A Pair of One-step Optimized Spectral Hybrid Block Methods for the Direct Solution of PDEs

Abstract

This paper introduces a pair of single-step, optimized spectral hybrid block methods for the direct solution of partial differential equations. The methods are derived using an optimization approach aimed at maximizing accuracy. The methods are consistent, zero stable, convergent, and A stable. Implementation is carried out using the linear partition iteration technique, where the PDE is solved in time t using the hybrid block method and in space x using the spectral collocation method, following a suitable linear transformation. The computed results are compared with those of other numerical methods to demonstrate the efficiency and accuracy of the proposed approach. The findings indicate that the methods are both reliable and efficient.

Keywords: Spectral collocation method; Optimized hybrid block method; Linear partitioning iteration method; Partial differential equations.

2010 Mathematics Subject Classification. 35A25; 65M12; 65M70