Picard Iteration Scheme Research Articles

Enriched Galerkin discretization for modeling poroelasticity and permeability alteration in heterogeneous porous media

In this paper, we utilize the enriched Galerkin (EG) finite element method for the flow equation in Biot's system, which provides a robust locally conservative flux in heterogeneous porous media. The computational algorithm to solve the coupled system with the permeability alteration is presented with the linearization and Picard's iterative scheme. The block structure is utilized for the linear system in numerical discretization, and the computer code is shared in the open-source platform. In the numerical experiments, we compare the proposed EG method with the classical continuous Galerkin (CG) and discontinuous Galerkin (DG) finite element methods in different scenarios, including the North sea reservoirs setup. While DG and EG methods provide similar approximations for the pressure solutions, the CG method produces spurious oscillations in fluid pressure and volumetric strain solutions near the material interfaces, especially for the soft materials. The difference of flux approximation between EG and DG methods is insignificant; still, the EG method demands approximately two and three times fewer degrees of freedom than the DG method for two- and three-dimensional geometries.

Oscillatory states and patterns formation in a two-cell cubic autocatalytic reaction-diffusion model subjected to the Dirichlet conditions

<p style='text-indent:20px;'>The approximate solutions of a two-cell reaction-diffusion model equation subjected to the Dirichlet conditions are obtained. The reaction is assumed to occur in the presence of cubic autocatalyst which decays to an inert compound in the first cell. Coupling with the reactant is assumed to be cubic in the concentrations. A linear exchange in the concentration of the reactant is taken between the two cells. The formal exact solution is found analytically. Here, in this investigation, use is made of the Picard iterative scheme which is constructed and applied after the exact one. The results obtained are compared with those found by means of a numerical method. It is observed that the solution obtained here is symmetric with respect to the mid-point of the container.The travelling wave is expected due to the parity of the space operator and the symmetric boundary conditions. Symmetric patterns, including among them a parabolic one, are observed for a large time.</p><p style='text-indent:20px;'>When the initial conditions are periodic, the most dominant modes travel at a constant speed for a large time. This phenomenon is highly affected by the rate of decay of the autocatalyst to an inert compound. The present work is of remarkably significant interest in chemical engineering as well as in other physical sciences. For example, in chemical industry, the objective is to achieve a great yield of a given product, which is carried by controlling the initial concentration of the reactant. Furthermore, in the last section on conclusions, we have cited many potentially useful recent works related to the subject-matter of this investigation in order to provide incentive and motivation for making further advances by using space-time fractional derivatives along the lines of the problem of finding approximate analytical solutions of the reaction-diffusion model equations which we have discussed in this article.</p>

Existence and uniqueness for the mild solution of the stochastic heat equation with non-Lipschitz drift on an unbounded spatial domain

We prove the existence and uniqueness of the mild solution for a nonlinear stochastic heat equation defined on an unbounded spatial domain. The nonlinearity is not assumed to be globally, or even locally, Lipschitz continuous. Instead the nonlinearity is assumed to satisfy a one-sided Lipschitz condition. First, a strengthened version of the Kolmogorov continuity theorem is introduced to prove that the stochastic convolutions of the fundamental solution of the heat equation and a spatially homogeneous noise grow no faster than polynomially. Second, a deterministic mapping that maps the stochastic convolution to the solution of the stochastic heat equation is proven to be Lipschitz continuous on polynomially weighted spaces of continuous functions. These two ingredients enable the formulation of a Picard iteration scheme to prove the existence and uniqueness of the mild solution.

A robust scheme based on novel‐operational matrices for some classes of time‐fractional nonlinear problems arising in mechanics and mathematical physics

AbstractIn this paper, we present a novel approach based on shifted Gegenbauer wavelets to attain approximate solutions of some classed of time‐fractional nonlinear problems. First, we present the approximation of a function of two variablesu(x,t) with help of shifted Gegenbauer wavelets and then some novel operational matrices are proposed with the help of piecewise functions to investigate the positive integer derivative (DxandDt), fractional‐order derivative (and), fractional‐order integration (and) and delay terms (and) of approximated functionu(x,t). In order to transform the discussed nonlinear problem into linear problem Picard iterative scheme has been adopt. The current scheme converts the discussed highly nonlinear time‐fractional problem into system of linear algebraic equation the help of developed operational matrices and Picard idea. Analysis on the error bound and convergence to authenticate the mathematical formulation of the computational algorithm. We solve various test problems, such as the van der Pol oscillator model, generalized Burger–Huxley, neutral delay parabolic differential equations, sine‐Gordon, parabolic integro‐differential equation and nonlinear Schrödinger equations to show the efficiency and accuracy of linearized shifted Gegenbauer wavelets method. A comprehensive comparative examination shows the credibility, accuracy, and reliability of the presently proposed computational approach. Also, this scheme can be extended conveniently to other multi‐dimensional physical problems of highly nonlinear fractional or variable order of complex nature.

