Explicit Discretization Research Articles

Linearly implicit variable step-size BDF schemes with Fourier pseudospectral approximation for incompressible Navier-Stokes equations

In this paper, linearly implicit backward differentiation formulas with variable step-sizes are proposed to solve numerically the two-dimensional incompressible Navier-Stokes equations (formulated in vorticity-stream function). With Fourier pseudospectral methods for spatial discretization, the diffusion term is discretized implicitly and the nonlinear convection term is treated by a combination of implicit and explicit discretizations. As a result, only linear solvers are needed at each time step to achieve the desired temporal accuracy. With the help of a priori assumption and aliasing error control techniques, the error estimates for one-step and two-step backward differentiation formulas are established in several norms under appropriate step-size constraints. Compared with the numerical results of implicit-explicit (the nonlinear convection term is treated explicitly) BDF2 method and fully implicit Crank-Nicolson method, it demonstrates that the proposed linearly implicit variable step-size BDF2 method is effective and robust.

RichardsFoam3: A new version of RichardsFoam for continental surfaces hydrogeology modelling

RichardsFoam3 is an updated version of the OpenFOAM® solver RichardsFoam, previously presented in “An open source massively parallel solver for Richards equation: Mechanistic modelling of water fluxes at the watershed scale” by L. Orgogozo, N. Renon, C. Soulaine, F. Hénon, S.K. Tomer, D. Labat, O.S. Pokrovsky, M. Sekhar, R. Ababou, M. Quintard (Comput. Phys. Commun. 185 (2014) 3358-3371, https://doi.org/10.1016/j.cpc.2014.08.004), and in the new version announcement “RichardsFOAM2: a new version of RichardsFOAM devoted to the modelling of the vadose zone” by L. Orgogozo (Comput. Phys. Commun. 196 (2015) 619-620, https://doi.org/10.1016/j.cpc.2015.07.009).This new version includes improvements of memory handling and of on-the-fly control of computations, a better integration in the OpenFOAM® framework, simplifications of the coding of some expressions, as well as new advanced boundary conditions. All together these developments allow to enhance the ease of application of the code to continental surfaces hydrogeology modelling, its computational performances and its readability.The description of the elements contained in this release may be found in the readMe file.Please note that you may also find RichardsFoam3 on the hydrology page of the develop.openfoam.com interface:https://develop.openfoam.com/Community/hydrology/ New version program summaryProgram title: RichardsFoam3CPC Library link to program files:https://doi.org/10.17632/vkr7sd6fhb.1Developer's repository link:https://develop.openfoam.com/Community/hydrology/Licensing provisions: GPLv3Programming language: C++Supplementary material: The full system of equations solved by RichardsFoam3 is presented in the documentation directory of the RichardsFoam3 package, with a numbering that is used in the comments of the main source file RichardsFoam3.C of the solver RichardsFoam3, in order to identify which lines of code are related to which equation.Journal reference of previous version: Comput. Phys. Commun. 196 (2015) 619-620Does the new version supersede the previous version?: YesReasons for the new version: Introducing new functionalities and implementing improvements for ergonomy, computational performances and readabilitySummary of revisions: The new features compared to RichardsFOAM [1] and RichardsFoam2 [2] are the following:(i) better handling of memory use during computation;(ii) possibility of on-the-fly modification of the control parameters of the linearisation procedure;(iii) new boundary conditions to be compiled along with the solver itself, and dedicated to the simulation of rain infiltration (rainFallFlux) or no infiltration (noFlux) conditions while allowing exfiltration with OpenFOAM® stand-alone (i.e. without mandatory use of swak4foam as it was the case in RichardsFoam2 [2]);(iv) Computation of the field of piezometric head along computation. NB: in the demonstration cases, a way to implement a constant piezometric head boundary conditions with OpenFOAM is proposed.(v) new functions that allow to build seasonally variable boundary conditions easily;(vi) use of postprocessing procedures with OpenFOAM® stand-alone (i.e. without mandatory use of swak4foam as it was the case in RichardsFoam2 [2]);(vii) several other minor rewritings/cleanings of the code.Nature of problem: This software solves the non-linear three-dimensional transient Richards equation, which is a very popular model for water transfer in variably saturated porous media (e.g.: soils). It is designed to take advantage of the massively parallel computing performance of OpenFOAM®. The goal is to be able to model natural hydrosystems on large temporal and spatial scales.Solution method: A mixed implicit (FVM in the object oriented OpenFOAM framework) and explicit (FVC in the object oriented OpenFOAM® framework) discretization of the equation with a backward time scheme is coupled with a linearization method (Picard algorithm). Due to the linearization loop the final solution of each time step tends towards a fully implicit solution. The implementation has been carried out with a concern for robustness and parallel efficiency.

