Semi-implicit Numerical Scheme Research Articles

Comparative analysis of three memory selection methods for time integration of Fractional Reaction-Diffusion Equations

By discretising in space, a non-linear time fractional reaction-diffusion equations (TFRDEs) can be converted into a system of time-fractional differential equations (TFDEs). The full memory method (FMM) and short memory method (SMM) are well-established memory selection methods used in the time integration of TFDEs. The main drawbacks of FMM and SMM are higher computational cost and uncontrollable error respectively. The only way to increase the accuracy of SMM is by increasing short memory length which causes an increase in computational cost. Especially when we apply these two methods to integrate TFRDEs, we have to solve a large system of TFDEs. Therefore, the drawbacks of these two methods affect seriously, when these are applied to solve TFRDEs. This paper aims to investigate the accuracy and efficiency of the memory selection method, Exponentially Decreasing Random Memory Method (EDRMM), and compare it with FMM and SMM when these methods apply to integrate TFRDEs. Based on these three memory selection methods, three semi-implicit numerical schemes namely semiimplicit scheme with full memory method (SI-FMM), semi-implicit scheme with short memory method (SI-SMM), and semi-implicit scheme with exponentially decreasing random memory method (SI-EDRMM)) are proposed and the accuracy and CPU time (computational time (CT)) of these three numerical schemes are compared. To do this comparison, these three numerical schemes are applied to four TFRDEs whose exact solutions are known. Numerical experiments confirm that the accuracy and efficiency of the SI-EDRMM are better than that of SI-SMM and the efficiency of SI-EDRMM is higher than that of SI-FMM. Therefore, EDRMM is better than SMM and FMM for the integration of TFRDEs.

Asymptotically preserving particle methods for strongly magnetized plasmas in a torus

We propose and analyze a class of particle methods for the Vlasov equation with a strong external magnetic field in a torus configuration. In this regime, the time step can be subject to stability constraints related to the smallness of Larmor radius. To avoid this limitation, our approach is based on higher-order semi-implicit numerical schemes already validated on dissipative systems [3] and for magnetic fields pointing in a fixed direction [10,11,13]. It hinges on asymptotic insights gained in [12] at the continuous level. Thus, when the magnitude of the external magnetic field is large, this scheme provides a consistent approximation of the guiding-center system taking into account curvature and variation of the magnetic field. Finally, we carry out a theoretical proof of consistency and perform several numerical experiments that establish a solid validation of the method and its underlying concepts.

Computational-Analysis of the Non-Isothermal Dynamics of the Gravity-Driven Flow of Viscoelastic-Fluid-Based Nanofluids Down an Inclined Plane

The paper explores the gravity-driven flow of the thin film of a viscoelastic-fluid-based nanofluids (VFBN) along an inclined plane under non-isothermal conditions and subjected to convective cooling at the free-surface. The Newton’s law of cooling is used to model the convective heat-exchange with the ambient at the free-surface. The Giesekus viscoelastic constitutive model, with appropriate modifications to account for non-isothermal effects, is employed to describe the polymeric effects. The unsteady and coupled non-linear partial differential equations (PDEs) describing the model problem are obtained and solved via efficient semi-implicit numerical schemes based on finite difference methods (FDM) implemented in Matlab. The response of the VFBN velocity, temperature, thermal-conductivity and polymeric-stresses to variations in the volume-fraction of embedded nanoparticles is investigated. It is shown that these quantities all increase as the nanoparticle volume-fraction becomes higher.

Chemical Potential-Based Modeling of Shale Gas Transport

Shale gas plays an increasingly important role in the current energy industry. Modeling of gas flow in shale media has become a crucial and useful tool to estimate shale gas production accurately. The second law of thermodynamics provides a theoretical criterion to justify any promising model, but it has been never fully considered in the existing models of shale gas. In this paper, a new mathematical model of gas flow in shale formations is proposed, which uses gas density instead of pressure as the primary variable. A distinctive feature of the model is to employ chemical potential gradient rather than pressure gradient as the primary driving force. This allows to prove that the proposed model obeys an energy dissipation law, and thus, the second law of thermodynamics is satisfied. Moreover, on the basis of energy factorization approach for the Helmholtz free energy density, an efficient, linear, energy stable semi-implicit numerical scheme is proposed for the proposed model. Numerical experiments are also performed to validate the model and numerical method.