A Fourier-based Picard-iteration approach for a class of McKean–Vlasov SDEs with Lévy jumps

We consider a prototype class of Lévy-driven stochastic differential equations (SDEs) with McKean–Vlasov (MK–V) interaction in the drift coefficient. It is assumed that the drift coefficient is affine in the state variable, and only measurable in the law of the solution. We study the equivalent functional fixed-point equation for the unknown time-dependent coefficients of the associated linear Markovian SDE. By proving a contraction property for the functional map in a suitable normed space, we infer existence and uniqueness results for the MK–V SDE, and derive a discretized Picard iteration scheme that approximates the law of the solution through its characteristic function. Numerical illustrations show the effectiveness of our method, which appears to be appropriate to handle the multi-dimensional setting.

A radiation hydrodynamics scheme on adaptive meshes using the Variable Eddington Tensor (VET) closure

AbstractWe present a new algorithm to solve the equations of radiation hydrodynamics (RHD) in a frequency-integrated, two-moment formulation. Novel features of the algorithm include i) the adoption of a non-local Variable Eddington Tensor (VET) closure for the radiation moment equations, computed with a ray-tracing method, ii) support for adaptive mesh refinement (AMR), iii) use of a time-implicit Godunov method for the hyperbolic transport of radiation, and iv) a fixed-point Picard iteration scheme to accurately handle the stiff nonlinear gas-radiation energy exchange. Tests demonstrate that our scheme works correctly, yields accurate rates of energy and momentum transfer between gas and radiation, and obtains the correct radiation field distribution even in situations where more commonly used – but less accurate – closure relations like the Flux-limited Diffusion and Moment-1 approximations fail. Our scheme presents an important step towards performing RHD simulations with increasing spatial and directional accuracy, effectively improving their predictive capabilities.

A Multilevel in Space and Energy Solver for 3-D Multigroup Diffusion and Coarse-Mesh Finite Difference Eigenvalue Problems

The Multilevel in Space and Energy Diffusion (MSED) method accelerates the iterative convergence of multigroup diffusion eigenvalue problems by performing work on lower-order equations with only one group and/or coarser spatial grids. It consists of two primary components: (1) a grey (one-group) diffusion eigenvalue problem that is solved via Wielandt-shifted power iteration (PI) and (2) a multigrid-in-space linear solver. In previous work, the efficiency of MSED was verified using Fourier analysis and numerical results from a one-dimensional multigroup diffusion code. Since that work, MSED has been implemented as a solver for the coarse-mesh finite difference (CMFD) system in the three-dimensional Michigan Parallel Characteristics Transport (MPACT) code. In this paper, the results from the implementation of MSED in MPACT are presented, and the changes needed to make MSED more suitable for MPACT are described. For problems without feedback, the results in this paper show that MSED can reduce the CMFD run time by an order of magnitude and the overall run time by a factor of 2 to 3 compared to the default CMFD solver in MPACT [PI with the generalized minimal residual (GMRES) method]. For problems with feedback, the convergence of the outer Picard iteration scheme is worsened by the well-converged CMFD solutions produced by the standard MSED method. To overcome this unintuitive deficiency, MSED may be run with looser convergence criteria (a modified version of the MSED method called MSED-L) to circumvent the issue until the multiphysics iteration in MPACT is improved. Results show that MSED-L can reduce the CMFD run time in MPACT by an order of magnitude, without negatively impacting the outer Picard iteration scheme.