Time‐discretizations of differentiators: Design of implicit algorithms and comparative analysis

AbstractA complete review of the known differentiators and their time‐discretizations have been addressed in this study. To resolve the drawbacks of the explicit (forward Euler) discretization, which is commonly utilized in sliding‐mode‐based differentiators, implicit time discretization methods are proposed to handle the set‐valued functions. The proposed schemes are supported by some analytical results to show their crucial properties, for example, finite‐time convergence, exactness, invariant sliding‐surface, chattering elimination, insensitivity to the gains during the sliding‐phase, and the well‐posedness. The causal implementation of the proposed implicit schemes has been addressed clearly and supported by flowcharts. Semi‐implicit schemes are also presented to provide a compromise between the performance and the ease of implementation. Finally, an exhaustive comparison is made using numerical experiments among 25 different state‐of‐the‐art differentiators to evaluate the behaviors of the differentiators facing the noise and initial error. Realistic conditions are considered in the simulations to provide useful information based on practical conditions. General conclusions are that implicit discretizations can supersede explicit and semi‐implicit ones.

Global Exponential Stability of a Neural Network for Inverse Variational Inequalities

We investigate the convergence properties of a projected neural network for solving inverse variational inequalities. Under standard assumptions, we establish the exponential stability of the proposed neural network. A discrete version of the proposed neural network is considered, leading to a new projection method for solving inverse variational inequalities, for which we obtain the linear convergence. We illustrate the effectiveness of the proposed neural network and its explicit discretization by considering applications in the road pricing problem arising in transportation science. The results obtained in this paper provide a positive answer to a recent open question and improve several recent results in the literature.

Tetrahedral spectral feature-Based bayesian manifold learning for grey matter morphometry: Findings from the Alzheimer's disease neuroimaging initiative.

Structural and anatomical analyses of magnetic resonance imaging (MRI) data often require a reconstruction of the three-dimensional anatomy to a statistical shape model. Our prior work demonstrated the usefulness of tetrahedral spectral features for grey matter morphometry. However, most of the current methods provide a large number of descriptive shape features, but lack an unsupervised scheme to automatically extract a concise set of features with clear biological interpretations and that also carries strong statistical power. Here we introduce a new tetrahedral spectral feature-based Bayesian manifold learning framework for effective statistical analysis of grey matter morphology. We start by solving the technical issue of generating tetrahedral meshes which preserve the details of the grey matter geometry. We then derive explicit weak-form tetrahedral discretizations of the Hamiltonian operator (HO) and the Laplace-Beltrami operator (LBO). Next, the Schrödinger's equation is solved for constructing the scale-invariant wave kernel signature (SIWKS) as the shape descriptor. By solving the heat equation and utilizing the SIWKS, we design a morphometric Gaussian process (M-GP) regression framework and an active learning strategy to select landmarks as concrete shape descriptors. We evaluate the proposed system on publicly available data from the Alzheimers Disease Neuroimaging Initiative (ADNI), using subjects structural MRI covering the range from cognitively unimpaired (CU) to full blown Alzheimer's disease (AD). Our analyses suggest that the SIWKS and M-GP compare favorably with seven other baseline algorithms to obtain grey matter morphometry-based diagnoses. Our work may inspire more tetrahedral spectral feature-based Bayesian learning research in medical image analysis.

Contrast-independent partially explicit time discretizations for multiscale flow problems

Many multiscale problems have a high contrast, which is expressed as a very large ratio between the media properties. The contrast is known to introduce many challenges in the design of multiscale methods and domain decomposition approaches. These issues to some extent are analyzed in the design of spatial multiscale and domain decomposition approaches. However, some of these issues remain open for time dependent problems as the contrast affects the time scales, particularly, for explicit methods. For example, in parabolic equations, the time step is dt=H2/κmax, where κmax is the largest diffusivity. In this paper, we address this issue in the context of parabolic equation by designing a splitting algorithm. The proposed splitting algorithm treats dominant multiscale modes in the implicit fashion, while the rest in the explicit fashion. The contrast-independent stability of these algorithms requires a special multiscale space design, which is the main purpose of the paper. We show that with an appropriate choice of multiscale spaces we can achieve an unconditional stability with respect to the contrast. This could provide computational savings as the time step in explicit methods is adversely affected by the contrast. We discuss some theoretical aspects of the proposed algorithms. Numerical results are presented.

