An optimized block hybrid spectral simple iteration methods for solving nonlinear evolution equations

Abstract

This study presents a new optimized block hybrid method and spectral simple iteration method (OBHM-SSIM) for solving nonlinear evolution equations. In this method, we employed a combination of the spectral collocation method in space and the optimized block hybrid method in time, along with a simple iteration scheme to linearize the equations. The performance of OBHM-SSIM is compared with other established numerical methods for various nonlinear evolution equations, including the Stokes' second problem equation, Burgers─Fisher equation, Burgers─Huxley equation, the FitzHugh─Nagumo equation with time-dependent coefficients, and coupled Burgers' equations. Furthermore, the proposed OBHM-SSIM is implemented to solve -dimensional problems, specifically the nonlinear Burgers' equation and the cubic Klein─Gordon equation, demonstrating its capability to solve nonlinear systems efficiently. The extension to two-dimensional cases further validates the flexibility and accuracy of the OBHM-SSIM method, achieved with a notably reduced computational cost. Unlike conventional spectral methods, the proposed OBHM-SSIM achieves high-order accuracy with fewer grid points by optimizing intra-step points and maintaining A-stability for large time domains. We demonstrate that the OBHM-SSIM method gives highly accurate solutions with fewer grid points. This results in enhanced computational efficiency and reduced complexity, particularly for large time domains of nonlinear evolution equations. The findings of this study offer a new approach for the application of the spectral block hybrid method, ultimately improving the accuracy and efficiency of computational solutions for nonlinear evolution equations.