Picard Method Research Articles

Switching the Richards’ equation for modeling soil water movement under unfavorable conditions

Simulation of variably saturated soil water flow requires the use of pressure head, or soil moisture, or a switching between the two, as the primary variable for solving Richards’ equation. Under unfavorable conditions, such as heterogeneity, rapidly changing atmospheric boundary, or sudden infiltration into dry soils, the traditional non-switching method suffers from numerical difficulties. Solving this problem with a primary variable switching method is less preferred due to the mathematical complexity. While the Picard method is more popular for solving the non-switching models due to its simplicity and stability, two different forms of Richards’ equation are combined into one numerical scheme for switching under specific hydraulic conditions. The method is successfully implemented in a one-dimensional model solved by a Picard iteration scheme. A threshold saturation based on the soil moisture retention relation is used for switching between either form of the Richards’ equation. The method developed here is applicable for simulating variably saturated subsurface flow in heterogeneous soils. Compared with traditional methods, the proposed model conserves mass well and is numerically more stable and efficient.

Approximate solution of intuitionistic fuzzy differential equations by using Picard's method

Approximate solution of intuitionistic fuzzy differential equations by using Picard's method

Fuzzy picard’s method for derivatives of second order

In this paper we proposed an iterative scheme for the solution of second order fuzzy differential equations with fuzzy initial conditions using fuzzy picard method. The convergence is also discussed under generalized H – differentiability. Finally this method evidenced is illustrated by solving some examples to show its efficiency

The Improvement of Coupling Method in TINTE by Fully Implicit Scheme

TINTE is a well-established code for the pebble-bed high-temperature gas-cooled reactor (HTR), including the complicated nuclear module and thermal-hydraulic module, which has been validated by experiments and widely used in the transient behavior simulation. However, only an operator splitting scheme is employed in TINTE to couple the neutronics and thermal hydraulics, and some physical quantities are not consistent in time. As a result, the accuracy and stability are limited by the additional error term derived from the unconverged physical term. In this paper, a fully implicit coupling method was investigated in which the coupled nonlinear fields at each time step are converged using Picard iterations. A physics-based preconditioning is proposed in the work here to further improve the computational performance of the fully implicit coupling method. Seven test problems are implemented based on a practical engineering model, rather than a simple model, to evaluate the performance of the Picard method. The numerical results show that the fully implicit Picard iteration method is more accurate and more stable, which permits longer time steps and a reduction of the computational burden for solving the coupled field equations. The computational efficiency is further enhanced when the physics-based preconditioning is utilized.

Coupling methods for HTR-PM primary circuit

The purpose of this paper is to analyze coupling methods for High-Temperature Gas-Cooled Reactor-Pebble bed Module (HTR-PM) primary circuit, including reactor core, steam generator, and other components. Each component has individual characteristics and is simulated by an independent code package. A feasible way to integrate these independent codes is by using Picard iteration. In practice, the primary circuit is divided into multiple parts based on physical fields or spatial domains. Another coupling method is Jacobian-free Newton-Krylov (JFNK), which is frequently applied for the simulation of coupled multi-physics systems due to its efficiency and capability for large-scale nonlinear problems. To demonstrate the efficiency of coupling method for tightly coupled primary circuit systems, several Picard and JFNK methods were compared consistently in this paper, where all the models, equations and codes for each method are the same. Especially for neutronics, we use the same additional equation for JFNK as the outer iteration step in power iteration. Results show that the equations in the reactor core and outside the reactor core should be solved simultaneously in Picard iteration, otherwise additional thermal-hydraulics iterations are necessary. JFNK methods have shown better convergence behavior than Picard iteration, especially for computational efficiency. Since the nonlinear residual is evaluated directly from nonlinear equation set without iteration, linear preconditioned JFNK has shown efficient convergence, which is recommended by the authors as the first choice for HTR-PM primary circuit if the linear preconditioned JFNK is applicable and the well-performing preconditioner is available. However, for utilizing physical iteration processes, nonlinear preconditioned JFNK is a practical choice when integrating legacy codes.