Stability of the Semi-Implicit Method for the Cahn-Hilliard Equation with Logarithmic Potentials

We consider the two-dimensional Cahn-Hilliard equation with logarithmic potentials and periodic boundary conditions. We employ the standard semi-implicit numerical scheme which treats the linear fourth-order dissipation term implicitly and the nonlinear term explicitly. Under natural constraints on the time step we prove strict phase separation and energy stability of the semi-implicit scheme. This appears to be the first rigorous result for the semi-implicit discretization of the Cahn-Hilliard equation with singular potentials.

Adaptive Euler methods for stochastic systems with non-globally Lipschitz coefficients

We present strongly convergent explicit and semi-implicit adaptive numerical schemes for systems of semi-linear stochastic differential equations (SDEs) where both the drift and diffusion are not globally Lipschitz continuous. Numerical instability may arise either from the stiffness of the linear operator or from the perturbation of the nonlinear drift under discretization, or both. Typical applications arise from the space discretization of an SPDE, stochastic volatility models in finance, or certain ecological models. Under conditions that include montonicity, we prove that a timestepping strategy which adapts the stepsize based on the drift alone is sufficient to control growth and to obtain strong convergence with polynomial order. The order of strong convergence of our scheme is (1 − ε)/2, for ε ∈ (0,1), where ε becomes arbitrarily small as the number of finite moments available for solutions of the SDE increases. Numerically, we compare the adaptive semi-implicit method to a fully drift-implicit method and to three other explicit methods. Our numerical results show that overall the adaptive semi-implicit method is robust, efficient, and well suited as a general purpose solver.

Stability of a simple scheme for the approximation of elastic knots and self-avoiding inextensible curves

We discuss a semi-implicit numerical scheme that allows for minimizing the bending energy of curves within certain isotopy classes. To this end we consider a weighted sum of the bending energy and the tangent-point functional. Based on estimates for the second derivative of the latter and a uniform bi-Lipschitz radius, we prove a stability result implying energy decay during the evolution as well as maintenance of arclength parametrization. Finally we present some numerical experiments exploring the energy landscape, targeted to the question how to obtain global minimizers of the bending energy in knot classes, so-called elastic knots.

Fast implicit active contour model for inverse lithography.

We combine the ideas from level-set methods in computer vision and inverse imaging to derive a generalized active contour model for inverse lithography problems endowed with a locally implemented semi-implicit difference scheme. We introduce a cognitive analogy to move an initial guess of the interesting pattern contour by image-driven forces to the boundaries of the desired layout pattern. We develop an efficient semi-implicit numerical scheme implemented in the vicinity of the zero level-set and apply additive operator splitting (AOS) with respect to coordinate axes to solve consecutive one-dimensional linear systems of equations with the Thomas method. We demonstrate with simulation results that computation and convergence efficiency are jointly improved with reduced optimization dimensionality and a sufficient large step-size.

Convergence analysis of Fourier pseudo-spectral schemes for three-dimensional incompressible Navier-Stokes equations

<p style='text-indent:20px;'>The stability and convergence of the Fourier pseudo-spectral method are analyzed for the three dimensional incompressible Navier-Stokes equation, coupled with a variety of time-stepping methods, of up to fourth order temporal accuracy. An aliasing error control technique is applied in the error estimate for the nonlinear convection term, while an a-priori assumption for the numerical solution at the previous time steps will also play an important role in the analysis. In addition, a few multi-step temporal discretization is applied to achieve higher order temporal accuracy, while the numerical stability is preserved. These semi-implicit numerical schemes use a combination of explicit Adams-Bashforth extrapolation for the nonlinear convection term, as well as the pressure gradient term, and implicit Adams-Moulton interpolation for the viscous diffusion term, up to the fourth order accuracy in time. Optimal rate convergence analysis and error estimates are established in details. It is proved that, the Fourier pseudo-spectral method coupled with the carefully designed time-discretization is stable provided only that the time-step and spatial grid-size are bounded by two constants over a finite time. Some numerical results are also presented to verify the established convergence rates of the proposed schemes.</p>

Damping, stabilization, and numerical filtering for the modeling and the simulation of time dependent PDEs

<p style='text-indent:20px;'>We present here different situations in which the filtering of high or low modes is used either for stabilizing semi-implicit numerical schemes when solving nonlinear parabolic equations, or for building adapted damping operators in the case of dispersive equation. We consider numerical filtering provided by mutigrid-like techniques as well as the filtering resulting from operator with monotone symbols. Our approach applies to several discretization techniques and we focus on finite elements and finite differences. Numerical illustrations are given on Cahn-Hilliard, Korteweig-de Vries and Kuramoto-Sivashinsky equations.