A hybrid Laplace‐Galerkin method for thermo‐hydro‐mechanical coupling in fluid saturated porous media: Application for borehole problems

AbstractA hybrid Laplace‐Galerkin method is proposed for modeling thermo‐hydro‐mechanical coupled problems of fluid saturated porous media in unsteady state regime. It consists of transforming the weak form equations into the Laplace space where they are solved by the Galerkin finite element method. Results in time space are then obtained by the Stehfest method of inverse Laplace transformation. This technique does not require any time stepping comparing to the traditional implicit or explicit time discretization method. Its exactitude is confirmed by comparing with existent analytical results as well as the one obtained by the implicit time stepping method.

Second Order Fully Semi-Lagrangian Discretizations of Advection-Diffusion-Reaction Systems

We propose a second order, fully semi-Lagrangian method for the numerical solution of systems of advection-diffusion-reaction equations, which is based on a semi-Lagrangian approach to approximate in time both the advective and the diffusive terms. The proposed method allows to use large time steps, while avoiding the solution of large linear systems, which would be required by implicit time discretization techniques. Standard interpolation procedures are used for the space discretization on structured and unstructured meshes. A novel extrapolation technique is proposed to enforce second-order accurate Dirichlet boundary conditions. We include a theoretical analysis of the scheme, along with numerical experiments which demonstrate the effectiveness of the proposed approach and its superior efficiency with respect to more conventional explicit and implicit time discretizations.

Hybrid Discontinuous Galerkin/Finite Volume Method with Subcell Resolution for Shocked Flows

A hybrid discontinuous Galerkin (DG)/finite volume (FV) method under the multistage boundary variation diminishing (BVD) reconstruction paradigm is developed for shocked inviscid and viscous flows. The methodology uses high-order unlimited polynomials and the THINC (which stands for tangent of hyperbola for interface capturing) function as the candidate reconstruction functions to perform FV computation on embedded subcells. The reconstruction scheme on the FV subcells can be designed in a way to keep the resulting approach as compact as the underlying DG method. The reconstruction information based on the modified BVD selection algorithm has been applied as an indicating strategy so that no additional discontinuity indicator is required for the hybrid DG/FV computation. The designed approach has been investigated through the coupling with the explicit and implicit time discretization scheme for both inviscid and viscous flows. Numerical results demonstrate the high-fidelity characteristics of the method by using a much coarser grid to capture both the discontinuity and the smooth flow structure with high resolution.

High Order Semi-Implicit WCNS for 1D Isentropic and Full Euler Equations

This paper designs a high order semi-implicit weighted compact nonlinear scheme (WCNS) to solve 1D isentropic and full Euler equations. The Euler equations are split into stiff and non-stiff terms that are solved by the implicit and explicit time discretization method, respectively, in order to improve the computational efficiency for low Mach flows. The fifth-order WCNS is applied for the spatial discretization. Several numerical examples are given to demonstrate the performance of the designed method.

Primal Dual Methods for Wasserstein Gradient Flows

Combining the classical theory of optimal transport with modern operator splitting techniques, we develop a new numerical method for nonlinear, nonlocal partial differential equations, arising in models of porous media, materials science, and biological swarming. Our method proceeds as follows: first, we discretize in time, either via the classical JKO scheme or via a novel Crank–Nicolson-type method we introduce. Next, we use the Benamou–Brenier dynamical characterization of the Wasserstein distance to reduce computing the solution of the discrete time equations to solving fully discrete minimization problems, with strictly convex objective functions and linear constraints. Third, we compute the minimizers by applying a recently introduced, provably convergent primal dual splitting scheme for three operators (Yan in J Sci Comput 1–20, 2018). By leveraging the PDEs’ underlying variational structure, our method overcomes stability issues present in previous numerical work built on explicit time discretizations, which suffer due to the equations’ strong nonlinearities and degeneracies. Our method is also naturally positivity and mass preserving and, in the case of the JKO scheme, energy decreasing. We prove that minimizers of the fully discrete problem converge to minimizers of the spatially continuous, discrete time problem as the spatial discretization is refined. We conclude with simulations of nonlinear PDEs and Wasserstein geodesics in one and two dimensions that illustrate the key properties of our approach, including higher-order convergence our novel Crank–Nicolson-type method, when compared to the classical JKO method.

