ABSTRACT
This chapter introduces nonlinear problems, the Picard and Newton methods and three space dimension problems, which uses high-performance computing (HPC). It discusses HPC and domain decomposition reordering for 2D and 3D models. The chapter presents two algorithms, Picard and Newton, which do generalize to nonlinear problems with more than one unknown. Newton’s algorithm is one of the most important numerical schemes because, under appropriate conditions, it has local and quadratic convergence properties. The Picard algorithm only has first-order convergence where the error at the next step is proportional to the error at the present step. But, the Picard algorithm may converge to a fixed point regardless of the initial guess. The chapter considers the nonlinear 2D problem where the thermal conductivity is a function of the temperature. The Picard nonlinear algorithm with a preconditioned conjugate gradient method for each of the linear solves are used.
