Explicit Discretization Research Articles

Local discontinuous Galerkin methods for parabolic interface problems with homogeneous and non-homogeneous jump conditions

In this paper, we present the local discontinuous Galerkin (LDG) methods for solving second order parabolic interface problems in two-dimensional convex polygonal domains with homogeneous and non-homogeneous jump conditions, respectively. We start analyzing from the homogeneous problems. After giving the semi discrete LDG method, we derive its stability and a prior error estimate. Then, after a series of derivation, we obtain the LDG method for the non-homogeneous problems, which takes the same formulation as the homogeneous case with a special choice of numerical fluxes that incorporate the jump conditions across interface. So its analysis can be easily obtained within the framework developed for the homogeneous case. It is an advantage of the LDG method that it provides a natural framework to enforce the discontinuities in both the potential and the flux weakly in the discrete formulation provided that the triangulation of the domain is fitted to the interface. Numerical experiments are conducted to confirm our theoretical analysis with the help of the explicit temporal discretization and show that the LDG methods have the optimal and suboptimal convergence rate in the L2 norms for the potential and the flux, respectively. In addition, they also indicate that the LDG methods possess the same convergence properties in the L∞ norms. What is more, to avoid the strict time-step restriction of explicit schemes, the implicit integration factor (IIF) method is employed, which not only relaxes the time step to Δt=O(hmin) but also remains the important property of LDG method that the computation can proceed element by element and avoid solving a global system of algebraic equations as the standard implicit schemes do. We provide numerical examples to illustrate that the numerical scheme combined LDG method with IIF method indeed reduces the computational cost greatly.

Non-asymptotic error bounds for the multilevel Monte Carlo Euler method applied to SDEs with constant diffusion coefficient

In this paper, we are interested in deriving non-asymptotic error bounds for the multilevel Monte Carlo method. As a first step, we deal with the explicit Euler discretization of stochastic differential equations with a constant diffusion coefficient. We prove that, as long as the deviation is below an explicit threshold, a Gaussian-type concentration inequality optimal in terms of the variance holds for the multilevel estimator. To do so, we use the Clark-Ocone representation formula and derive bounds for the moment generating functions of the squared difference between a crude Euler scheme and a finer one and of the squared difference of their Malliavin derivatives.

Randomly-fluctuating heterogeneous continuum model of a ballasted railway track

This paper proposes a description of a granular medium as a stochastic heterogeneous continuum medium. The heterogeneity of the material properties field recreates the heterogeneous stress field in a granular medium. The stochastic approach means that only statistical information, easily available, is required to construct the model. The heterogeneous continuum model is Calibrated with respect to discrete simulations of a set of railway ballast samples. As they are continuum-based, the equilibrium equations can be solved on a large scale using a parallel implementation of an explicit time discretization scheme for the Finite Element Method. Simulations representative of the influence on the environment of the passage of a train on a ballasted railway track clearly show the influence of the heterogeneity. These simulations seem to correlate well with previously unexplained overly damped measurements in the free field.

Semi-implicit discontinuous Galerkin methods for the incompressible Navier–Stokes equations on adaptive staggered Cartesian grids

