Effective Preconditioner Updates for Sequences of Linear Systems Derived from Nonlinear Partial Differential Equations

With the advance of science and technology, people deal with problems more precisely and accurately in many fields like Physics, Seismology, Aerodynamics, Chemistry and so on and so forth. Therefore scientific computing should be highly concerned. Effective solving sequence of linear systems with large and sparse matrices plays a very important role in scientific computing. With the advance of science and technology, people deal with problems more precisely and accurately in many fields like Physics, Seismology, Aerodynamics, Chemistry and so on and so forth. Therefore scientific computing should be highly concerned. Effective solving sequence of linear systems with large and sparse matrices plays a very important role in scientific computing. In our article, we will take a two-dimensional nonlinear convection-diffusion model problem to be our example. We present a brief introduction of finite difference method, Newton-Raphson method and line search method. After applying these ideas, we will have a sequence of linear systems needed to be solve. And then, we will discuss three interesting methods for approximate updates of factorized preconditioners for solving sequences of linear systems. Numerical experiments show that these three method are profitable, that is, they have fewer number of iterations of preconditioned iterative methods for solving sequent systems of a sequence than freezing the preconditioner from the first system of the sequence. Since the interesting updates mainly cost less and straightforward, they may substitute for recomputing preconditioners which may take lots of time. To complete our work, we mainly consult [1], [2], [3], [7], [8] and [10]. And we also redesign and rearrange [7] in order to introduce everything as explicit as we can.