User:Egm6322.s14.team2/ElizabethTermPaper2

=The Spectral Method and the Application of Legendre Polynomials in the Solution to Partial Differential Equations=

Elizabeth Bartlett Department of Mechanical and Aerospace Engineering University of Florida Principles of Engineering Analysis II EGM6322

History of the Spectral Method and the Legendre Differential Equation
A partial differential equation (PDE) relates some function of more than one variable to its partial derivatives, like the example homogeneous second order partial differential equation in $$. Partial differential equations can be used to solve several engineering problems including the heat equation ($$) and Newton's equation of motion ($$). Partial differential equations can be solved using either analytic (for example: separation of variables and the superposition principle) or numerical methods (e.g. variation methods and methods of weighted residuals).5  The spectral method is a method of weighted residuals (MWR) where the solution is approximated by a truncated expansion of orthogonal functions. Some other examples of numerical methods are finite difference, finite element, and finite volume.

Three of the most common general spectral schemes are the Galerkin, collocation, and tau approaches. The Galerkin method is named after the Russian mathematician and engineer Boris Galerkin.10 Silbermann used the spectral Galerkin method in 1954 in a meteorological PDE model.2  The tau approach is a modification of the Galerkin method similar to the Petrov-Galerkin method. Finally, the collocation approach, also called the point collocation approach or pseudo spectral approach, is similar to the finite difference method in the use of gridpoints, called collocation points. The collocation spectral method was used in 1972 by Orszag to solve periodic PDEs. The collocation method is widely used in several disciplines because it is ideal for non-linear problems. The most frequently used trial functions are trigonometric polynomials, Chebyshev polynomials ($$), and Legendre polynomials ($$). In general a fourier series is selected for periodic problems and orthogonal polynomials, like the Chebyshev and Legendre, for nonperiodic problems.

The most general definition of the Fourier Series, introduced by Jean-Baptiste Joseph Fourier in his 1807 Treatise on the Propagation of Heat in Solid Bodies, is:3,7

where the Fourier coefficients are defined in Equations 7 and 8 below.

The Chebyshev polynomials, named after Pafnuty Chebyshev who is also well known for the Chebyshev Bias, Bertrand-Chebyshev Theorem, and the Chebyshev Inequality, are related to de Moivre's formula ($$) in complex number theory and are the solution to the Chebyshev differential equation ($$).9 The Legendre differential equation and polynomials are named for the French mathematician Adrien-Marie Legendre. Legendre found the polynomial solution to the Legendre differential equation ($$) while studying Newtonian potential in 1782.8 In 1816 Benjamin Olinde Rodrigues discovered the following generating function ($$). The Rodrigues formula was previously known as the Ivory-Jacobi formula for Sir James Ivory and Carl Jacobi for their independent, but later, discovery of the same equation.11 The Legendre polynomials have several applications in mathematics, physics, and engineering.

A Sturm-Liouville equation, named after Jacques Charles Francois Sturm and Joseph Liouville for their work in 1929,12 is shown in ($$). The Sturm-Liouville problem is an eigenvalue problem where the set of eigenvalues (λ) are known as the spectrum and w(x) is the weight function. A detailed discussion of the Sturm-Liouville problem is presented by King et al in Differential Equations, Linear, Nonlinear, Ordinary, Partial. The Legendre differential equation ($$) can be rewritten as a Sturm-Liouville problem where $$p(x)=(1-x^{2})$$, q(x)=0 and w(x)=1. Consequently, the Legendre polynomials, like the Chebyshev polynomials, are orthogonal ($$) where $${\delta}_{mn}$$ is the dirac or kronecker delta function.

Legendre Polynomials
Because the Legendre polynomials are orthogonal and complete on the interval from -1 to 1, any continuous function can be expanded as a linear combination of the Legendre polynomials ($$). Where the first eight Legendre polynomials are presented in Table 1 below.

To determine the coefficients of the Legendre polynomial expansion, the inner product with Pm(x) is:

Using the orthogonality equation ($$), the above expression becomes:

And solving the algebraic equation for the coefficients and then plugging the coefficients back into the Legendre polynomial expansion ($$), Equation 18 is called the Fourier-Legendre series.

Spectral Methods
Expansion functions, or trial functions, (φ) and weighting functions(W) (or test functions (ψ)) as well as the residual (R), are key in the spectral method.2 The trial functions are the basis functions for the truncated series expansion of the solution, ($$). The trial function is often an inner product of the eigenfunctions of singular Sturm-Liouville problems.1 The test function is used to minimize the residual of the truncated series ($$). The difference between the three spectral methods covered in this article: Galerkin, collocation, and tau, is the selection of the test function. In the Galerkin method the test functions are the same as the trial functions.2 In the collocation method, the test function is a dirac (or Kronecker) delta function centered at collocation points. The test function for the tau method is similar to the Galerkin method test function, except they need not satisfy the boundary conditions. A supplementary set of equations is instead used to satisfy the boundary equations. The trial functions for spectral methods are infinitely differentiable global functions, unlike the local trial functions of both the finite element and finite difference methods. Therefore, spectral methods are very accurate using fewer points by comparison.3 Table 2 is an example of error decay using a second order finite difference method and a Fourier collocation spectral method.2 Several examples of all three different spectral methods, Galerkin, collocation, and Tau, of increasing complexity are presented in the following sections.