Coupling brain-tumor biophysical models and diffeomorphic image registration

We present SIBIA (Scalable Integrated Biophysics-based Image Analysis), a framework for joint image registration and biophysical inversion and we apply it to analyze MR images of glioblastomas (primary brain tumors). We have two applications in mind. The first one is normal-to-abnormal image registration in the presence of tumor-induced topology differences. The second one is biophysical inversion based on single-time patient data. The underlying optimization problem is highly non-linear and non-convex and has not been solved before with a gradient-based approach.Given the segmentation of a normal brain MRI and the segmentation of a cancer patient MRI, we determine tumor growth parameters and a registration map so that if we “grow a tumor” (using our tumor model) in the normal brain and then register it to the patient image, then the registration mismatch is as small as possible. This “coupled problem” two-way couples the biophysical inversion and the registration problem. In the image registration step we solve a large-deformation diffeomorphic registration problem parameterized by an Eulerian velocity field. In the biophysical inversion step we estimate parameters in a reaction–diffusion tumor growth model that is formulated as a partial differential equation (PDE). In SIBIA, we couple these two sub-components in an iterative manner. We first presented the components of SIBIA in “Gholami et al., Framework for Scalable Biophysics-based Image Analysis, IEEE/ACM Proceedings of the SC2017”, in which we derived parallel distributed memory algorithms and software modules for the decoupled registration and biophysical inverse problems. In this paper, our contributions are the introduction of a PDE-constrained optimization formulation of the coupled problem, and the derivation of a Picard iterative solution scheme. We perform extensive tests to experimentally assess the performance of our method on synthetic and clinical datasets. We demonstrate the convergence of the SIBIA optimization solver in different usage scenarios. We demonstrate that using SIBIA, we can accurately solve the coupled problem in three dimensions (2563 resolution) in a few minutes using 11 dual-x86 nodes.

Numerical approximation of general Lipschitz BSDEs with branching processes

We extend the branching process based numerical algorithm of Bouchard et al. [3], that is dedicated to semilinear PDEs (or BSDEs) with Lipschitz nonlinearity, to the case where the nonlinearity involves the gradient of the solution. As in [3], this requires a localization procedure that uses a priori estimates on the true solution, so as to ensure the well-posedness of the involved Picard iteration scheme, and the global convergence of the algorithm. When, the nonlinearity depends on the gradient, the later needs to be controlled as well. This is done by using a face-lifting procedure. Convergence of our algorithm is proved without any limitation on the time horizon. We also provide numerical simulations to illustrate the performance of the algorithm.

The nonlinear Schrödinger equation on the half-line with Neumann boundary conditions

The initial-boundary value problem on the half-line for the nonlinear Schrödinger equation supplemented with a Neumann boundary condition is shown to be locally well-posed in the sense of Hadamard for data in Sobolev spaces Hs. The proof takes advantage of a novel explicit solution formula for the forced linear counterpart of the nonlinear problem, which is obtained via Fokas' unified transform method. This formula provides the basis for setting up a Picard iteration scheme and for deriving the linear estimates required for proving local well-posedness of the nonlinear problem via a contraction mapping argument.

Existence of a regular solution for 1D Green\u2013Naghdi equations with surface tension at a large time instant

In this paper the model 1D-GNσ is considered, which concerns the 1D Green–Naghdi equations with non-flat bottom and under the influence of surface tension, to be widely used in coastal oceanography to describe the propagation of large-wave amplitudes. The purpose of this paper is to show that the solution of 1D-GNσ can be made by the Picard iterative scheme, which proves that there is no loss of regularity of the solution relative to the initial condition.

Meshless local Petrov Galerkin method for 2D/3D nonlinear convection–diffusion equations based on LS-RBF-PUM

In this article, we propose a meshless local Petrov Galerkin (MLPG) method based on least square radial basis function partition of unity method (LS-RBF-PUM), which is applied to the nonlinear convection–diffusion equations. The proposed method is not sensitive to the node layout, and has good stability and flexibility to complex domain. In order to treat nonlinear term, Picard iterative scheme is employed to confirm the convergence of iterative process. Error estimates are derived by the radial basis function interpolation method and convergence rate is proven to be second order. Numerical examples are performed for the nonlinear convection–diffusion equations in two and three space dimensions (2D/3D), which not only supports the theoretical results but also finds out superconvergence of third order.

Derivation and well-posedness of the extended Green-Naghdi equations for flat bottoms with surface tension

In this paper, we will derive the two-dimensional extended Green-Naghdi system {see Matsuno [Proc. R. Soc. A 472, 20160127 (2016)] for determination in a various way} for flat bottoms of order three with respect to the shallowness parameter μ. Then we consider the 1D extended Green-Naghdi equations taking into account the effect of small surface tension. We show that the construction of solution with a standard Picard iterative scheme can be accomplished in which the well-posedness in Xs=Hs+2(R)×Hs+2(R) for some s>32 of the new extended 1D system for a finite large time existence t=O(1ε) is demonstrated.

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.