Computer simulation of hyperthermia with nanoparticles using an OcTree finite volume technique

This paper develops an efficient finite volume approach using OcTree mesh for simulation of hyperthermia treatment by injection of nanoparticles. A three-dimensional nonlinear Pennes' model is employed to simulate the heat transfer in a heterogeneous blood perfused living tissue. The OcTree mesh is a good compromise between regular grid from finite difference method and unstructured meshes from finite element method due to the possibility of applying local refinement using a nonconforming mesh in a straightforward manner. Implicit time-marching scheme coupled with the Picard method and explicit method are used. A numerical example is analyzed, illustrating the advantages of the approach.

The application of Picard method in identification of the heat transfer coefficient in flow boiling in a minichannel

This paper summarizes the procedure of experimental determination of flow boiling heat transfer coefficient in the rectangular minichannel 2 mm wide, 1.8 mm deep and 193 mm long. In this experiment, the basic thermal and flow parameters were measured and the temperature distribution on the foil insulating the heater was recorded using an infrared camera. A high speed video camera observed the vapor patterns forming during the boiling process. The void fraction was determined with the image processing technique. Stationary heat transfer process in the insulating foil, the heater and the flowing liquid was described using two-dimensional Laplace’s equation (for the foil), the Poisson equation (for the heater) and energy equation (for the liquid) – all complemented with an appropriate system of boundary conditions. The system of the differential equations with adopted boundary conditions gives the three consecutive inverse problems in elements of the test section containing minichanel. Trefftz method was used to determine two dimensional temperature distribution of the foil and the heater. The liquid temperature was calculated using the Picard-Trefftz method. The values of the experimentally obtained heat transfer coefficients with the values of these coefficients calculated on the basis of correlations known from the literature were also compared.

Efficient simultaneous solution of multi-physics multi-scale nonlinear coupled system in HTR reactor based on nonlinear elimination method

Pebble bed HTR reactor core is a globally coupled system involving neutronics and thermal-hydraulic field. At the same time, the locally coupled relations exist in fuel sphere temperature, local neutron flux and pebble bed temperature. What’s more, several types of fuel spheres with different burnup and fission power must be treated separately. Due to the essential difference between local and global coupling properties, the method, named nonlinear elimination (NlEm), is developed in this paper to solve the multi-physics/multi-scale system of HTR in a tightly coupled, two-level nonlinear form. In the NlEm framework, the local coupled variables and the global coupled variables are treated separately at two different levels to enhance the computational efficiency. For comparison, the Jacobian-free Newton Krylov (JFNK) method and the Picard method are utilized to solve the multi-physics/multi-scale system respectively. The numerical results reveal that the computational efficiency of NlEm is 6.7 times higher than that of Picard method and 3.5 times higher than that of JFNK method for five transient cases of a two-dimension simplified reactor model. And with the increase of the computational scale and time step size, the speedup ratio of NlEm is further raised.

Modified methods for solving two classes of distributed order linear fractional differential equations

This paper introduces two methods for the numerical solution of distributed order linear fractional differential equations. The first method focuses on initial value problems (IVPs) and based on the αth Caputo fractional definition with the shifted Chebyshev operational matrix of fractional integration. By applying this method, the IVPs are converted into simple linear differential equations which can be easily handled. The other method focuses on boundary value problems (BVPs) based on Picard's method frame. This method is based on iterative formula contains an auxiliary parameter which provides a simple way to control the convergence region of solution series. Several numerical examples are used to illustrate the accuracy of the proposed methods compared to the existing methods. Also, the response of mechanical system described by such equations is studied.

Efficacy of Coefficients on Rate of Convergence of Some Iteration Methods for Quasi-Contractions

By comparing rate of convergence for some iteration methods, we show that the coefficients have important role in rate of convergence for quasi-contractions in Picard, Mann, Ishikawa, Noor, Agarwal and Thakur iteration methods. Using the Matlab, we provide some numerical examples for our results.

Nonlinear Effects on the Convergence of Picard and Newton Iteration Methods in the Numerical Solution of One-Dimensional Variably Saturated–Unsaturated Flow Problems

Finite element discretization of the pressure head form of the Richards equation leads to a nonlinear model, which yields numerical convergence difficulties. When the numerical solution to this problem has either an extremely sharp moving front, infiltration into dry soil, flow domains containing materials with spatially varying properties, or involves time-dependent boundary conditions, the corrector iteration used in many time integrators can terminate prematurely, which leads to incorrect results. While the Picard and Newton iteration methods can solve this problem through tightening the tolerances provided to the solvers, there is a more efficient approach to overcome the convergence difficulties. Four tests examples are examined, and each test case is solved with five sufficiently small tolerances to demonstrate the effectiveness of convergence. The numerical results illustrate that the methods greatly improve the convergence and stability. Test experiments show that the Newton method is more complex and expensive on a per iteration basis than the Picard method for simulating variably saturated–unsaturated flow in one spatial dimension. Consequently, it is suggested that the resulting local and global mass balance is exact within the minimum specified accuracy.