Three-Dimensional Unstructured Grid Finite-Volume Model for Coastal and Estuarine Circulation and Its Application

We developed a three-dimensional unstructured grid coastal and estuarine circulation model, named the General Ocean Model (GOM). Combining the finite volume and finite difference methods, GOM achieved both the exact conservation and computational efficiency. The propagation term was implemented by a semi-implicit numerical scheme, the so-called θ scheme, and the time-explicit Eulerian–Lagrangian method was used to discretize the nonlinear advection term to remove the major limitation of the time step, which appears when solving shallow water equations, by the Courant–Friedrichs–Lewy stability condition. Because the GOM uses orthogonal unstructured computational grids, allowing both triangular and quadrilateral grids, considerable flexibility to resolve complex coastal boundaries is allowed without any transformation of governing equations. The GOM was successfully verified with five analytical solutions, and it was also validated when applied to the Texas coast, showing an overall skill value of 0.951. The verification results showed that the algorithm used in GOM was correctly coded, and it is efficient and robust.

Convergence analysis for a stabilized linear semi-implicit numerical scheme for the nonlocal Cahn–Hilliard equation

In this paper, we provide a detailed convergence analysis for a first order stabilized linear semi-implicit numerical scheme for the nonlocal Cahn–Hilliard equation, which follows from consistency and stability estimates for the numerical error function. Due to the complicated form of the nonlinear term, we adopt the discrete H − 1 H^{-1} norm for the error function to establish the convergence result. In addition, the energy stability obtained by Du et al., [J. Comput. Phys. 363 (2018), pp. 39–54] requires an assumption on the uniform ℓ ∞ \ell ^\infty bound of the numerical solution, and such a bound is figured out in this paper by conducting the higher order consistency analysis. Taking the view that the numerical solution is indeed the exact solution with a perturbation, the error function is ℓ ∞ \ell ^\infty bounded uniformly under a loose constraint of the time step size, which then leads to the uniform maximum-norm bound of the numerical solution.

A New Mass-Conservative, Two-Dimensional, Semi-Implicit Numerical Scheme for the Solution of the Navier-Stokes Equations in Gravel Bed Rivers with Erodible Fine Sediments

The selective trapping and erosion of fine particles that occur in a gravel bed river have important consequences for its stream ecology, water quality, and overall sediment budgeting. This is particularly relevant in water bodies that experience periodic alternation between sediment supply-limited conditions and high sediment loads, such as downstream from a dam. While experimental efforts have been spent to investigate fine sediment erosion and transport in gravel bed rivers, a comprehensive overview of the leading processes is hampered by the difficulties in performing flow field measurements below the gravel crest level. In this work, a new two-dimensional, semi-implicit numerical scheme for the solution of the Navier-Stokes equations in the presence of deposited and erodible sediment is presented, and tested against analytical solutions and performing numerical tests. The scheme is mass-conservative, computationally efficient, and allows for a fine discretization of the computational domain. Overall, this makes the model suitable to appreciate small-scales phenomena such as inter-grain circulation cells, thus offering a valid alternative to evaluate the shear stress distribution, on which erosion and transport processes depend, compared to traditional experimental approaches. In this work, we present proof-of-concept of the proposed model, while future research will focus on its extension to a three-dimensional and parallelized version, and on its application to real case studies.

Implicit and Semi-implicit Numerical Schemes for the Gradient Flow of the Formation of Biological Transport Networks

Implicit and semi-implicit time discretizations are developed for the Cai–Hu model describing the formation of biological transport networks. The model couples a nonlinear elliptic equation for the pressure with a nonlinear reaction-diffusion equation for the network conductance vector. Numerical challenges include the nonlinearity and the stiffness, thus an explicit discretization puts severe constraints on the time step. We propose an implicit and a semi-implicit discretizations, which decays the energy unconditionally or under a condition independent of the mesh size respectively, as will be proven in 1D and verified numerically in 2D.

