![]() | This article includes a list of general references, but it lacks sufficient corresponding inline citations .(August 2013) |
Spectral methods are a class of techniques used in applied mathematics and scientific computing to numerically solve certain differential equations. The idea is to write the solution of the differential equation as a sum of certain "basis functions" (for example, as a Fourier series which is a sum of sinusoids) and then to choose the coefficients in the sum in order to satisfy the differential equation as well as possible.
Spectral methods and finite-element methods are closely related and built on the same ideas; the main difference between them is that spectral methods use basis functions that are generally nonzero over the whole domain, while finite element methods use basis functions that are nonzero only on small subdomains (compact support). Consequently, spectral methods connect variables globally while finite elements do so locally. 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). [1] In the finite-element community, a method where the degree of the elements is very high or increases as the grid parameter h increases is sometimes called a spectral-element method.
Spectral methods can be used to solve differential equations (PDEs, ODEs, eigenvalue, etc) and optimization problems. 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 were developed in a long series of papers by Steven Orszag starting in 1969 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 . For very small problems, the spectral method is unique in that solutions may be written out symbolically, yielding a practical alternative to series solutions for differential equations.
Spectral methods can be computationally less expensive and easier to implement than finite element methods; they shine best when high accuracy is sought in simple domains with smooth solutions. However, because of their global nature, the matrices associated with step computation are dense and computational efficiency will quickly suffer when there are many degrees of freedom (with some exceptions, for example if matrix applications can be written as Fourier transforms). For larger problems and nonsmooth solutions, finite elements will generally work better due to sparse matrices and better modelling of discontinuities and sharp bends.
Here we presume an understanding of basic multivariate calculus and Fourier series. If is a known, complex-valued function of two real variables, and g is periodic in x and y (that is, ) then we are interested in finding a function f(x,y) so that
where the expression on the left denotes the second partial derivatives of f in x and y, respectively. 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.
If we write f and g in Fourier series:
and substitute into the differential equation, we obtain this equation:
We have exchanged partial differentiation with an infinite sum, which is legitimate if we assume for instance that f has a continuous second derivative. By the uniqueness theorem for Fourier expansions, we must then equate the Fourier coefficients term by term, giving
| (*) |
which is an explicit formula for the Fourier coefficients aj,k.
With periodic boundary conditions, the Poisson equation possesses a solution only if b0,0 = 0. Therefore, we can freely choose a0,0 which will be equal to the mean of the resolution. This corresponds to choosing the integration constant.
To turn this into an algorithm, only finitely many frequencies are solved for. This introduces an error which can be shown to be proportional to , where and is the highest frequency treated.
Since we're only interested in a finite window of frequencies (of size n, say) this can be done using a fast Fourier transform algorithm. Therefore, globally the algorithm runs in time O(n log n).
We wish to solve the forced, transient, nonlinear Burgers' equation using a spectral approach.
Given on the periodic domain , find such that
where ρ is the viscosity coefficient. In weak conservative form this becomes
where following inner product notation. Integrating by parts and using periodicity grants
To apply the Fourier–Galerkin method, choose both
and
where . This reduces the problem to finding such that
Using the orthogonality relation where is the Kronecker delta, we simplify the above three terms for each to see
Assemble the three terms for each to obtain
Dividing through by , we finally arrive at
With Fourier transformed initial conditions and forcing , this coupled system of ordinary differential equations may be integrated in time (using, e.g., a Runge Kutta technique) to find a solution. The nonlinear term is a convolution, and there are several transform-based techniques for evaluating it efficiently. See the references by Boyd and Canuto et al. for more details.
One can show that if is infinitely differentiable, then the numerical algorithm using Fast Fourier Transforms will converge faster than any polynomial in the grid size h. That is, for any n>0, there is a such that the error is less than for all sufficiently small values of . We say that the spectral method is of order , for every n>0.
Because a spectral element method is a finite element method of very high order, there is a similarity in the convergence properties. However, whereas the spectral method is based on the eigendecomposition of the particular boundary value problem, the finite element method does not use that information and works for arbitrary elliptic boundary value problems.
In theoretical physics, a Feynman diagram is a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic particles. The scheme is named after American physicist Richard Feynman, who introduced the diagrams in 1948. The interaction of subatomic particles can be complex and difficult to understand; Feynman diagrams give a simple visualization of what would otherwise be an arcane and abstract formula. According to David Kaiser, "Since the middle of the 20th century, theoretical physicists have increasingly turned to this tool to help them undertake critical calculations. Feynman diagrams have revolutionized nearly every aspect of theoretical physics." While the diagrams are applied primarily to quantum field theory, they can also be used in other areas of physics, such as solid-state theory. Frank Wilczek wrote that the calculations that won him the 2004 Nobel Prize in Physics "would have been literally unthinkable without Feynman diagrams, as would [Wilczek's] calculations that established a route to production and observation of the Higgs particle."
The uncertainty principle, also known as Heisenberg's indeterminacy principle, is a fundamental concept in quantum mechanics. It states that there is a limit to the precision with which certain pairs of physical properties, such as position and momentum, can be simultaneously known. In other words, the more accurately one property is measured, the less accurately the other property can be known.
The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics. Furthermore, it is one of the few quantum-mechanical systems for which an exact, analytical solution is known.
