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B-spline adaptive collocation/Runge-Kutta software with interpolation-based spatial error estimation for the error-controlled numerical solution of PDEs

Banff International Research Station for Mathematical Innovation and Discovery

An essential component of a high quality numerical software package is a framework that provides an error-controlled computation of the approximate solution. This means that the package returns a numerical solution such that an associated error estimate satisfies a user-prescribed tolerance. This type of computation has two important advantages: (i) the user can have reasonable confidence that the numerical solution has an error that is consistent with the requested tolerance, and (ii) the cost of the computation will be consistent with the requested accuracy. In this talk, we introduce a new software package, called BACOLRI, for the error-controlled numerical solution of 1D PDEs. This code employs a high order B-spline collocation algorithm to discretize the spatial domain and then uses a high order error control Runge-Kutta package to solve the resulting system of time-dependent Differential-Algebraic Equations. BACOLRI estimates the spatial error of the approximate solution on each time-step using an interpolation-based scheme and employs adaptive mesh refinement to provide control of the spatial error. We will briefly survey the family of solvers from which BACOLRI has evolved, describe the algorithms implemented in BACOLRI, and review numerical results that demonstrate the superior performance of BACOLRI compared to other comparable solvers.

Mathematics; Numerical analysis; Partial differential equations

video/mp4

Author affiliation: Saint Mary's University

2019-03-25

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/69210

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/