Newton's method serves as the primary tool for solving systems of nonlinear algebraic equations encountered in numerical simulators that discretize partial differential equations. The method is formally second order and efficient unless the algebraic system contains nonlinear stiffness, in the sense that a subset of equations imposes strong constraints on the solution update, limiting the effectiveness of the full increment computed from the linearized system.
Nonlinear preconditioning addresses nonlinearly stiff problems. Left preconditioners like ASPEN, MSPIN, RASPIN, NEPIN, etc., modify the algebraic system to achieve improved balance of nonlinearities. In contrast, right preconditioners maintain the original Jacobian, instead focusing on adjusting a subset of "problematic" variables in an inner relaxation step to enhance the initial guess for the subsequent global Newton step.
Many nonlinear preconditioners are domain decomposition methods and enhance scalability and efficiency for large-scale problems by decomposing the problem into smaller, manageable systems that can be distributed and solved across multiple processors or cores, leveraging parallel computing capabilities.
We have expertise in applying such methods to systems arising from multiphase, multicomponent flows in porous media, which can be notoriously difficult to solve due to discontinuities in solution variables and coefficients, many orders of variations in media properties, abrupt changes in drive mechanisms, complex grids, strong nonlinear coupling among different subequations, etc.