Iterative acceleration methods with secondorder time accuracy for nonlinear diffusion equations For a twolayer finite difference scheme with secondorder time accuracy of nonlinear diffusion equations, we present three iterative solving algorithms, including Picard, PicardNewton and derivativefree PicardNewton iterations. The main purpose of this paper is to solve the twolayer scheme efficiently and accurately, and give strict theoretical proofs of the convergence and efficiency of the three iterative methods. For the three iterative methods, two strategies are adopted to offer iterative initial evaluations, one is to use the value at the previous time step with firstorder accuracy, the other is an extrapolation with secondorder accuracy. With an induction reasoning technique, we prove that the solutions of the three iterative methods all converge to the exact solution of the diffusion problem with secondorder accuracy both in space and time after two nonlinear iteration steps, even if the firstorder initial value is used. It is also proved that the solutions of Picard iteration converge linearly to the solution of the discrete scheme, while PicardNewton and derivativefree PicardNewton iterative solutions converge with a quadratic speed. Moreover, no difference occurs in convergent speed with the two different initial values. Finally, some numerical tests are presented to verify our theoretical results, which show that compared with Picard iteration, PicardNewton and derivativefree PicardNewton iterations are more efficient for solving nonlinear problems.

Nonoverlapping Localized Exponential Time Differencing Methods for Diffusion Problems In this paper, we propose nonoverlapping localized exponential time differencing (ETD) methods for diffusion problems. The model timedependent diffusion equation is first reformulated on subdomains based on the nonoverlapping domain decomposition, in which Neumann boundary conditions are imposed on the interfaces for the subdomain problems and Dirichlet type conditions are enforced to form a spacetime interface problem. After spatial discretization by standard central finite differences and temporal integration with the first or second order ETD methods, the fully discrete interface problem is obtained. Such an interface problem is then solved iteratively either at each time step or over the whole time interval: the former involves the solution of stationary problems in each subdomain at each iteration while the latter involves the solution of timedependent subdomain problems at each iteration. For both approaches, we prove that localized ETD solutions conserve mass exactly and converge in time to the exact space semidiscrete solution. Numerical experiments in two dimensions are also presented to illustrate the performance of the proposed methods.