On the application of the method of fundamental solutions to nonlinear partial differential equations

This paper addresses the application of a domain-type method of fundamental solutions (MFS-D) together with a Picard iteration scheme for solving nonlinear elliptic partial differential equations. A mathematical justification of the iterative process and an a posteriori error estimate are provided for a class of nonlinear problems. Numerical simulations illustrate the convergence and accuracy of the method, in several examples, including a case for a sine-Gordon equation.

Numerical approximation of BSDEs using local polynomial drivers and branching processes

Abstract We propose a new numerical scheme for Backward Stochastic Differential Equations (BSDEs) based on branching processes. We approximate an arbitrary (Lipschitz) driver by local polynomials and then use a Picard iteration scheme. Each step of the Picard iteration can be solved by using a representation in terms of branching diffusion systems, thus avoiding the need for a fine time discretization. In contrast to the previous literature on the numerical resolution of BSDEs based on branching processes, we prove the convergence of our numerical scheme without limitation on the time horizon. Numerical simulations are provided to illustrate the performance of the algorithm.

Well-posedness of the stochastic neural field equation with discontinuous firing rate

We study the existence and uniqueness of mild solutions to the deterministic and the stochastic neural field equation with Heaviside firing rate. Since standard well-posedness results do not apply in case of a discontinuous firing rate, we present a monotone Picard iteration scheme to show the existence of a maximal mild solution. Further, we illustrate that general uniqueness does not hold, and therefore investigate uniqueness under suitable additional properties of the solutions. Here a novel criterion, the so-called absolute continuity condition is introduced. Moreover, we observe regularisation by noise: with a suitable choice of spatially correlated additive noise uniqueness is restored without imposing any additional structural assumptions.

Simulation of seepage flow of non-aqueous phase liquid in vadose zone

A comprehensive numerical method for simulating transport of non-aqueous phase liquids (NAPLs) in unsaturated subsurface was developed based on a novel capillary pressure-saturation model for three-phase water-NAPL-air in soils. In the method, weak forms of the mass balance equations for water, NAPL and air were discretised in space using the Galerkin method and were discretised in time using the implicit backward Euler method. A modified Picard iteration scheme was implemented to deal with the non-linearity of the governing equations and to keep the mass balance of three fluids. The method properly takes into account the interactions among three-phase fluids in soils and is capable of describing the capillary behaviour of NAPL in unsaturated soils. The methodology developed was applied to some initial value problems of NAPL leakage into the vadose zone. A one-dimensional leakage simulation of NAPL that is less dense than water (LNAPL) validated that the proposed method predicts the realistic penetration behaviour of NAPLs into an unsaturated subsurface. Furthermore, two-dimensional simulations on leakage of LNAPL and NAPLs that are denser than water (DNAPL) into an unsaturated ground containing impermeable layers confirmed that the proposed analysis method is capable of representing the contamination process in shallow ground, including differences in the penetration mechanisms of LNAPL and DNAPL.

A modified Picard iteration scheme for overcoming numerical difficulties of simulating infiltration into dry soil

Numerical models based on Richards’ equation are often employed to simulate the soil water dynamics. Among them, those Picard iteration models which use the head as primary variable are widely adopted due to their simplicity and capability for handling partially saturated flow conditions. However, it is well-known that those models are prone to convergence failure in some unfavorable flow conditions, especially when simulating infiltration into initially dry soils. Here we analyze the reasons that give rise to the numerical difficulty. Moreover, several modifications to the mass-conservative Picard iteration method are proposed so that numerical difficulty is avoided in these unfavorable flow conditions. Our proposed modifications do not degrade the simulated results, while they lead to more robust convergence performances and cost-effective simulations.

Batch Solvent Extraction of Bitumen from Oil Sand. Part 1: Theoretical Basis

A model for flow through a porous medium with variable saturation is presented in this work to predict the volume and flow rate of extracted solvent-diluted bitumen from mineral solids during centrifugal filtration. The numerical method used for solving the flow equation is based on the mass-conservative-modified Picard iteration scheme and the Anderson acceleration algorithm. It is found that bitumen extraction involves filtration and desaturation phases, and the transition between them can be identified from the evolution with time of extracted liquid volume and flow rate. An analytical description of the filtration phase is developed to predict the evolution of the liquid layer thickness above the cake during centrifugation. Numerical and analytical models presented in this work can be used in a fitting procedure to determine hydraulic properties of oil sand cake. Effects of centrifugal force and solvent mass fraction on the predicted fraction of extracted diluted bitumen are investigated.