Implicit Time Scheme Research Articles

Implementation of the vehicular occupancy-emission relation using a cubic B-splines collocation method

The complexity and non-linearity of flow phenomena are explained by numerous criteria, including the interactions of the large number of vehicles occupying the road, which influence the road density. This density under certain conditions, leads to traffic congestion which has dangerous effects on the environment such as; resources consumption; noise and the effect caused by greenhouse gas emissions of the \begin{document}$ CO_{2} $\end{document} and other pollutants. In this paper we consider working in an uniform, homogeneous road where the traffic is described by the Lighthill Whitham-Richard (LWR) model resolved using a cubic B-spline collocation scheme in space and an implicit Runge Kutta scheme in time. We also shed light on the relation between vehicle occupancy and vehicle emissions.

A new implicit hybridizable discontinuous Galerkin time-domain method for solving the 3-D electromagnetic problems

Based on an implicit time scheme, this paper presents a new implicit hybridizable discontinuous Galerkin time-domain (imHDGTD) method for solving the 3-D electromagnetic problems with locally refined meshes. Compared with the classical discontinuous Galerkin (DG) method, the proposed imHDGTD method has a larger suitable time step and the reduction in number of degrees of freedom (DOFs). Besides, we adopt the nodal hierarchical basis functions and further propose a matrix solving technology with an effective p-type multigrid preconditioner, which decreases significantly imHDGTD’s matrix factorization time and memory usage. Some numerical results show the accuracy and a higher computational performance of the imHDGTD, which illustrates this method more promising and effective for the complex multiscale electromagnetic problem.

Productivity evaluation of unconventional reservoir development with three-dimensional fracture networks

The evaluation of oil productivity in unconventional reservoirs with complex fracture networks is essential in practical design for improving the production performance. A three-dimensional unified pipe-network method (3D UPM) is developed to understand oil flow behavior in fractured unconventional reservoirs by considering the compressibility of both fluids and solids. Governing equations for transient pressure analysis are discretized based on a pipe equivalence technique. The complex flow transfer between the rock matrix and fractures is solved using the superposition principle at the media interface, greatly simplifying the matrix-fracture coupling process in the 3D pipe-network system. The results, calculated in an implicit time scheme, are verified against commercial software COMSOL to demonstrate the reliability of this method for evaluating well performance. A single multi-fractured horizontal well (MFHW) is numerically investigated to evaluate the influences of natural/hydraulic fracture numbers and properties on the well productivity, which also indicates the necessity of using a 3D model for oil production simulation. Finally, a double MFHWs system was adopted to analyze the influences of fracture networks formed by the interconnection between natural fractures and hydraulic fractures, demonstrating that natural fractures greatly contribute to high well productivity.

Implicit method and slope limiter in AHMR procedure for high order discontinuous Galerkin methods for compressible flows

In this paper the robustness and the performance of adaptive hierarchical mesh refinement (AHMR) for high order Discontinuous Galerkin (DG) finite element method with slope limiting procedure combined with an implicit time scheme for the 2D non-linear Euler equations are shown. A slope limiting procedure based on triangular meshes is implemented and has been extended and amended accordingly to suit quadrilateral elements. The combination of DG methods and slope limiters is generally used with explicit time schemes. Here, the slope limiter implemented is incorporated into a quasi implicit time scheme procedure combined with an automatic h-adaptive hierarchical mesh refinement allowing non-conforming meshes. The time scheme is the implicit Second Order Backward Difference Formula (BDF2) with varying time step. The numerical test cases including subsonic, transsonic and supersonic flows show that the current slope limiting with quadrilateral meshes process together with the implicit time scheme is able to remove overshoots and undershoots around high gradient regions while preserving the high accuracy of the DG method. While combining this procedure with the automatic h-adaptive mesh refinement, one can improve the accuracy of the solutions and be able to capture quite precisely the features of the flows under consideration. The AHMR automatic procedure presented can easily be implemented in the numerical resolution of any physical models. Furthermore the limiting method used in this paper can be generalized to any type of mesh in two dimensions.

