The automatic selection of an appropriate time step size has been considered extensively in the literature. However, most of the strategies developed operate under the assumption that the computational cost (per time step) is independent of the step size. This assumption is reasonable for non-stiff ordinary differential equations and for partial differential equations where the linear systems of equations resulting from an implicit integrator are solved by direct methods. It is, however, usually not satisfied if iterative (for example, Krylov) methods are used.
In this paper, we propose a step size selection strategy that adaptively reduces the computational cost (per unit time step) as the simulation progresses, constraint by the tolerance specified. We show that the proposed approach yields significant improvements in performance for a range of problems (diffusion-advection equation, Burgers' equation with a reaction term, porous media equation, viscous Burgers' equation, Allen--Cahn equation, and the two-dimensional Brusselator system). While traditional step size controllers have emphasized a smooth sequence of time step sizes, we emphasize the exploration of different step sizes which necessitates relatively rapid changes in the step size.
Adaptive iterative linearised finite element methods for implicitly constituted incompressible fluid flow problems and its application to Bingham fluids
Weak and strong convergence of an implicit iterative process with errors for a finite family of asymptotically quasi $I-$nonexpansive mappings in Banach space
International Islamic University Malaysia（马来西亚国际伊斯兰大学）
Fast iterative method with a second order implicit difference scheme for time-space fractional convection-diffusion equations
University of Electronic Science and Technology of China（电子科技大学）；University of Groningen（格罗宁根大学）；Southeast University（东南大学）；Nottingham Trent University（诺丁汉特伦特大学）；Russian Academy of Sciences（俄罗斯科学院）
Numerical Modeling of Nanoparticles Transport with Two-Phase Flow in Porous Media Using Iterative Implicit Method
School of Mathematics and Statistics, Hubei Engineering University（湖北工程学院数学与统计学院）；Computational Transport Phenomena Laboratory (CTPL), Division of Physical Sciences and Engineering (PSE), King Abdullah University of Science and Technology (KAUST)（阿卜杜拉国王科技大学物理科学与工程学部计算流动现象实验室）
Semi-Implicit Neural Solver for Time-dependent Partial Differential Equations
Technical University Munich（慕尼黑工业大学）；École normale supérieure, Paris（巴黎高等师范学院）；deepc GmbH, Munich（慕尼黑deepc有限公司）；Helmholtz Zentrum München（慕尼黑亥姆霍兹研究中心）；University of Zurich（苏黎世大学）
