A Comparative Study of Integration-Based and Trigonometric Collocation based Spectral Methods for High-Order Differential Equations

Abstract

Many models arising in structural mechanics, aero elasticity, and ballistic systems lead to high-order differential equations whose numerical solution remains challenging due to severe ill-conditioning in conventional spectral collocation methods. This paper compares two Chebyshev spectral collocation formulations for the numerical solution of high-order differential equations. The first is an integration-based approach in which the highest-order derivative is discretized and the solution is recovered through successive integration, with integration constants enforced via boundary conditions. This formulation alters the conventional error propagation mechanism, as numerical integration smooth discretization errors while differentiation amplifies them. The second approach is a trigonometric collocation method based on the cosine transformation, which maps Chebyshev polynomials to trigonometric functions and permits direct differentiation using analytical chain-rule expressions without constructing differentiation matrices.

The methods are evaluated on representative test problems, including fourth-order boundary value problems, eigenvalue problems, singularly perturbed equations, and sixth-order differential equations. Numerical experiments demonstrate that both approaches yield substantially improved accuracy and conditioning relative to classical differentiation-based Chebyshev methods. The integration-based formulation exhibits robust numerical stability, with condition numbers scaling as   independent of the differential order, making it well suited for stiff and very high-order problems. The trigonometric collocation method attains comparable accuracy with reduced system size and lower assembly cost, offering computational advantages for eigenvalue calculations.