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**On the equivalence of the time domain differential quadrature method and the dissipative Runge-Kutta collocation method.**
*(English)*
Zbl 0995.65085

Summary: Numerical solutions for initial value problems can be evaluated accurately and efficiently by the differential quadratic 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 nonlinear 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.

### MSC:

65L60 | Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations |

65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |

65L20 | Stability and convergence of numerical methods for ordinary differential equations |

34A34 | Nonlinear ordinary differential equations and systems |

### Keywords:

dissipative Runge-Kutta collocation method; single-step time marching schemes; higher-order accurate algorithms; controllable numerical dissipation; nonlinear transient analysis; initial value problems; differential quadratic method; numerical results; algorithms; implicit Runge-Kutta methods; Gauss methods; Radau IIA methods; stability
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\textit{T. C. Fung}, Int. J. Numer. Methods Eng. 53, No. 2, 409--431 (2002; Zbl 0995.65085)

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