Efficient spectral-Galerkin methods for systems of coupled second-order equations and their applications

We construct in this paper two efficient spectral-Galerkin algorithms for solving systems of n coupled second-order equations. The computational complexity of these algorithms is essentially n times the cost of solving one second-order equation. We present numerical results which illustrate the accuracy and flexibility of these algorithms, as well as several interesting and challenging applications, including in particular a number of high-order nonlinear parabolic type equations.