Accelerated implicitization: robust fixed-point iterations arising from an explicit scheme

Purpose This paper aims to propose a general strategy for solving the possibly nonlinear problems arising from implicit time discretizations as a sequence of explicit solutions. This sequence can be as unstable as the base explicit discretization, which can be improved through Anderson acceleration (AA). Design/methodology/approach The authors propose using explicit fixed-point sub-iterations for nonlinear problems combined with the AA technique to improve convergence and speed, verifying its usability and scalability in three nonlinear differential equations. This study provides an error analysis to establish the expected properties of the proposed strategy, both for time and space-time based problems. Through several examples, this study shows how simple it is to setup this method and then expose its strengths by doing parameter sensitivity. Findings The proposed method is simple to implement and yields acceptable performance in a wide range of problems. This type of method is best suited for situations in which matrix assembly can be prohibitively expensive, or whenever a good preconditioner for the implicit problem is out of reach, such as in highly convective fluid flows. In tests where nonaccelerated iterations are convergent, acceleration provides an overall reduction in the iterations of up to a 96%. Originality/value This work formalizes the delay of implicit terms in an implicit time discretization, provides a succinct error analysis and further enhances the strategy through AA. The results are encouraging and well founded in existing theory, thus establishing the foundations for interesting future research.