Solving initial value problems by differential quadrature method - part 1: first-order equations

In this paper, the di erential quadrature method is used to solve rst-order initial value problems. The initial condition is given at the beginning of a time interval. The time derivative at a sampling grid point within the time interval can be expressed as a weighted linear sum of the given initial condition and the function values at the sampling grid points within the time interval. The order of accuracy and the stability property of the quadrature solutions depend on the locations of the sampling grid points. It is shown that the order of accuracy of the quadrature solutions at the end of a time interval can be improved to 2n − 1 or 2n if the n sampling grid points are chosen carefully. In fact, the approximate solutions are equivalent to the generalized Pad e approximations. The resultant algorithms are therefore unconditionally stable with controllable numerical dissipation. The corresponding sampling grid points are found to be given by the roots of the modi ed shifted Legendre polynomials. From the numerical examples, the accuracy of the quadrature solutions obtained by using the proposed sampling grid points is found to be better than those obtained by the commonly used uniformly spaced or Chebyshev{Gauss{Lobatto sampling grid points.