In physics, engineering and mathematics, the Fourier transform (FT) is an integral transform that converts a function into a form that describes the frequencies present in the original function. The output of the transform is a complex-valued function of frequency. The term Fourier transform refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made the Fourier transform is sometimes called the frequency domain representation of the original function. The Fourier transform is analogous to decomposing the sound of a musical chord into the intensities of its constituent pitches.
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. Its discovery was a significant landmark in the development of quantum mechanics. It is named after Erwin Schrödinger, who postulated the equation in 1925 and published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933.
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and returns another function.
Pseudo-spectral 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. They are closely related to spectral methods, but complement the basis by an additional pseudo-spectral basis, which allows representation of functions on a quadrature grid. This simplifies the evaluation of certain operators, and can considerably speed up the calculation when using fast algorithms such as the fast Fourier transform.
In mathematics and its applications, a Sturm–Liouville problem is a second-order linear ordinary differential equation of the form:
In mathematics, the Poisson summation formula is an equation that relates the Fourier series coefficients of the periodic summation of a function to values of the function's continuous Fourier transform. Consequently, the periodic summation of a function is completely defined by discrete samples of the original function's Fourier transform. And conversely, the periodic summation of a function's Fourier transform is completely defined by discrete samples of the original function. The Poisson summation formula was discovered by Siméon Denis Poisson and is sometimes called Poisson resummation.
In linear algebra, the Gram matrix of a set of vectors in an inner product space is the Hermitian matrix of inner products, whose entries are given by the inner product . If the vectors are the columns of matrix then the Gram matrix is in the general case that the vector coordinates are complex numbers, which simplifies to for the case that the vector coordinates are real numbers.
In mathematics, specifically the theory of elliptic functions, the nome is a special function that belongs to the non-elementary functions. This function is of great importance in the description of the elliptic functions, especially in the description of the modular identity of the Jacobi theta function, the Hermite elliptic transcendents and the Weber modular functions, that are used for solving equations of higher degrees.
In mathematics, the Riesz–Fischer theorem in real analysis is any of a number of closely related results concerning the properties of the space L2 of square integrable functions. The theorem was proven independently in 1907 by Frigyes Riesz and Ernst Sigismund Fischer.
In mathematics, the Rogers–Ramanujan identities are two identities related to basic hypergeometric series and integer partitions. The identities were first discovered and proved by Leonard James Rogers, and were subsequently rediscovered by Srinivasa Ramanujan some time before 1913. Ramanujan had no proof, but rediscovered Rogers's paper in 1917, and they then published a joint new proof. Issai Schur independently rediscovered and proved the identities.
In mathematics, Bochner's theorem characterizes the Fourier transform of a positive finite Borel measure on the real line. More generally in harmonic analysis, Bochner's theorem asserts that under Fourier transform a continuous positive-definite function on a locally compact abelian group corresponds to a finite positive measure on the Pontryagin dual group. The case of sequences was first established by Gustav Herglotz
The Debye–Waller factor (DWF), named after Peter Debye and Ivar Waller, is used in condensed matter physics to describe the attenuation of x-ray scattering or coherent neutron scattering caused by thermal motion. It is also called the B factor, atomic B factor, or temperature factor. Often, "Debye–Waller factor" is used as a generic term that comprises the Lamb–Mössbauer factor of incoherent neutron scattering and Mössbauer spectroscopy.
In mathematics, the simplest real analytic Eisenstein series is a special function of two variables. It is used in the representation theory of SL(2,R) and in analytic number theory. It is closely related to the Epstein zeta function.
This article relates the Schrödinger equation with the path integral formulation of quantum mechanics using a simple nonrelativistic one-dimensional single-particle Hamiltonian composed of kinetic and potential energy.
In mathematics, Hilbert spaces allow the methods of linear algebra and calculus to be generalized from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that induces a distance function for which the space is a complete metric space.
The theory of causal fermion systems is an approach to describe fundamental physics. It provides a unification of the weak, the strong and the electromagnetic forces with gravity at the level of classical field theory. Moreover, it gives quantum mechanics as a limiting case and has revealed close connections to quantum field theory. Therefore, it is a candidate for a unified physical theory. Instead of introducing physical objects on a preexisting spacetime manifold, the general concept is to derive spacetime as well as all the objects therein as secondary objects from the structures of an underlying causal fermion system. This concept also makes it possible to generalize notions of differential geometry to the non-smooth setting. In particular, one can describe situations when spacetime no longer has a manifold structure on the microscopic scale. As a result, the theory of causal fermion systems is a proposal for quantum geometry and an approach to quantum gravity.
In mathematics, the Weil–Brezin map, named after André Weil and Jonathan Brezin, is a unitary transformation that maps a Schwartz function on the real line to a smooth function on the Heisenberg manifold. The Weil–Brezin map gives a geometric interpretation of the Fourier transform, the Plancherel theorem and the Poisson summation formula. The image of Gaussian functions under the Weil–Brezin map are nil-theta functions, which are related to theta functions. The Weil–Brezin map is sometimes referred to as the Zak transform, which is widely applied in the field of physics and signal processing; however, the Weil–Brezin Map is defined via Heisenberg group geometrically, whereas there is no direct geometric or group theoretic interpretation from the Zak transform.