Volume 3, Issue 3
The Discrete Orthogonal Polynomial Least Squares Method for Approximation and Solving Partial Differential Equations

Anne Gelb, Rodrigo B. Platte & W. Steven Rosenthal

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Commun. Comput. Phys., 3 (2008), pp. 734-758.

Published online: 2008-03

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  • Abstract

We investigate numerical approximations based on polynomials that are orthogonal with respect to a weighted discrete inner product and develop an algorithm for solving time dependent differential equations. We focus on the family of super Gaussian weight functions and derive a criterion for the choice of parameters that provides good accuracy and stability for the time evolution of partial differential equations. Our results show that this approach circumvents the problems related to the Runge phenomenon on equally spaced nodes and provides high accuracy in space. For time stability, small corrections near the ends of the interval are computed using local polynomial interpolation. Several numerical experiments illustrate the performance of the method. 

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@Article{CiCP-3-734, author = {Anne Gelb, Rodrigo B. Platte and W. Steven Rosenthal}, title = {The Discrete Orthogonal Polynomial Least Squares Method for Approximation and Solving Partial Differential Equations}, journal = {Communications in Computational Physics}, year = {2008}, volume = {3}, number = {3}, pages = {734--758}, abstract = {

We investigate numerical approximations based on polynomials that are orthogonal with respect to a weighted discrete inner product and develop an algorithm for solving time dependent differential equations. We focus on the family of super Gaussian weight functions and derive a criterion for the choice of parameters that provides good accuracy and stability for the time evolution of partial differential equations. Our results show that this approach circumvents the problems related to the Runge phenomenon on equally spaced nodes and provides high accuracy in space. For time stability, small corrections near the ends of the interval are computed using local polynomial interpolation. Several numerical experiments illustrate the performance of the method. 

}, issn = {1991-7120}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/cicp/7873.html} }
TY - JOUR T1 - The Discrete Orthogonal Polynomial Least Squares Method for Approximation and Solving Partial Differential Equations AU - Anne Gelb, Rodrigo B. Platte & W. Steven Rosenthal JO - Communications in Computational Physics VL - 3 SP - 734 EP - 758 PY - 2008 DA - 2008/03 SN - 3 DO - http://dor.org/ UR - https://global-sci.org/intro/cicp/7873.html KW - AB -

We investigate numerical approximations based on polynomials that are orthogonal with respect to a weighted discrete inner product and develop an algorithm for solving time dependent differential equations. We focus on the family of super Gaussian weight functions and derive a criterion for the choice of parameters that provides good accuracy and stability for the time evolution of partial differential equations. Our results show that this approach circumvents the problems related to the Runge phenomenon on equally spaced nodes and provides high accuracy in space. For time stability, small corrections near the ends of the interval are computed using local polynomial interpolation. Several numerical experiments illustrate the performance of the method. 

Anne Gelb, Rodrigo B. Platte & W. Steven Rosenthal. (1970). The Discrete Orthogonal Polynomial Least Squares Method for Approximation and Solving Partial Differential Equations. Communications in Computational Physics. 3 (3). 734-758. doi:
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