Fast Solvers for Two-Dimensional Fractional Diffusion Equations Using Rank Structured Matrices

We consider the discretization of time-space diffusion equations with fractional derivatives in space and either one-dimensional (1D) or 2D spatial domains. The use of an implicit Euler scheme in time and finite differences or finite elements in space leads to a sequence of dense large scale linear systems describing the behavior of the solution over a time interval. We prove that the coefficient matrices arising in the 1D context are rank structured and can be efficiently represented using hierarchical formats ($\mathcal H$-matrices, HODLR). Quantitative estimates for the rank of the off-diagonal blocks of these matrices are presented. We analyze the use of HODLR arithmetic for solving the 1D case and we compare this strategy with existing methods that exploit the Toeplitz-like structure to precondition the GMRES iteration. The numerical tests demonstrate the convenience of the HODLR format when at least a reasonably low number of time steps is needed. Finally, we explain how these properties can be leveraged to design fast solvers for problems with 2D spatial domains that can be reformulated as matrix equations. The experiments show that the approach based on the use of rank-structured arithmetic is particularly effective and outperforms current state of the art techniques.

Estimation of Hydrochemical Unsaturated Soil Parameters Using a Multivariational Objective Analysis

Numerical experiment involving both moisture and solute transport predictions is performed to estimate the hydrochemical characteristics of unsaturated porous soil. The moisture and solute transport in the soil are described by the flow and advection–dispersion transport equations. These transport equations are solved by the spectral element method, which is based on Legendre–Gauss–Lobatto quadrature rule and the fully implicit time scheme using the modified Picard iterative procedure constructed with the standard chord slope approximation. The estimation of hydraulic and solute transport parameters has been conducted using the Levenberg–Marquardt method. The goals of the inverse problem were to develop soil hydrochemical characteristics estimation strategies based on combined two of the following functional cost measurements: moisture content, pressure head, hydraulic conductivity, cumulative outflow, and solute concentration. The performance of the inverse algorithm was evaluated using the coefficient of determination, the root-mean-square error, and the relative error analysis which provide an optimal scheme for parameters estimation. The spectral element method was shown to provide good results with negligible error when compared to analytical values. The obtained results indicate excellent agreement of the method for estimating hydraulic and transport parameters with negligible relative error when compared estimated parameters and true values. The choice and the order of combination of objective functions affect crucially the inverse solution especially in case of large hydrochemical parameters estimates.

Application of carbon dioxide as working fluid in geothermal development considering a complex fractured system

The advantages of using supercritical carbon dioxide as working fluid in geothermal development (potential carbon dioxide geological storage and relatively high mobility) have attracted considerable interest in the simulation of a carbon dioxide-based enhanced geothermal system. This study proposed a numerical model based on the three-dimensional unified pipe-network method to simulate the two-phase flow and thermal-hydraulic coupled process in a carbon dioxide-based enhanced geothermal system considering complex fracture networks. The van Genuchten capillary model and relative permeability model are adopted to characterize the displacement process in both the fractures and the rock matrix. A pipe equivalent technique is employed to discretize the governing equations for the two-phase flow and the heat transmission with the local thermal non-equilibrium concept. Two-phase fluid transfer at material interfaces is solved based on a pipe superposition principle in a pipe-network system. Discretized forms are solved by a sequential implicit time scheme alleviating the Courant-Friedrichs-Lewy condition in fracture networks. The proposed model is verified against analytical results for the Buckley-Leverett problem. Saturation evolution curves at different time steps converge to the analytical solutions as the node number increases. Sensitivity analyses using a doublet system horizontally embedded with one large fracture indicate that the capillary contrast between the rock matrix and the fractures and the injection flow rate pose impacts on the produced carbon dioxide saturation and the outlet fluid temperature drawdown. The simulation of circulating processes considering randomly generated fracture networks demonstrates that the more promising potential of heat extraction can be achieved using the pure carbon dioxide as the working fluid instead of cooled water. Analyzing the effects of the initial reservoir carbon dioxide saturation shows that increasing the reservoir carbon dioxide saturation before heat excavation can improve the heat production. Both increase the matrix permeability and the fracture aperture enhances the performance of the heat extraction and the carbon dioxide sequestration.

