EFFICIENT PRECONDITIONER FOR IMPLICIT RUNGE-KUTTA METHODS OF PARABOLIC EQUATIONS

Abstract

This research talks about the numerical solution of the one-dimensional heat equation using the Finite Element Method (FEM) for spatial discretization using two high-order implicit Runge–Kutta schemes for time integration Gauss–Legendre and Radau IIA. The aim of the study is to evaluate the stability, accuracy and computational efficiency of these fully implicit collocation methods when applied to parabolic partial differential equations PDEs with study the effectiveness of preconditioning in reducing the iterations and improving the convergence of GMRES. The numerical experiments reveal that Gauss–Legendre, while it is slightly less energy wasting, keep high time-related accuracy and exhibits reliable performance with cut-price computational cost. Radau IIA supplies excellent stability and strong damping of high-frequency numerical modes, leading to smooth and highly exact solutions even for relatively large time steps. Comparisons with analytical solutions evidence the correctness of the spatial FEM scheme and the high-order convergence of both time integrators. In general, the outcomes highlight the effectiveness of combining FEM with advanced implicit Runge–Kutta schemes for long-time heat diffusion simulations. The study shows an evident assessment of the advantages and disadvantages between Radau IIA and Gauss–Legendre, offering useful insight for selecting robust and efficient time-integration strategies for parabolic PDEs.