Using the picard method to calculate covariance matrices in the discrete Kalman filters

A new method of calculating covariance matrices for transition from a system of continuous linear stochastic differential equations to its discrete multidimensional stochastic analog has been developed. The proposed method is based on the use of the Picard iterative process. The comparison of the proposed method with more widespread analogs has shown a significant computational advantage of the new method when applied to the Kalman algorithms.

Anderson acceleration and application to the three-temperature energy equations

The Anderson acceleration method is an algorithm for accelerating the convergence of fixed-point iterations, including the Picard method. Anderson acceleration was first proposed in 1965 and, for some years, has been used successfully to accelerate the convergence of self-consistent field iterations in electronic-structure computations. Recently, the method has attracted growing attention in other application areas and among numerical analysts.Compared with a Newton-like method, an advantage of Anderson acceleration is that there is no need to form the Jacobian matrix. Thus the method is easy to implement. In this paper, an Anderson-accelerated Picard method is employed to solve the three-temperature energy equations, which are a type of strong nonlinear radiation-diffusion equations. Two strategies are used to improve the robustness of the Anderson acceleration method. One strategy is to adjust the iterates when necessary to satisfy the physical constraint. Another strategy is to monitor and, if necessary, reduce the matrix condition number of the least-squares problem in the Anderson-acceleration implementation so that numerical stability can be guaranteed. Numerical results show that the Anderson-accelerated Picard method can solve the three-temperature energy equations efficiently. Compared with the Picard method without acceleration, Anderson acceleration can reduce the number of iterations by at least half. A comparison between a Jacobian-free Newton–Krylov method, the Picard method, and the Anderson-accelerated Picard method is conducted in this paper.

A vertex-centered and positivity-preserving scheme for anisotropic diffusion problems on arbitrary polygonal grids

We suggest a new positivity-preserving finite volume scheme for anisotropic diffusion problems on arbitrary polygonal grids. The scheme has vertex-centered, edge-midpoint and cell-centered unknowns. The vertex-centered unknowns are primary and have finite volume equations associated with them. The edge-midpoint and cell-centered unknowns are treated as auxiliary ones and are interpolated by the primary unknowns, which makes the final scheme a pure vertex-centered one. Unlike most existing positivity-preserving schemes, the construction of the scheme is based on a special nonlinear two-point flux approximation that has a fixed stencil and does not require the convex decomposition of the co-normal. In order to solve efficiently the nonlinear systems resulting from the nonlinear scheme, Picard method and its Anderson acceleration are discussed. Numerical experiments demonstrate the second-order accuracy and well positivity of the solution for heterogeneous and anisotropic problems on severely distorted grids. The high efficiency of the Anderson acceleration is also shown on reduction of the number of nonlinear iterations. Moreover, the proposed scheme does not have the so-called numerical heat-barrier issue suffered by most existing cell-centered and hybrid schemes. However, further improvements have to be made if the solution is very close to the machine precision and the mesh distortion is very severe.

Analysis of a linearization scheme for an interior penalty discontinuous Galerkin method for two‐phase flow in porous media with dynamic capillarity effects

SummaryWe present a linearization scheme for an interior penalty discontinuous Galerkin method for two‐phase porous media flow model, which includes dynamic effects on the capillary pressure. The fluids are assumed immiscible and incompressible, and the solid matrix non‐deformable. The physical laws are approximated in their original form, without using the global or complementary pressures. The linearization scheme does not require any regularization step. Furthermore, in contrast with Newton or Picard methods, there is no computation of derivatives involved. We prove rigourously that the scheme is robust and linearly convergent. We make an extensive parameter study to compare the behaviour of the L‐scheme with the Newton method. Copyright © 2017 John Wiley & Sons, Ltd.