Positive control volume finite element scheme for a degenerate compressible two-phase flow in anisotropic porous media

In this paper, we are concerned with the convergence analysis of a positive control volume finite element scheme (CVFE) for a degenerate compressible two-phase flow model in anisotropic porous media. For this, we consider the global pressure saturation formulation. We next use an implicit Euler scheme in time and a CVFE discretization in space. This approach rests on a particular choice of the mean value of the gas density on the interfaces, a centered scheme of the total mobility, and the upwind approximation of fractional fluxes according to the total velocity. Thus, the maximum principle is fulfilled without any constraint on the stiffness coefficients. Moreover, uniform estimates on the discrete gradient of the global pressure and the dissipative term are derived. As the mesh size is sent to zero, we establish that the sequence of approximate solutions converges to a weak solution of the continuous problem. Numerical tests are presented in two dimensions to exhibit the behavior of the gas pressure and the water saturation through the medium.

The Influence of Velocity Field Approximations in Tracer Injection Processes

Although the concentration is the most important variable in tracer injection processes, an efficient and accurate velocity field approximation is crucial to obtain a good physical behaviour for the problem. In this paper we analyse a Stabilized Dual Hybrid Mixed (SDHM) method to solve the Darcy's system in the velocity and pressure variables that involves the conservation of mass and Darcy's law. This approach is locally conservative, free of compromise between the finite element approximation spaces and capable of dealing with heterogeneous media with discontinuous properties. The tracer concentration is solved via a combination of the Streamline Upwind Petrov-Galerkin (SUPG) method in space with an implicit finite difference scheme in time. We also employ a semi-analytical approach (Abbaszadeh-Dehghani analytical solution) to integrate the transport equation. A numerical comparative study using the SDHM formulation, the Galerkin method and a post-processing technique to calculate the velocity field in combination with those concentration approximation methodologies are presented. In all comparisons, the SDHM formulation appears as the most efficient, accurate and almost free of spurious oscillations.

A fully transient novel thermal model for in-field photovoltaic modules using developed explicit and implicit finite difference schemes

Three fully transient numerical thermal models have been developed for photovoltaic (PV) modules in MATLAB environment, using 2-D finite difference (FD) method. One of the thermal FD model is based on explicit time scheme, while the other two are based on implicit time schemes. Out of the two implicit FD models, one has been modeled using preexisting toolboxes of MATLAB, while the other has been modeled using a self-developed novel method. All the three FD models are based on energy balance of different control volumes, which as a whole constitute the complete solid domain of the PV panel. The models have been tested against a variety of experimental data, ranging from sunny clear days, sunny cloudy days, rainy overcast days and consecutive sunny clear days. All the three models are found to agree very well with experimental results, i.e. the errors between the modeled and experimental data ranges between 0.2–0.7 °C. The main difference is between their computational speeds. In terms of average execution time per iteration for transient analyses, the self-developed novel implicit method (referred to as implicit BB throughout the work) was about 1200 times slower than the explicit method. However, overall, the implicit BB method took less time for the entire transient analyses, as it requires less number of iterations due to its tolerance to adapt longer time steps for each iteration. In other words, the explicit method although slightly more accurate, took approximately 900 times more CPU time to simulate the same time span test compared to implicit BB method. Thus, the self-developed novel method (implicit BB) has been recommended for these types of thermal models.

A priori error analysis of the implicit Euler, spectral discretization of a nonlinear equation for a flow in a partially saturated porous media

AbstractThe aim of this work is the numerical study of a nonlinear equation, which models the water flow in a partially saturated underground porous medium under the surface. We propose a discretization of this equation that combines Euler’s implicit scheme in time and spectral methods in space. We prove optimal error estimates between the continuous and discrete solutions. Some numerical experiments confirm the interest of this approach. We present numerical experiments which are in perfect coherence with the analysis.

