When local difficulties slow down the entire simulation
Many physical models are described by partial differential equations. Discretising these equations in space and time produces large nonlinear algebraic systems, commonly solved using Newton's method. The method can converge very rapidly near the solution, but strong nonlinearities can make reaching that regime difficult. A small region of the model may restrict the update for the entire system, leading to excessive iterations, repeated time-step cuts or convergence failure.
Nonlinear preconditioning addresses these difficulties by reformulating the equations or using local nonlinear solves to improve the global iteration. The aim is to reduce the impact of the most challenging nonlinearities, allowing the solver to make larger and more effective steps towards the solution.
How we can help
- Analyse convergence problems: Identify which parts of the model, equation system or solution algorithm limit progress.
- Develop and adapt solution methods: Design nonlinear preconditioners and combine them with Newton's method, relaxation and adaptive time stepping.
- Integrate methods into simulators: Adapt algorithms to the simulator's discretisation, coupled physics and software architecture.
- Evaluate robustness and performance: Assess total run time, convergence and parallel scalability on representative models, and determine whether the benefits outweigh the cost of additional local solves.
Methods and numerical expertise
We work with nonlinear preconditioning and domain decomposition, dividing a large problem into smaller subproblems. These can be solved locally, often in parallel, before their results are combined in a global iteration. The decomposition may follow spatial subdomains, different physical processes or groups of variables.
Left preconditioning applies Newton's method to a transformed residual system. Right preconditioning introduces a nonlinear mapping of the variables. A related practical approach uses local nonlinear solves to improve the state before a global Newton step. Our expertise covers Newton–Schwarz methods and related approaches, including ASPEN, MSPIN, RASPIN and NEPIN.
Performance depends on how the subproblems are selected and coupled, how accurately they are solved, and how the method interacts with the global solver. We therefore consider nonlinear preconditioning together with linear solvers, convergence criteria, relaxation and time-step control.
Particular expertise in flow through porous media
Our expertise is particularly relevant to multiphase, multicomponent flow in porous media. Large contrasts in material properties, sharp fronts, phase transitions, complex grids and strong coupling between equations can make these systems difficult to solve. Changes in well conditions and operating strategies introduce further challenges.
These difficulties arise in applications such as reservoir simulation and CO₂ storage modelling. Through numerical method and simulator development, we investigate how local nonlinear solves can improve robustness and reduce the need for expensive global iterations and time-step cuts.
Work with us
Are you developing a simulator, or finding that demanding models cause slow or unreliable convergence? Contact our Applied Computational Science research group to discuss collaboration on analysis, method development and implementation.