Solving initial value problems by differential quadrature method - part 2: second- and higher-order equations

In Part 1 of this paper, the sampling grid points for the di erential quadrature method to give unconditionally stable higher-order accurate time step integration algorithms are proposed to solve rst-order initial value problems. In this paper, the di erential quadrature method is extended to solve second-order initial value problems. The conventional approaches to impose the given initial conditions are discussed. A new approach to impose the given initial conditions is then presented. It is found that the proposed approach could generate unconditionally stable higher-order accurate time step integration algorithms for second-order equations directly. Furthermore, the procedure can be generalized to construct unconditionally stable higher-order accurate time step integration algorithms for third- and higher-order initial value problems without any di culties. The computational procedures for multi-degree-of-freedom systems and non-linear problems are also discussed. As demonstrated by the numerical examples, the di erential quadrature method using the proposed sampling grid points and the proposed method to impose the given initial conditions is found to be more e cient than the conventional di erential quadrature method in solving initial value problems