Legendre–Gauss collocation methods for ordinary differential equations

Abstract

In this paper, we propose two efficient numerical integration processes for initial value problems of ordinary differential equations. The first algorithm is the Legendre-Gauss collocation method, which is easy to be implemented and possesses the spectral accuracy. The second algorithm is a mixture of the collocation method coupled with domain decomposition, which can be regarded as a specific implicit Legendre-Gauss Runge-Kutta method, with the global convergence and the spectral accuracy. Numerical results demonstrate the spectral accuracy of these approaches and coincide well with theoretical analysis.

Summation-by-parts in time

“…The fully discrete version of (12) is obtained by discretizing (15) or equivalently (18) in time using the SBP-SAT technique. The use of the Kronecker product rules (21)(22)(23) and (19) yield…”

“…In global methods, the whole time interval from zero to the final time T is considered. Global methods using collocation and spectral approximations have been considered previously (see [13], [14], [15], [16]) but have often been considered unpractical. However, the unconditional stability in combination with the very high accuracy cannot be matched by the local methods.…”

Laguerre-Gauss collocation method for initial value problems of second order ODEs

“…Remark 2 We can also use (19) and (20) to evaluate the values of u N (t N β,k ). By (2.16) of [25], we know t N β,N ∼ = 4β −1 N. Thus, we obtain the values of the numerical solution at the large interpolation nodes, even for the moderate mode N .…”

“…When f (z 1 , z 0 , t) is nonlinear, we can use a certain iteration process to solve (19) and obtain u N β,l (0 l N + 2). It is shown in the last part of Appendix A of this paper that, if some conditions are fulfilled, then the related iteration is convergent.…”

“…Remark 4 The mechanism of the algorithm (18) is not the same as that of the commonly used collocation method (see Remark 2.1 of [19]). In fact, in the derivation of our new process (18), we approximate the solution U (t) directly by the Laguerre-Gauss interpolation I β,N , which is stable for the large N .…”