problems for ODEs are similarly converted to matrix eigenvalue problems [ citation needed ].
Spectral Methods
Spectral methods were developed in a long series of papers by Steven Orszag starting in including, but not limited to, Fourier series methods for periodic geometry problems, polynomial spectral methods for finite and unbounded geometry problems, pseudospectral methods for highly nonlinear problems, and spectral iteration methods for fast solution of steady state problems. The implementation of the spectral method is normally accomplished either with collocation or a Galerkin or a Tau approach.
Spectral methods are computationally less expensive than finite element methods, but become less accurate for problems with complex geometries and discontinuous coefficients. This increase in error is a consequence of the Gibbs phenomenon. Here we presume an understanding of basic multivariate calculus and Fourier series.
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This is the Poisson equation , and can be physically interpreted as some sort of heat conduction problem, or a problem in potential theory, among other possibilities. We have exchanged partial differentiation with an infinite sum, which is legitimate if we assume for instance that f has a continuous second derivative.