Galerkin Method
Cook et al provided an example of the Galerkin weighted residual method assuming a second degree polynomial solution in Concepts and Applications of Finite Element Analysis4 of a uniform bar under an axial load as shown in Figure 1. This simplified example is presented here to introduce the Galerkin method. Using the free body diagram, Newton's and Hooke's laws were used to find the governing equation ($$) with the natural boundary conditions of $$x_{0}=0$$ and $$P=AE\frac{du}{dx}$$.

Where cx is a linearly varying axial force, A is the cross section, E is the modulus of elasticity, u is the displacement, and x is the distance from the fixed end of the bar. Using the Galerkin method, a solution $$\tilde{u}$$ is assumed and substituted into the governing equation to get a Galerkin statement of the problem.

A simple second order polynomial solution was assumed to introduce the Galerkin method, $$\tilde{u}=a_{n}{\varphi}_{n}(x)=a_{1}x+a_{2}x^{2}$$ with the derivative $$\frac{d \tilde{u}}{dx}=a_{1}+2a_{2}x$$. Using integrations by parts on one term ($$\int u dv = uv- \int v du$$) with $$u={\psi}_i$$ and $$dv=\frac{d^{2} \tilde{u}}{dx^{2}}dx$$ (note that u, here, is not the solution to the differential equation but one of the two factors in the integrand of the familiar integration by parts equation), the integral can be expressed in $$.

In the Galerkin method the test functions are the trial functions $${\psi}_{1}={\varphi}_{1}=x$$ and $${\psi}_{2}={\varphi}_{2}=x^{2}$$ with derivatives $$\frac{d{\psi}_{1}}{dx}=1$$ and $$\frac{d{\psi}_{2}}{dx}=2x$$, the system of equations can be written as follows:

Equations 26 and 27 for the coefficients a1 and a2 are found by solving the algebraic system of equations.

Substituting the coefficients back into the equation for $$\tilde{u}$$ give an approximation of the displacement and, using Hooke's Law again, the stress.

Canuto et al provide a one dimensional example of the Galerkin spectral method using the fourier series for the test and trial functions for the wave partial differential equation ($$).2

The trial and test functions for this example problem are presented in Equations 31 and 32.

Substituting the trial and test functions into the equation for weighted residuals:

with the initial coefficients:

Collocation Method
The collocation method is similar to the Galerkin method, but residuals are set to zero at n different locations, "collocation points," xi.4

where j<n. An example of the Chebyshev collocation method for the heat equation is also presented by Canuto, Hussaini, Quarteroni, and Zang.2

Tau Method
An example of the Tau method utilizing the Legendre polynomials will be presented in the next section.

Spectral Method Solution Using Legendre Polynomials
In spectral methods, the solution of a PDE is approximated by a linear combination of a truncated set of orthogonal functions. Because the Legendre differential equation is a singular Sturm-Liouville problem, the Legendre polynomial is an ideal choice for the trial function, φ, as shown in ($$).

Where $$a_{k}(x)$$ is the spectral coefficient defined in ($$).

A few examples of spectral methods using Legendre polynomials are presented below.

Legendre Tau Method for the Poisson Equations
An example of the Legendre Tau method for the Poisson equation in two dimensional Euclidean space was presented by Canuto et al where the general equation and boundary conditions on (-1,1) x (-1,1) are defined in ($$) and ($$).2

The trial and test functions are defined in ($$) and ($$), respectively, where k and l = 0, 1, ...,N.

Legendre Galerkin with Numerical Integration (G-NI) Method for the Advection-Diffusion-Reaction Equation
Canuto et. al provide a second example of a spectral method using the Legendre polynomial in the solution of a PDE. The problem, the Advection-Diffusion-Reaction Equation, is a PDE ($$) for the species concentration of a classical advection-diffusion-reaction in Chemical processes.2,13

Additional Examples
Several more examples are available in journal articles (like Patera's Spectral Element Method for Fluid Dynamics, Laminar Flow in a Channel Expansion, in the Journal of Computational Physics and Costa's Spectral Methods for Partial Differential Equations and Spectral Methods, Fundamentals in Single Domains in Scientific Computation), online, (https://en.wikipedia.org/wiki/Heat_equation and https://en.wikipedia.org/wiki/Spectral_method), and in the textbooks (including Spectral Methods in Fluid Dynamics, Spring Series in Computational Physics and King Differential Equations, Linear, Nonlinear, Ordinary and Partial) because of the wide application of partial differential equations to engineering, mathematics, and physics.