Iterative acceleration methods with second-order time accuracy for nonlinear diffusion equations

Abstract

For a two-layer finite difference scheme with second-order time accuracy of nonlinear diffusion equations, we present three iterative solving algorithms, including Picard, Picard-Newton and derivative-free Picard-Newton iterations. The main purpose of this paper is to solve the two-layer 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 first-order accuracy, the other is an extrapolation with second-order 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 second-order accuracy both in space and time after two nonlinear iteration steps, even if the first-order initial value is used. It is also proved that the solutions of Picard iteration converge linearly to the solution of the discrete scheme, while Picard-Newton and derivative-free Picard-Newton 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, Picard-Newton and derivative-free Picard-Newton iterations are more efficient for solving nonlinear problems.