An accurate and efficient Chebyshev expansion method for large-scale transient heat conduction problems

Abstract

In this paper, an efficient and accurate Chebyshev expansion method is presented for solving large-scale transient heat conduction problems. Based on the Chebyshev expansion method, the matrix exponential is approximated with a series of Chebyshev matrix polynomials. Furthermore, according to the characteristics of practical thermal loads, an efficient method is developed to decrease the computational cost of temperature response induced by heat sources and nonhomogeneous boundary conditions. A theoretical method is developed to investigate the relationship of the computational cost of the proposed method and the time step, and the results indicate that under the given truncation criterion, the computational cost decreases with the increasing of the time step. Since the computational cost is sparse matrix–vector multiplications and only a few of vectors are stored in the computer memory, the proposed method has great advantages both in computational cost and storage requirement for large-scale transient heat conduction problems. In addition, a stability analysis is developed and the results show that the proposed method is unconditionally stable. Numerical examples exhibit that the proposed method has excellent efficiency and accuracy.