In this paper a new high order semi-implicit discontinuous Galerkin method (SI-DG) is presented for the solution of the incompressible Navier–Stokes equations on staggered space-time adaptive Cartesian grids (AMR) in two and three space dimensions. The pressure is written in the form of piecewise polynomials on the main grid, which is dynamically adapted within a cell-by-cell AMR framework. According to the time dependent main grid, different face-based spatially staggered dual grids are defined for the piece-wise polynomials of the respective velocity components. Although the resulting adaptive staggered grids are more complex than classical uniform Cartesian meshes, the numerical scheme can still be written in a rather compact form. Thanks to the use of a tensor-product formulation for the definition of the nodal basis in the d-dimensional space (d=2,3), all the discrete operators can be efficiently written as a combination of linear one-dimensional operators acting in the d space directions separately.Arbitrary high order of accuracy is achieved in space, while a very simple semi-implicit time discretization is obtained via an explicit discretization of the nonlinear convective terms, and an implicit discretization of the pressure gradient in the momentum equation and of the divergence of the velocity field in the continuity equation. The real advantages of the staggered grid arise in the solution of the Schur complement associated with the saddle point problem of the discretized incompressible Navier–Stokes equations, i.e. after substituting the discrete momentum equations into the discrete continuity equation. This leads to a linear system for only one unknown, the scalar pressure. Indeed, the resulting linear pressure system is shown to be symmetric and positive-definite. In order to avoid a quadratic stability condition for the parabolic terms given by the viscous stress tensor, an implicit discretization is also used for the diffusive terms in the momentum equation. A particular feature of our staggered DG approach is that the viscous stress tensor is discretized on the dual mesh. This corresponds to the use of a lifting operator, but on the staggered grid. Both linear systems for pressure and velocity are very efficiently solved by means of a classical matrix-free conjugate gradient method, for which fast convergence is observed. Moreover, it should be noticed that all test cases shown in this paper have been performed without the use of any preconditioner. Due to the explicit discretization of the nonlinear convective terms, the final algorithm is stable within the classical CFL-type time step restriction for explicit DG methods based on the fluid velocity. Moreover, by using a suitable local time stepping technique based on the ADER-DG framework, each element can use its optimal local time step also within the AMR context.The new space-time adaptive staggered DG scheme has been thoroughly verified for polynomial degrees up to N=9 for a large set of non-trivial test problems in two and three space dimensions, for which analytical, numerical or experimental reference solutions exist. The high order of accuracy of the scheme is shown by means of a numerical convergence table obtained by comparing the numerical solution against a smooth analytical solution.To the knowledge of the authors, this is the first staggered semi-implicit DG scheme for the incompressible Navier–Stokes equations on space-time adaptive meshes in two and three space dimensions.

An Antidiffusive HLL Scheme for the Electronic $M_1$ Model in the Diffusion Limit

In this work, an asymptotic-preserving scheme is proposed for the electronic $M_1$ model in the diffusion limit. A very simple modification of the HLL numerical viscosity is considered in order to capture the correct asymptotic limit in the diffusion limit. This alteration also ensures the realizability of the numerical solution under a suitable CFL condition. Interestingly, it is proved that the new scheme can also be understood as a Godunov-type scheme based on a suitable approximate Riemann solver. Various numerical test cases are performed and the results are compared with a standard HLL scheme and an explicit discretization of the limit diffusion equation.

A numerical scheme based on radial basis function finite difference (RBF-FD) technique for solving the high-dimensional nonlinear Schrödinger equations using an explicit time discretization: Runge–Kutta method

In this research, we investigate the numerical solution of nonlinear Schrödinger equations in two and three dimensions. The numerical meshless method which will be used here is RBF-FD technique. The main advantage of this method is the approximation of the required derivatives based on finite difference technique at each local-support domain as Ωi. At each Ωi, we require to solve a small linear system of algebraic equations with a conditionally positive definite matrix of order 1 (interpolation matrix). This scheme is efficient and its computational cost is same as the moving least squares (MLS) approximation. A challengeable issue is choosing suitable shape parameter for interpolation matrix in this way. In order to overcome this matter, an algorithm which was established by Sarra (2012), will be applied. This algorithm computes the condition number of the local interpolation matrix using the singular value decomposition (SVD) for obtaining the smallest and largest singular values of that matrix. Moreover, an explicit method based on Runge–Kutta formula of fourth-order accuracy will be applied for approximating the time variable. It also decreases the computational costs at each time step since we will not solve a nonlinear system. On the other hand, to compare RBF-FD method with another meshless technique, the moving kriging least squares (MKLS) approximation is considered for the studied model. Our results demonstrate the ability of the present approach for solving the applicable model which is investigated in the current research work.

Fully implicit methodology for the dynamics of biomembranes and capillary interfaces by combining the level set and Newton methods

This framework is concerned with the numerical modeling of the dynamics of individual biomembranes and capillary interfaces in a surrounding Newtonian fluid. A level set approach helps to follow the interface motion. Our method features the use of high order fully implicit time integration schemes that enable to overcome stability issues related to the explicit discretization of the highly non-linear bending force or capillary force. At each time step, the tangent systems are derived and the resulting nonlinear problems are solved by a Newton–Raphson method. Based on the signed distance assumption, several inexact Newton strategies are employed to solve the capillary and vesicle problems and guarantee the second-order convergence behavior. We address in detail the main features of the proposed method, and we report several experiments in the two-dimensional case with the aim of illustrating its accuracy and efficiency. Comparative investigations with respect to the fully explicit scheme depict the stabilizing effect of the new method, which allows to use significantly larger time step sizes.