Implicit and explicit discrete‐time realizations of homogeneous differentiators

AbstractThis article presents explicit and implicit discrete‐time realizations for a class of homogeneous sliding mode‐based differentiators. The proposed approach relies on the method of exact discretization for linear systems with a zero‐order holder. The two discrete‐time schemes present the homogeneity property and, after a finite time, the accuracy of its continuous‐time counterpart, even in the presence of bounded noise. In comparison to the explicit realization, which makes it possible to determine the state of the system later from the state at present, for this case, the implicit method requires finding a solution by solving generalized equations that involve the current state of the system and two support variables. Therefore, this document proves that the polynomial's unique positive root defines the required solution as part of the main results. Hence, it is possible to introduce a non‐anticipative method to implement the implicit discrete‐time realization, including an appropriate root‐finding method for the polynomial. Finally, the simulation results include comparisons between the proposed implicit and explicit discrete methods with other existing schemes. Numerical studies clearly show that the implicit method supersedes the explicit one, consistent with the implicit and explicit time discretization of other continuous‐time algorithms.

Correction to: An implicit–explicit time discretization scheme for second-order semilinear wave equations with application to dynamic boundary conditions

Correction to: An implicit–explicit time discretization scheme for second-order semilinear wave equations with application to dynamic boundary conditions

Numerical comparison of mathematical and computational models for the simulation of stochastic neutron kinetics problems

This paper concerns numerical comparisons between five mathematical models capable of modelling the stochastic behaviour of neutrons in low extraneous (extrinsic or fixed) neutron source applications. These models include analog Monte-Carlo (AMC), forward probability balance equations (FPB), generating function form of the forward probability balance equations (FGF), generating function form of the backward probability balance equations (Pál-Bell), and an Itô calculus model using both an explicit and implicit Euler-Maruyama discretization scheme. Results such as the survival probability, extinction probability, neutron population mean and standard deviation, and neutron population cumulative distribution function have all been compared. The least computationally demanding mathematical model has been found to be the use of the Pál-Bell equations which on average take four orders of magnitude less time to compute than the other methods in this study. The accuracy of the AMC and FPB models have been found to be strongly linked to the computational efficiency of the models. The computational efficiency of the models decrease significantly as the maximum allowable neutron population is approached. The Itô calculus methods, utilising explicit and implicit Euler-Maruyama discretization schemes, have been found to be unsuitable for modelling very low neutron populations. However, improved results, using the Itô calculus methods, have been achieved for systems containing a greater number of neutrons.

Full space‐time adaptive method based on collocation strategy and implicit multirate time stepping

AbstractIn this paper, we present a full space‐time Adaptive Fup Collocation Method (AFCM) for solving initial‐boundary value problems with particular application on advection‐dominated advection‐diffusion problems. The spatial adaptive strategy dynamically changes the computational grid at each global time step, while the adaptive time‐marching algorithm uses different local time steps for different collocation points based on temporal accuracy criteria. Contrary to the existing space‐time adaptive methods that are based on explicit temporal discretization and scale‐dependent local time stepping, the presented method is implicit and resolves all spatial and temporal scales independently of each other. This means that smaller local time steps are used only for spatial zones where temporal solution changes are intensive, which are not necessarily the zones with finest spatial discretization. In this manner, the computationally efficient full space‐time adaptive strategy accurately resolves small‐scale features and controls numerical error and spurious oscillations. The efficiency and accuracy of AFCM are verified with some classical one‐ and two‐dimensional benchmark test cases.

Body-fitted topology optimization of 2D and 3D fluid-to-fluid heat exchangers

We present a topology optimization approach for the design of fluid-to-fluid heat exchangers which rests on an explicit meshed discretization of the phases at stake, at every iteration of the optimization process. The considered physical situations involve a weak coupling between the Navier–Stokes equations for the velocity and the pressure in the fluid, and the convection–diffusion equation for the temperature field. The proposed framework combines several recent techniques from the field of shape and topology optimization, and notably a level-set based mesh evolution algorithm for tracking shapes and their deformations, an efficient optimization algorithm for constrained shape optimization problems, and a numerical method to handle a wide variety of geometric constraints such as thickness constraints and non-penetration constraints. Our strategy is applied to the optimization of various types of heat exchangers. At first, we consider a simplified 2D cross-flow model where the optimized boundary is the section of the hot fluid phase flowing in the transverse direction, which is naturally composed of multiple holes. A minimum thickness constraint is imposed on the cross-section so as to account for manufacturing and maximum pressure drop constraints. In a second part, we optimize the design of 2D and 3D heat exchangers composed of two types of fluid channels (hot and cold), which are separated by a solid body. A non-mixing constraint between the fluid components containing the hot and cold phases is enforced by prescribing a minimum distance between them. Numerical results are presented on a variety of test cases, demonstrating the efficiency of our approach in generating new, realistic, and unconventional heat exchanger designs.