High‐order Padé and singly diagonally Runge‐Kutta schemes for linear ODEs, application to wave propagation problems

In this article, we address the problem of constructing high‐order implicit time schemes for wave equations. We consider two classes of one‐step A‐stable schemes adapted to linear Ordinary Differential Equation (ODE). The first class, which is not dissipative is based on the diagonal Padé approximant of exponential function. For this class, the obtained schemes have the same stability function as Gauss Runge‐Kutta (Gauss RK) schemes. They have the advantage to involve the solution of smaller linear systems at each time step compared to Gauss RK. The second class of schemes are constructed such that they require the inversion of a unique linear system several times at each time step like the Singly Diagonally Runge‐Kutta (SDIRK) schemes. While the first class of schemes is constructed for an arbitrary order of accuracy, the second‐class schemes is given up to order 12. The performance assessment we provide shows a very good level of accuracy for both classes of schemes, and the great interest of considering high‐order time schemes that are faster. The diagonal Padé schemes seem to be more accurate and more robust.

A scalable nonlinear fluid–structure interaction solver based on a Schwarz preconditioner with isogeometric unstructured coarse spaces in 3D

Nonlinear fluid–structure interaction (FSI) problems on unstructured meshes in 3D appear in many applications in science and engineering, such as vibration analysis of aircrafts and patient-specific diagnosis of cardiovascular diseases. In this work, we develop a highly scalable, parallel algorithmic and software framework for FSI problems consisting of a nonlinear fluid system and a nonlinear solid system, that are coupled monolithically. The FSI system is discretized by a stabilized finite element method in space and a fully implicit backward difference scheme in time. To solve the large, sparse system of nonlinear algebraic equations at each time step, we propose an inexact Newton–Krylov method together with a multilevel, smoothed Schwarz preconditioner with isogeometric coarse meshes generated by a geometry preserving coarsening algorithm. Here “geometry” includes the boundary of the computational domain and the wet interface between the fluid and the solid. We show numerically that the proposed algorithm and implementation are highly scalable in terms of the number of linear and nonlinear iterations and the total compute time on a supercomputer with more than 10,000 processor cores for several problems with hundreds of millions of unknowns.

A MFE method combined with L1-approximation for a nonlinear time-fractional coupled diffusion system

In this paper, a nonlinear time-fractional coupled diffusion system is solved by using a mixed finite element (MFE) method in space combined with L1-approximation and implicit second-order backward difference scheme in time. The stability for nonlinear fully discrete finite element scheme is analyzed and a priori error estimates are derived. Finally, some numerical tests are shown to verify our theoretical analysis.

Parallelization of an Unsteady ALE Solver with Deforming Mesh Using OpenACC

This paper presents a parallel, GPU-based, deforming mesh-enabled unsteady numerical solver for solving moving body problems by using OpenACC. Both the 2D and 3D parallel algorithms based on spring-like deforming mesh methods are proposed and then implemented through OpenACC programming model. Furthermore, these algorithms are coupled with an unstructured mesh based, implicit time scheme integrated numerical solver, which makes the full GPU version of the solver capable of handling unsteady calculations with deforming mesh. Experiments results show that the proposed parallel deforming mesh algorithm can achieve over 2.5x speedup on K20 GPU card in comparison with 20 OpenMP threads on Intel E5-2658 V2 CPU cores. And both 2D and 3D cases are conducted to validate the efficiency, correctness, and accuracy of the present solver.

High-order implicit residual smoothing time scheme for direct and large eddy simulations of compressible flows

Restrictions on the maximum allowable time step of explicit time integration methods for direct and large eddy simulations of compressible turbulent flows at high Reynolds numbers can be very severe, because of the extremely small space steps used close to solid walls to capture tiny and elongated boundary layer structures. A way of increasing stability limits is to use implicit time integration schemes. However, the price to pay is a higher computational cost per time step, higher discretization errors and lower parallel scalability. In quest for an implicit time scheme for scale-resolving simulations providing the best possible compromise between these opposite requirements, we develop a Runge–Kutta implicit residual smoothing (IRS) scheme of fourth-order accuracy, based on a bilaplacian operator. The implicit operator involves the inversion of scalar pentadiagonal systems, for which efficient parallel algorithms are available. The proposed method is assessed against two explicit and two implicit time integration techniques in terms of computational cost required to achieve a threshold level of accuracy. Precisely, the proposed time scheme is compared to four-stages and six-stages low-storage Runge–Kutta method, to the second-order IRS and to a second-order backward scheme solved by means of matrix-free quasi-exact Newton subiterations. Numerical results show that the proposed IRS scheme leads to reductions in computational time by a factor 3 to 5 for an accuracy comparable to that of the corresponding explicit Runge–Kutta scheme.

