We investigate effectiveness of an acceleration method applied to the modified Picard iteration for simulations of variably saturated flow. We solve nonlinear systems using both unaccelerated and accelerated modified Picard iteration as well as Newton’s method. Since Picard iterations can be slow to converge, the advantage of acceleration is to provide faster convergence while maintaining advan...

For a class of quasi-contractive operators defined on an arbitrary Banach space, it has been shown that the Picard iteration technique converges faster than the Mann iteration technique. In this paper we make a comparison of the Picard and Mann iterations with respect to their convergence rate for a more general class of operators called quasi-contractions in metrizable topological vector space...

In this article, we propose a new multigrid-based algorithm for solving integral equations of the reference interactions site model (RISM). We also investigate the relationship between the parameters of the algorithm and the numerical accuracy of the hydration free energy calculations by RISM. For this purpose, we analyzed the performance of the method for several numerical tests with polar and...

Recently Bai and Yang in [On HSS-based iteration methods for weakly nonlinear systems, Appl. Numer. Math. 59 (2009) 2923–2936.] proposed the Picard-HSS iteration method to solve the system of nonlinear equations Ax = φ(x), where A ∈ Cn×n is a nonHermitian positive definite matrix and φ : D ⊂ C → C is continuously differentiable function defined on the open complex domain D in the n-dimensional ...

Let X, d be a complete metric space, and let T be a self-map of X. If T has a unique fixed point, which can be obtained as the limit of the sequence {pn}, where pn Tp0, p0 any point of X, then T is called a Picard operator see, e.g., 1 , and the iteration defined by {pn} is called Picard iteration. One of the most general contractive conditions for which a map T is a Picard operator is that of ...

converge to a unique solution of the IVP up to the boundary of U [1]. In general, φn may converge slowly to the exact solution. The Picard iteration based integrator described in this paper has three main advantages. The the integrator has arbitrary order, is time adaptive, and has dense output. Dense output refers to the integrator being able to take time steps of variable length without havin...

This paper discusses the diffusion equation with a damping term as follows ut = div(|Dum|p−2Dum)− u1 |Du|1 , where p > 2,m > 1, and p > p1, q1+p1m > m(p−1) > 1. By the standard Picard iteration method, a sufficient condition is given to the existence of the singular self-similar solutions. Moreover, the paper gives a classification for these singular self-similar solutions. Key–Words: Diffusion...

In this paper, we introduce and study a class of new Picard-Mann iterative methods with mixed errors for common fixed points of two different nonexpansive and contraction operators. We also give convergence and stability analysis of the new Picard-Mann iterative approximation and propose numerical examples to show that the new Picard-Mann iteration converges more effectively than the Picard ite...

A standard method for solving coupled multiphysics problems in light water reactors is Picard iteration, which sequentially alternates between solving single physics applications. This solution approach is appealing due to simplicity of implementation and the ability to leverage existing software packages to accurately solve single physics applications. However, there are several drawbacks in t...

Recently, two families of HSS-based iteration methods are constructed for solving the system of absolute value equations (AVEs), which is a class of non-differentiable NP-hard problems. In this study, we establish the Picard-CSCS iteration method and the nonlinear CSCS-like iteration method for AVEs involving the Toeplitz matrix. Then, we analyze the convergence of the Picard-CSCS iteration met...