Spectral Methods in MATLAB

Trefethen, Lloyd N.

doi:10.1137/1.9780898719598

Cited by 3,240 publications

(2,882 citation statements)

References 8 publications

Supporting

Mentioning

2,827

Contrasting

Unclassified

Order By: Relevance

“…Since the problem is nonperiodic, the discrete equations (3.27) are augmented with appropriate Dirichlet-or Neumann-type boundary conditions at x = ±1. We now seek a spectral approximant in terms of algebraic polynomials [93,214]. We proceed by expressing the spectral approximant in terms of the orthogonal family of Legendre polynomials, {p k (x)} k≥0 :…”

Section: Spectral Methodsmentioning

confidence: 99%

A review of numerical methods for nonlinear partial differential equations

Tadmor

2012

Bull. Amer. Math. Soc.

146

View full text Add to dashboard Cite

Abstract. Numerical methods were first put into use as an effective tool for solving partial differential equations (PDEs) by John von Neumann in the mid1940s. In a 1949 letter von Neumann wrote "the entire computing machine is merely one component of a greater whole, namely, of the unity formed by the computing machine, the mathematical problems that go with it, and the type of planning which is called by both." The "greater whole" is viewed today as scientific computation: over the past sixty years, scientific computation has emerged as the most versatile tool to complement theory and experiments, and numerical methods for solving PDEs are at the heart of many of today's advanced scientific computations. Numerical solutions found their way from financial models on Wall Street to traffic models on Main Street. Here we provide a bird's eye view on the development of these numerical methods with a particular emphasis on nonlinear PDEs.

show abstract

Section: Spectral Methodsmentioning

confidence: 99%

A review of numerical methods for nonlinear partial differential equations

Tadmor

2012

Bull. Amer. Math. Soc.

146

View full text Add to dashboard Cite

show abstract

“…To write (12) even in the form c T X − c = 0, we express dX/dt into X. Spectral differentiation [12] provides dX/dt = D · X with good accuracy using some matrix D. This results in a choice c T = w T · C · D and c = 1. We observe that we always can compare v with dX/dt for convergence.…”

Section: Using Generalized Eigenvalue Methodsmentioning

confidence: 99%

Robust Periodic Steady State Analysis of Autonomous Oscillators Based on Generalized Eigenvalues

Mirzavand

Maten

Beelen

et al. 2011

Scientific Computing in Electrical Engineering SCEE 2010

View full text Add to dashboard Cite

“…This very nice property has been used in several applications, see [9,10] or [11] for instance. However, due to the clustering of the points near the extremities, the information (that is the f j 's) is badly distributed over the interval and could lead to mediocre approximation of functions with shocks close to the center.…”

Section: From Rational To Polynomial Interpolationmentioning

confidence: 99%

“…However, due to the clustering of the points near the extremities, the information (that is the f j 's) is badly distributed over the interval and could lead to mediocre approximation of functions with shocks close to the center. Moreover, there is an ill-conditioning of the derivatives of P N [ f ] near the extremities, see [10] for instance.…”

Section: From Rational To Polynomial Interpolationmentioning

confidence: 99%