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

Abstract: This work proposes a general strategy for solving possibly nonlinear problems arising from implicit time discretizations as a sequence of explicit solutions. The resulting sequence may exhibit instabilities similar to those of the base explicit scheme, which can be mitigated through Anderson acceleration. The approach uses explicit fixed-point subiterations for nonlinear problems, combined with Anderson acceleration to improve convergence and computational efficiency. Its usability and scalability are verified on three nonlinear differential equations. An error analysis is presented to establish the expected properties of the proposed strategy for both time and space-time formulations. Several examples illustrate the simplicity of the implementation and reveal the influence of parameter choices. The method proves simple to implement and performs well across a range of problems, particularly when matrix assembly is expensive or a good preconditioner for the implicit system is unavailable, such as in highly convective fluid flows. This work formalizes the delay of implicit terms in time discretization, provides a concise error analysis, and enhances the approach using Anderson acceleration. The results are encouraging and well supported by existing theory, laying the groundwork for further research.