Implementation of the fractional-step, artificial compressibility with pressure projection (FSAC-PP) method into openfoam for unsteady flows

Commercial and open-source CFD solvers rely mostly on incompressible approximate projection methods to overcome the pressure-velocity decoupling, such as the SIMPLE (Patankar, 1980) or PISO (Issa, 1986) algorithm. Incompressible methods based on the Artificial Compressibility method (Chorin, 1967) lack a mechanism to evolve in time and need to be supplemented by a real time derivative through the dual time scheme. The current study investigates the implementation of the explicit dual time discretization of the Artificial Compressibility method into OpenFOAM and extends on that by applying the dual time scheme to the incompressible FSAC-PP method (Könözsy, 2012). Applied to the Couette 2D flow at Re=100 and Re=1000, results show that for both methods accurate time evolutions of the velocity profiles are presented, where the FSAC-PP methods seemingly produces smoother profiles compared to the AC method, especially during the start-up of the simulation.

Lyapunov--Krasovskii Functional for Discretized Homogeneous Systems

The paper is devoted to stability analysis of discrete-time time-delay systems, obtained after explicit Euler discretization of (locally) homogeneous continuous-time models. The results are derived by applying the Lyapunov--Krasovskii theory. A generic structure of the functional is given that suits for any homogeneous system of nonzero degree (and it can also be used for any dynamics admitting a homogeneous approximation). The obtained conditions are utilized to demonstrate stability for a discretized delayed locally homogeneous planar system with negative and positive degrees.

Finite element analysis of coupled phase-field and thermoelasticity equations at large strains for martensitic phase transformations based on implicit and explicit time discretization schemes

In this paper, a staggered, nonlinear finite element procedure is developed to solve the large-strain based coupled system of time dependent Ginzburg-Landau (GL) and thermoelasticity equations for phase transformations at the nanoscale. Geometrical nonlinearities are included based on the total Lagrangian description where the total deformation gradient is defined as the multiplicative decomposition of elastic, transformational and thermal deformation gradient tensors. The principle of virtual work is utilized to obtain the integral form of the Lagrangian equation of motion whose discretized form is solved using the iterative methods which gives the thermal and transformational deformation gradient tensors. Next, the elastic deformation gradient tensor and the first Piola-Kirchhoff stress are calculated and substituted in the GL equation which gives the phase order parameter. The weighted residual method is used to derive the corresponding finite element form which allows for the phase-dependent surface energy BCs. Explicit and different implicit methods belonging to the generalized trapezoidal family are used for time discretization of the GL equation. Various examples of phase transformations are simulated and discussed. The effect of different time integration schemes are also investigated. The current study allows for a proper modeling of various phase field problems at large and small strains.

A High-Order Conservative Semi-Lagrangian Solver for 3D Free Surface Flows with Sediment Transport on Voronoi Meshes

In this paper, we present a conservative semi-Lagrangian scheme designed for the numerical solution of 3D hydrostatic free surface flows involving sediment transport on unstructured Voronoi meshes. A high-order reconstruction procedure is employed for obtaining a piecewise polynomial representation of the velocity field and sediment concentration within each control volume. This is subsequently exploited for the numerical integration of the Lagrangian trajectories needed for the discretization of the nonlinear convective and viscous terms. The presented method is fully conservative by construction, since the transported quantity or the vector field is integrated for each cell over the deformed volume obtained at the foot of the characteristics that arises from all the vertexes defining the computational element. The semi-Lagrangian approach allows the numerical scheme to be unconditionally stable for what concerns the advection part of the governing equations. Furthermore, a semi-implicit discretization permits to relax the time step restriction due to the acoustic impedance, hence yielding a stability condition which depends only on the explicit discretization of the viscous terms. A decoupled approach is then employed for the hydrostatic fluid solver and the transport of suspended sediment, which is assumed to be passive. The accuracy and the robustness of the resulting conservative semi-Lagrangian scheme are assessed through a suite of test cases and compared against the analytical solution whenever is known. The new numerical scheme can reach up to fourth order of accuracy on general orthogonal meshes composed by Voronoi polygons.