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On the equivalence of the time domain differential quadrature method and the dissipative Runge–Kutta collocation method
Author(s) -
Fung T. C.
Publication year - 2001
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.283
Subject(s) - runge–kutta methods , mathematics , dissipative system , gaussian quadrature , gauss , numerical integration , numerical methods for ordinary differential equations , equivalence (formal languages) , quadrature (astronomy) , collocation method , numerical analysis , differential equation , mathematical analysis , nyström method , ordinary differential equation , integral equation , physics , pure mathematics , quantum mechanics , optics
Numerical solutions for initial value problems can be evaluated accurately and efficiently by the differential quadrature method. Unconditionally stable higher order accurate time step integration algorithms can be constructed systematically from this framework. It has been observed that highly accurate numerical results can also be obtained for non‐linear problems. In this paper, it is shown that the algorithms are in fact related to the well‐established implicit Runge–Kutta methods. Through this relation, new implicit Runge–Kutta methods with controllable numerical dissipation are derived. Among them, the non‐dissipative and asymptotically annihilating algorithms correspond to the Gauss methods and the Radau IIA methods, respectively. Other dissipative algorithms between these two extreme cases are shown to be B‐stable (or algebraically stable) as well and the order of accuracy is the same as the corresponding Radau IIA method. Through the equivalence, it can be inferred that the differential quadrature method also enjoys the same stability and accuracy properties. Copyright © 2001 John Wiley & Sons, Ltd.