On the equivalence of the time domain differential quadrature method and the dissipative Runge-Kutta collocation method

摘要

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.