Sequential-implicit Newton method for multiphysics simulation

Efficient simulation of multiphysics problems is a challenging task. This is often due to the multiscale nature of the physics and nonlinear coupling between the different processes. One approach to this problem is to solve the entire multiphysics problem simultaneously in a fully coupled manner. However, due to the strong coupling and multiphysics interactions, it is difficult to design and analyze fully coupled solvers, which often entail the construction and solution of global fully coupled Jacobian systems. Another approach is the sequential-implicit method, whereby the full multiphysics problem is split into different subproblems. Each subproblem is constructed and solved separately; then the solutions of the subproblems are stitched together in a specific sequence. Isolation of each subproblem allows for the design of specialized solvers that tackle the complexity of the particular subproblem efficiently. The sequential-implicit approach offers wide flexibility and extensibility. However, these advantages are often offset by slow convergence rates when there is strong coupling between the subproblems. This slow convergence rate of the overall coupled system is directly linked to the use of a fixed-point outer-loop iteration in sequential-implicit methods. We present a Sequential Implicit Newton (SIN)approach, whereby the sequence of implicit subproblems is wrapped with an outer full Newton scheme. We demonstrate that the SIN formulation improves the overall convergence rate from linear to quadratic. The SIN method uses the same sequential-implicit scheme as the fixed-point method, but after each sequential iteration a Newton update is computed. Wrapping a Newton loop around the traditional sequential-implicit scheme leads to significant improvements in the convergence rate. The SIN method allows for the ability to split a multiphysics problem into individual subproblems while taking advantage of the quadratic convergence rate of the Newton method. We demonstrate the effectiveness of SIN using two different multiphysics porous-media problems: flow-thermal in geothermal reservoir simulation and flow-mechanics in geomechanics reservoir simulation. Just as with the fixed-point iteration version of the sequential-implicit method, SIN benefits from careful design of the constraints used to stitch the sequence of the subproblems. Our numerical experiments show that the SIN approach improves the overall convergence rate for all the nonlinear multiphysics problems considered. For specific cases where there is strong nonlinear coupling between the subproblems, we see up to two orders of magnitude decrease in the number of sequential iterations when using SIN compared with the fixed-iteration scheme.