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Collocation and Galerkin Time-Stepping Methods by Nasa Technical Reports Server (Ntrs), H. T. Huynh

Collocation and Galerkin Time-Stepping Methods

by Nasa Technical Reports Server (Ntrs) , H. T. Huynh
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Book cover type: Paperback
  • ISBN13: 9781289031084
  • Binding: Paperback
  • Subject: N/A
  • Publisher: Bibliogov
  • Publisher Imprint: Bibliogov
  • Publication Date:
  • Pages: 42
  • Original Price: USD 15.75
  • Language: English
  • Edition: N/A
  • Item Weight: 96 grams
  • BISAC Subject(s): General

We study the numerical solutions of ordinary differential equations by one-step methods where the solution at tn is known and that at t(sub n+1) is to be calculated. The approaches employed are collocation, continuous Galerkin (CG) and discontinuous Galerkin (DG). Relations among these three approaches are established. A quadrature formula using s evaluation points is employed for the Galerkin formulations. We show that with such a quadrature, the CG method is identical to the collocation method using quadrature points as collocation points. Furthermore, if the quadrature formula is the right Radau one (including t(sub n+1)), then the DG and CG methods also become identical, and they reduce to the Radau IIA collocation method. In addition, we present a generalization of DG that yields a method identical to CG and collocation with arbitrary collocation points. Thus, the collocation, CG, and generalized DG methods are equivalent, and the latter two methods can be formulated using the differential instead of integral equation. Finally, all schemes discussed can be cast as s-stage implicit Runge-Kutta methods.

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