A spectral collocation method for solving initial value problems of first order ordinary differential equations

Legendre–Gauss-type spectral collocation algorithms for nonlinear ordinary/partial differential equations

“…These approaches are more stable and more accurate for large N. We also notice that Guo and Wang [15] derived a Legendre-Gauss collocation algorithm for Equation (16).…”

“…Let M and N m , 1 ≤ m ≤ M be any positive integers. We first decompose the interval First, replacing T and N by τ 1 and N 1 in Equation (16) and all other formulas in Sections 2.1 and 2.2, we can derive an alternative algorithm, with which we obtain the numerical solution…”

“…Indeed, Lambert [22] designed an algorithm to solve the discrete system (16). Moreover, Wright [39] and Hairer and Wanner [18] studied the relationship between the method in [22] and the implicit Runge-Kutta method.…”

“…An important problem is how to solve Equation (16) efficiently. Indeed, Lambert [22] designed an algorithm to solve the discrete system (16).…”

“…We consider the following problem: We first use the the single-step scheme (16) to resolve the problem (32). Let y N (t) be the global numerical solution and denote the numerical errors by…”