• Journal of the Korean Society for Industrial and Applied Mathematics
• /
• v.20 no.3
• /
• pp.243-259
• /
• 2016

The Richards equation for water movement in unsaturated soil is highly nonlinear partial differential equations which are not solvable analytically unless unrealistic and oversimplifying assumptions are made regarding the attributes, dynamics, and properties of the physical systems. Therefore, conventionally, numerical solutions are the only feasible procedures to model flow in partially saturated porous media. The standard Finite element numerical technique is usually coupled with an Euler time discretizations scheme. Except for the fully explicit forward method, any other Euler time-marching algorithm generates nonlinear algebraic equations which should be solved using iterative procedures such as Newton and Picard iterations. In this study, lumped mass and distributed mass in the frame of Picard and Newton iterative techniques were evaluated to determine the most efficient method to solve the Richards equation with finite element model. The accuracy and computational efficiency of the scheme and of the Picard and Newton models are assessed for three test problems simulating one-dimensional flow processes in unsaturated porous media. Results demonstrated that, the conventional mass distributed finite element method suffers from numerical oscillations at the wetting front, especially for very dry initial conditions. Even though small mesh sizes are applied for all the test problems, it is shown that the traditional mass-distributed scheme can still generate an incorrect response due to the highly nonlinear properties of water flow in unsaturated soil and cause numerical oscillation. On the other hand, non oscillatory solutions are obtained and non-physics solutions for these problems are evaded by using the mass-lumped finite element method.

• Korean Journal of Mathematics
• /
• v.8 no.1
• /
• pp.1-17
• /
• 2000

The existence of density of process, which is given by canonical stochastic differential equation, can be proved by the Picard's method([5]) also.

• Proceedings of the KIEE Conference
• /
• 1999.11c
• /
• pp.485-487
• /
• 1999

We propose the algorithm with which one can solve the problem of the two-level hierarchical optimal control of nonlinear systems by repeatedly updating the state vectors using the haar function and Picard's iteration methods. Using the simple operation of the coefficient vectors from the fast haar transformation in the upper level and applying that vectors to Picard iteration methods in the independently lower level allow us to obtain the another method except the inversion matrix operation of the high dimention and the kronecker product in the optimal control algorithm.

• Journal of the Korean Mathematical Society
• /
• v.50 no.5
• /
• pp.1129-1163
• /
• 2013

Based on finite element discretization, two linearization approaches to the defect-correction method for the steady incompressible Navier-Stokes equations are discussed and investigated. By applying $m$ times of Newton and Picard iterations to solve an artificial viscosity stabilized nonlinear Navier-Stokes problem, respectively, and then correcting the solution by solving a linear problem, two linearized defect-correction algorithms are proposed and analyzed. Error estimates with respect to the mesh size $h$, the kinematic viscosity ${\nu}$, the stability factor ${\alpha}$ and the number of nonlinear iterations $m$ for the discrete solution are derived for the linearized one-step defect-correction algorithms. Efficient stopping criteria for the nonlinear iterations are derived. The influence of the linearizations on the accuracy of the approximate solutions are also investigated. Finally, numerical experiments on a problem with known analytical solution, the lid-driven cavity flow, and the flow over a backward-facing step are performed to verify the theoretical results and demonstrate the effectiveness of the proposed defect-correction algorithms.

• Nonlinear Functional Analysis and Applications
• /
• v.27 no.4
• /
• pp.785-795
• /
• 2022

In this paper, we introduce a Mann-like three step iteration method and show that it can be used to approximate the fixed point of a weak contraction mapping. Furthermore, we prove that this scheme is equivalent to the Mann iterative scheme. A comparison is made with the other third order iterative methods. Results are presented in a table to support our conclusion.

• Proceedings of the KIEE Conference
• /
• 2007.10a
• /
• pp.267-268
• /
• 2007

This paper presents the new algebraic iterative algorithm of the bilinear system analysis using triangular orthogonal functions(TR) and the Picard's method. TR representation does not need any integration to evaluate the coefficients, thereby reducing a lot of computational burden. the proposed algorithm is more accuracy than BPF's. it is verified through simulation.

• Journal of Korean Port Research
• /
• v.13 no.1
• /
• pp.167-174
• /
• 1999

Numerical analysis of two-fluid flows for both water and air is carried out. Free-Surface flows with an arbitrary deformation have been simulated around two dimensional submerged hydrofoil. The computation is performed using a finite volume method with unstructured meshes and an interface capturing scheme to determine the shape of the free surface. The method uses control volumes with an arbitrary number of faces and allows cell-wise local mesh refinement. the integration in space is of second order based on midpoint rule integration and linear interpolation. The method is fully implicit and uses quadratic interpolation in time through three time levels The linear equation systems are solved by conjugate gradient type solvers and the non-linearity of equations is accounted for through picard iterations. The solution method is of pressure-correction type and solves sequentially the linearized momentum equations the continuity equation the conservation equation of one species and the equations or two turbulence quantities.

• Journal of Mechanical Science and Technology
• /
• v.14 no.7
• /
• pp.789-795
• /
• 2000

Free-surface flows with an arbitrary deformation, induced by a submerged hydrofoil, are simulated numerically, considering two-fluid flows of both water and air. The computation is performed by a finite volume method using unstructured meshes and an interface capturing scheme to determine the shape of the free surface. The method uses control volumes with an arbitrary number of faces and allows cell wise local mesh refinement. The integration in space is of second order, based on midpoint rule integration and linear interpolation. The method is fully implicit and uses quadratic interpolation in time through three time levels. The linear equations are solved by conjugate gradient type solvers, and the non-linearity of equations is accounted for through Picard iterations. The solution method is of pressure-correction type and solves sequentially the linearized momentum equations, the continuity equation, the conservation equation of one species, and the equations for two turbulence quantities. Finally, a comparison is quantitatively made at the same speed between the computation and experiment in which the grid sensitivity is numerically checked.

• Nonlinear Functional Analysis and Applications
• /
• v.29 no.1
• /
• pp.179-195
• /
• 2024

In this article, we study the Picard-Ishikawa iterative method for approximating the fixed point of generalized α-Reich-Suzuki nonexpanisive mappings. The weak and strong convergence theorems of the considered method are established with mild assumptions. Numerical example is provided to illustrate the computational efficiency of the studied method. We apply our results to the solution of a nonlinear delay integral equation. The results in this article are improvements of well-known results.

• East Asian mathematical journal
• /
• v.32 no.1
• /
• pp.129-138
• /
• 2016

In this paper, we propose an efficient regularization technique (The Method of Regularized Ratios) for the reconstruction of the shape of a rigid elastic scatterer from far field measurements. The approach used is based on the factorization method and creates via Picard's condition ratios, baptized Regularized Ratios, that serve to effectively remove unwanted singular values that may lead to poor reconstructions. This is achieved through the use of a sophisticated algorithm that progressively adjusts an initially set moderate tolerance. In comparison with the well established Tikhonov-Morozov regularization techniques our new algorithm appears to be more computationally efficient as it doesn't require computation of the regularization parameter for each point in the grid.