A Novel Energy Factorization Approach for the Diffuse-Interface Model with Peng--Robinson Equation of State

The Peng--Robinson equation of state (PR-EoS) has become one of the most extensively applied equations of state in chemical engineering and the petroleum industry due to its excellent accuracy in predicting the thermodynamic properties of a wide variety of materials, especially hydrocarbons. Although great effort has been made to construct efficient numerical methods for the diffuse interface models with PR-EoS, there is still not a linear numerical scheme that can be proved to preserve the original energy dissipation law. In order to pursue such a numerical scheme, we propose a novel energy factorization (EF) approach, which first factorizes an energy function into a product of several factors and then treats the factors using their properties to obtain the semi-implicit linear schemes. We apply the EF approach to deal with the Helmholtz free energy density determined by PR-EoS, and then propose a linear semi-implicit numerical scheme that inherits the original energy dissipation law. Moreover, the proposed scheme is proved to satisfy the maximum principle in both time semidiscrete form and cell-centered finite difference fully discrete form under certain conditions. Numerical results are presented to demonstrate the stability and efficiency of the proposed scheme.

Improvement of MARS-KS sub-channel module using CTF code

In this work, the sub-channel analysis module of MARS-KS nuclear safety analysis code was upgraded using CTF sub-channel analysis code for realistic prediction of the multi-component and multi-dimensional nuclear core thermal hydraulics. The semi-implicit pressure coupling method was used to couple these two codes. Both codes use the finite difference method and semi-implicit numerical scheme and hence they were coupled together by exchanging volume/cell pressures through the boundary interfaces between 1D (i.e., system) and 3D (i.e., sub-channel) computational regions via implicit phasic velocities at the boundary. The pressure linkage was simply accomplished by designating the 1D and 3D regions as implicit sinks in the CTF sub-channel module and time dependent volumes in the MARS-KS code, respectively. The heat structure component and point kinetics models of the MARS-KS code were also coupled with the CTF core sub-channel module for the analysis of the whole reactor system. Automatic initialization of the 3D vessel was performed with CTF null-transient calculation to initialize all computational cells using a single set of initial and boundary conditions. The improved MARS-KS code has been verified and validated using various problems and tests. This paper selectively presents the results of three analyses, including a conceptual manometer oscillation, a blowdown test, and a 3 × 3 hot rod sub-channel experiments, with the improved code. All the analyses showed good predictions and this confirms the semi-implicit pressure coupling as well as the capability of the improved MARS-KS code. In the future, parallel processing should be implemented for the improved MARS-KS core sub-channel module to perform full core sub-channel analysis.

A continuum approach for consolidation modeling in composites processing

During composite processing, consolidation refers to the application of pressure and/or temperature cycle to the fiber preforms and the uncured liquid resin to increase the fiber volume fraction. The goal is to eliminate voids before the resin solidifies. Consolidation process involves two stages. In the first stage voids are filled and reinforcement compressed due to the application of pressure. Elevated temperatures facilitate this by reducing the viscosity of the resin. In the second stage, the resin cures and the structure solidifies.This paper describes the modeling approach for the first stage of the consolidation process. Governing equations are presented for reinforcement deformation, resin redistribution and porosity evaluation and a semi-implicit numerical scheme well suited to handle complex material models is formulated to track the evolution of part properties during the consolidation process. The applicability of the model is demonstrated with the help of two diverse examples: (1) Consolidation of prepreg facesheets during co-cure of a sandwich panel and (2) Corner consolidation during autoclave processing of a prepreg-based panel. The effects of geometry (such as radius of curvature) and materials (such as initial porosity) are discussed.

Accelerated Variational PDEs for Efficient Solution of Regularized Inversion Problems.

We further develop a new framework, called PDE acceleration, by applying it to calculus of variation problems defined for general functions on , obtaining efficient numerical algorithms to solve the resulting class of optimization problems based on simple discretizations of their corresponding accelerated PDEs. While the resulting family of PDEs and numerical schemes are quite general, we give special attention to their application for regularized inversion problems, with particular illustrative examples on some popular image processing applications. The method is a generalization of momentum, or accelerated, gradient descent to the PDE setting. For elliptic problems, the descent equations are a nonlinear damped wave equation, instead of a diffusion equation, and the acceleration is realized as an improvement in the CFL condition from Δt ~ Δx 2 (for diffusion) to Δt ~ Δx (for wave equations). We work out several explicit as well as a semi-implicit numerical scheme, together with their necessary stability constraints, and include recursive update formulations which allow minimal-effort adaptation of existing gradient descent PDE codes into the accelerated PDE framework. We explore these schemes more carefully for a broad class of regularized inversion applications, with special attention to quadratic, Beltrami, and total variation regularization, where the accelerated PDE takes the form of a nonlinear wave equation. Experimental examples demonstrate the application of these schemes for image denoising, deblurring, and inpainting, including comparisons against primal-dual, split Bregman, and ADMM algorithms.

