Application of the Krylov subspace method to numerical heat transfer

Five algorithms of the Krylov subspace method and three preconditioning techniques are presented. Applications of the Krylov subspace method were illustrated in solving four example problems. The matrix inversion example showed that the Krylov subspace method can be viewed as a direct method. It was demonstrated that, in solving an example problem of heat conduction with a constant source term, incomplete lower-upper (LU) (ILU) decomposition and polynomial preconditioning could substantially reduce the number of iterations. Also, linear relationships were observed between the iteration number and the equation number. It was also found that double preconditioning using a fifth-order polynomial and ILU decomposition could further reduce the computing time. The generalized minimal residual (GMRES) method with double preconditioning was compared with such iterative methods as alternating direction implicit (ADI) and Gauss-Seidel. The results showed that the GMRES method only required fractions of the computing time required by ADI or Gauss-Seidel method. Example problems of heat conduction with an Arrhenius source term and cavity flow were also solved by the GMRES method with preconditioning. Converged solutions were obtained with one or two iterations for the momentum equation of cavity flow considered, and three to six iterations for the pressure Poisson equation. Furthermore » effort seems to be warranted to explore the implementation of the Krylov subspace method for the finite difference modeling of heat transfer and fluid flow problems.« less