Parameter free hybrid numerical method for solving modified Burgers’ equations on a nonuniform mesh

In this paper, the modified Burgers’ equation is considered. These kinds of problems come from the field of sonic boom and explosions theories. At big Reynolds’ number there is a boundary layer in the right side of the domain. From numerical point of view, the major difficulty in dealing with this type of problem is that the smooth initial data can give rise to solution varying regions i.e. boundary layer regions. To tackle this situation, we propose a numerical method on nonuniform mesh of Shishkin type, which works well at high as well as low Reynolds number. The proposed method comprises of Euler implicit scheme and hybrid scheme in time and space direction, respectively. First, we discretize the continuous problem in temporal direction by Euler implicit method, which yields a set of ode’s at each time level. The resulting set of differential equations are approximated by a hybrid scheme on Shishkin mesh i.e. upwind in regular region (nonboundary layer region) and central difference in boundary layer regions. The convergence of proposed method has been shown parameter uniform. Some numerical experiments have been carried out to corroborate the theoretical results.

Analysis of a Time Implicit Scheme for the Oseen Model Driven by Nonlinear Slip Boundary Conditions

This work is concerned with the time discrete analysis of the Oseen system of equations driven by nonlinear slip boundary conditions of friction type. We study the existence of solutions of the time discrete model and derive several a priori estimates needed to recover the solution of the continuous problem by means of weak compactness. Moreover, for the difference between the exact and approximate solutions, we obtain the rate of convergence of order one with respect to the time step without imposing extra regularity on the weak solution.

A geometrical multi-scale numerical method for coupled hygro-thermo-mechanical problems in photovoltaic laminates

A comprehensive computational framework based on the finite element method for the simulation of coupled hygro-thermo-mechanical problems in photovoltaic laminates is herein proposed. While the thermo-mechanical problem takes place in the three-dimensional space of the laminate, moisture diffusion occurs in a two-dimensional domain represented by the polymeric layers and by the vertical channel cracks in the solar cells. Therefore, a geometrical multi-scale solution strategy is pursued by solving the partial differential equations governing heat transfer and thermo-elasticity in the three-dimensional space, and the partial differential equation for moisture diffusion in the two dimensional domains. By exploiting a staggered scheme, the thermo-mechanical problem is solved first via a fully implicit solution scheme in space and time, with a specific treatment of the polymeric layers as zero-thickness interfaces whose constitutive response is governed by a novel thermo-visco-elastic cohesive zone model based on fractional calculus. Temperature and relative displacements along the domains where moisture diffusion takes place are then projected to the finite element model of diffusion, coupled with the thermo-mechanical problem by the temperature and crack opening dependent diffusion coefficient. The application of the proposed method to photovoltaic modules pinpoints two important physical aspects: (i) moisture diffusion in humidity freeze tests with a temperature dependent diffusivity is a much slower process than in the case of a constant diffusion coefficient; (ii) channel cracks through Silicon solar cells significantly enhance moisture diffusion and electric degradation, as confirmed by experimental tests.

Spectral discretization of an unsteady flow through a porous solid

We consider the non stationary flow of a viscous incompressible fluid in a rigid homogeneous porous medium provided with mixed boundary conditions. Since the boundary pressure can present high variations, the permeability of the medium also depends on the pressure, so that the problem is nonlinear. We propose a discretization of this equation that combines Euler's implicit scheme in time and spectral methods in space. We prove optimal a priori error estimates and present some numerical experiments which confirm the interest of the discretization.