Phase equilibrium calculations in shale gas reservoirs

Compositional multiphase flow in subsurface porous media is becoming increasingly attractive due to issues related with enhanced oil recovery, CO 2 sequestration and the urgent need for development in unconventional oil/gas reservoirs. One key effort to construct the mathematical model governing the compositional flow is to determine the phase compositions of the fluid mixture, and then calculate other related physical properties. In this paper, recent progress on phase equilibrium calculations in unconventional reservoirs has been reviewed and concluded with authors’ own analysis, especially focusing on the special mechanisms involved. Phase equilibrium calculation is the main approach to investigate phase behaviors, which could be conducted using different variable specifications, such as the NPT flash and NVT flash. Recently, diffuse interface models, which have been proved to possess a high consistency with thermodynamic laws, have been introduced in the phase equilibrium calculation, incorporating the realistic equation of state (EOS), e.g. Peng-Robinson EOS. In the NVT flash, the Helmholtz free energy is minimized instead of the Gibbs free energy used in NPT flash, and this thermodynamic state function is decomposed into two terms using the convex-concave splitting technique. A semi-implicit numerical scheme is applied to the dynamic model, which ensures the thermodynamic stability and then preserves the fast convergence property. A positive definite coefficient matrix is designed to meet the Onsager reciprocal principle so as to keep the entropy increasing property in the presence of capillary pressure, which is required by the second law of thermodynamics. The robustness of the proposed algorithm is demonstrated by using two numerical examples, one of which has up to seven components. In the complex fluid mixture, special phenomena could be captured from the global minimum of tangent plane distance functions and the phase envelope. It can be found that the boundary between the single-phase and vapor-liquid two phase regions shifts in the presence of capillary pressure, and then the area of each region changes accordingly. Furthermore, the effect of the nanopore size distribution on the phase behavior has been analyzed and a multi-scale scheme is presented based on literature reviews. Fluid properties including swelling factor, criticality, bubble point and volumetrics have been investigated thoroughly by comparing with the bulk fluid flow in a free channel. Cited as : Zhang, T., Li, Y., Sun, S. Phase equilibrium calculations in shale gas reservoirs. Capillarity, 2019, 2(1): 8-16, doi: 10.26804/capi.2019.01.02

Correcting the Semi-Implicit Numerical Scheme Incorporated in the KORSAR Code Two-Liquid Model

Two algorithms used to correct the semi-implicit numerical scheme for time integration of the conservation equations incorporated into the KORSAR/GP computer code two-liquid model are presented. The KORSAR/GP code has been developed jointly by specialists of the Federal State Unitary Enterprise Aleksandrov Research Institute of Technology and Gidropress Experimental Design Office; in 2009, the code was certified at the Federal Service for Environmental, Technological, and Nuclear Supervision (Rostekhnadzor) as applied to numerical safety assessment of VVER-based reactor plants. In the semi-implicit scheme, the convective terms appearing in the phase momentum conservation equations are written in an explicit form. The mass and energy flows of the phases are represented implicitly with respect to phase velocities, and the transferred donor quantities are calculated based on the parameters taken from the previous time layer. Owing to linearization of unsteady and source terms, the linear system of finite difference equations is solved in a noniteration manner. The first of the presented algorithms compensates for numerical imbalances of phase masses and energies resulting from linearization of unsteady terms appearing in the discrete equations and ensures conservativeness of the scheme. The second algorithm introduces correction for the nonphysical redistribution of coolant mass and energy over the computational cells if a change occurs in the phase motion direction within a time step when the scheme of approximating the convective terms becomes “antidonor” with respect to flow. The numerical imbalances and refinements taking into account the correction of donor quantities are computed at each time step according to the proposed correlations and are used for making compensation at the next time step as supplementary sources in the conservation equations. Results from testing the correction algorithms in the KORSAR/GP code are presented. The effectiveness of the imbalance compensation algorithm is confirmed by solving a problem involving the natural circulation loop heating process. The adequacy of the “antidonor” scheme correction algorithm is demonstrated on problems involving steplike initial distribution of the scalar parameters of a stagnant single-phase gaseous or water coolant in a horizontal tube when the flowrate at the tube inlet has different signs at different time steps.