Controlled Picard Method for Solving Nonlinear Fractional Reaction–Diffusion Models in Porous Catalysts

This paper discusses the diffusion and reaction behaviors of catalyst pellets in the fractional-order domain as well as the case of nth-order reactions. Two generic models are studied to calculate the concentration of reactant in a porous catalyst in the case of a spherical geometric pellet and a flat-plate particle with different examples. A controlled Picard analytical method is introduced to obtain an approximated solution for these systems in both linear and nonlinear cases. This method can cover a wider range of problems due to the extra auxiliary parameter, which enhances the convergence and is suitable for higher-order differential equations. Moreover, the exact solution in the linear fractional-order system is obtained using the Mittag–Leffler function where the conventional solution is a special case. For nonlinear models, the proposed method gives matched responses with the homotopy analysis method (HAM) solutions for different fractional orders. The effect of fractional-order parameter on the dimensionless concentration of the reactant in a porous catalyst is analyzed graphically for different cases of order reactions and Thiele moduli. Moreover, the proposed method has been applied numerically for different cases to predict and calculate the dual solutions of a nonlinear fractional model when the reaction order n = −1.

CAS Picard method for fractional nonlinear differential equation

In this paper, a computational method for solving the fractional nonlinear differential equation is introduced. We proposed a method by utilizing the CAS wavelets in conjunction with Picard technique. We call the method as CAS Picard method. The fractional nonlinear differential equations are transformed into a system of discrete fractional differential equations by Picard technique and then transformed into a system of algebraic equations by using the operational matrices of CAS wavelets. The error and supporting analysis of the method are also investigated. The comparison analysis of method with other existing numerical methods is also performed.

Block triangular preconditioners for linearization schemes of the Rayleigh–Bénard convection problem

SummaryIn this paper, we compare two block triangular preconditioners for different linearizations of the Rayleigh–Bénard convection problem discretized with finite element methods. The two preconditioners differ in the nested or nonnested use of a certain approximation of the Schur complement associated to the Navier–Stokes block. First, bounds on the generalized eigenvalues are obtained for the preconditioned systems linearized with both Picard and Newton methods. Then, the performance of the proposed preconditioners is studied in terms of computational time. This investigation reveals some inconsistencies in the literature that are hereby discussed. We observe that the nonnested preconditioner works best both for the Picard and for the Newton cases. Therefore, we further investigate its performance by extending its application to a mixed Picard–Newton scheme. Numerical results of two‐ and three‐dimensional cases show that the convergence is robust with respect to the mesh size. We also give a characterization of the performance of the various preconditioned linearization schemes in terms of the Rayleigh number.

A comparative study of finite element method for the numerical solution of reaction-diffusion problem

This paper presents a least-squares finite element approach for the solution of reaction-diffusion pellet problem for slab geometry by considering some special types of reactions which are applicable for chemical transformation of natural gas components. The nonlinear terms are treated with Newton's linearisation technique. A detailed comparison with other weighted residual methods such as collocation, sub-domain and Galerkin methods are also examined. The issues related to chemical reactor problems and their error analysis is discussed both theoretically and numerically. The least-squares method suffers convergency effect by using the Picard's method instead of Newton's iteration method as obtained by Galerkin and orthogonal collocation methods.

Algorithms for the orientation of a moving object with separation of the integration of fast and slow motions

Abstract Equations and algorithms for determining the orientation of a moving object in inertial and normal geographic coordinate systems are considered with separation of the integration of the fast and slow motions into ultrafast, fast and slow cycles. Ultrafast cycle algorithms are constructed using a Riccati-type kinematic quaternion equation and the Picard method of successive approximations, and the increments in the integrals of the projections of the absolute angular velocity vector of the object onto the coordinate axes (quasicoordinates) associated with them are used as input information. The fast cycle algorithm realizes the calculation of the classical rotation quaternion of an object on a step of the fast cycle in an inertial system of coordinates. The slow cycle algorithm is used in calculating the orientation quaternion of an object in the normal geographic coordinate system and aircraft angles. Results of modelling different versions of the fast and ultrafast cycle algorithms for calculating the inertial orientation of an object are presented and discussed. The experience of the authors in developing algorithms for determining the orientation of moving objects using a strapdown inertial navigation system is described and results obtained by them earlier in this field are developed and extended.