The [formula omitted]-theorem for the physico-chemical kinetic equations with explicit time discretization

There is demonstrated in the present paper, that the H-theorem in the case of explicit time discretization of the physico-chemical kinetic equations, generally speaking, is not valid. We prove the H-theorem, when the system of the physico-chemical kinetic equations with explicit time discretization has the form of non-linear analogue of the Markov process with doubly stochastic matrix, and for more general cases. In these cases the proof is reduced to the proof of the H-theorem for Markov chains. The simplest discrete velocity models of the Boltzmann equation with explicit time discretization –the Carleman and Broadwell models are discussed and the H-theorem for them in the case of discrete time is proved.

OpenMP parallel computing of 2D TiC combustion synthesis process using an explicit finite-volume scheme

This paper analyzes from a numerical point of view the ignition and propagation of the combustion front during the exothermic TiC combustion synthesis of a material made of pressed titanium and carbide particles when thermophysical properties are either assumed constant or temperature and conversion rate dependent. A two-dimensional cylindrical geometry is considered. The heat supply is prescribed on one, two or three sides of the physical domain. A one-step kinetics is used to describe the reaction Ti+C→TiC in a solid phase and leads to the computation of the conversion rate. A coupling with a non-linear heat equation which takes into account the heat generated by the exothermic kinetics and the two allotropic phase-changes is considered. An explicit finite-volume discretization of the coupled system is constructed and analyzed. Time-step’s stability condition is given for a general expression of the thermo-physical characteristics. A discrete maximum principle is reported. Open MP API was used to parallelize the numerical software written in C. An average speedup of three was obtained on an intel quad-core processor i7-2600. The ignition time and the fraction of unreacted material are systematically computed and compared for several heat supply scenario.

Basic parameters test and 3D modeling of bond between high‐strength concrete and ribbed steel bar after elevated temperatures

This paper presents basic parameters test and an efficient numerical model for the bond between reinforcing bars and concrete after elevated temperatures. The chemical adhesive and friction coefficient are measured by self‐designed setups, and the variation law of the parameters is summarized. In the 3D modeling, bar‐to‐concrete interface has been modeled by means of an explicit discretization of the bar ribs. Surface‐to‐surface contact pair elements are employed for the interface, and the tangential contact behavior is elastic‐plastic with a Coulomb‐type friction model. Comparison between experimental and numerical results shows that the proposed model describes the behavior of steel bar relatively well in the pull‐out process after elevated temperatures. The 3D modeling makes up some deficiencies of macro pull‐out test, such as the detailed stress distributions and crack patterns.

Fully implicit finite element method for the modeling of free surface flows with surface tension effect

A new approach based on the use of the Newton and level set methods allows to follow the motion of interfaces with surface tension immersed in an incompressible Newtonian fluid. Our method features the use of a high-order fully implicit time integration scheme that circumvents the stability issues related to the explicit discretization of the capillary force when capillary effects dominate. A strategy based on a consistent Newton–Raphson linearization is introduced, and performances are enhanced by using an exact Newton variant that guarantees a third-order convergence behavior without requiring second-order derivatives. The problem is approximated by mixed finite elements, while the anisotropic adaptive mesh refinements enable us to increase the computational accuracy. Numerical investigations of the convergence properties and comparisons with benchmark results provide evidence regarding the efficacy of the methodology. The robustness of the method is tested with respect to the standard explicit method, and stability is maintained for significantly larger time steps compared with those allowed by the stability condition. Copyright © 2016 John Wiley & Sons, Ltd.

Advances in computation of local problems for a flow-based upscaling in fractured reservoirs

In this article we present some numerical techniques to increase efficiency and applicability of a flow-based upscaling procedure used to solve single and multi-phase flows in naturally fractured reservoirs. These geological formations may be characterized by hundreds up to hundreds of thousands of fractures, ranging from small to medium scales, spreading all over the reservoir. In these scenarios numerical simulations using an explicit discretization of all fractures become rapidly unfeasible.The problem may be overcome by upscaling procedures. In this work we assume that the flow in the reservoir is simulated by using a corner-point grid completely unrelated to the fracture network. The presence of the latter is accounted for by a numerical upscaled procedure that employs an embedded discrete fracture model (EDFM).To characterize the scaled up problem, we consider a flow-based upscaling procedure where multiple sub-regions are used to derive transmissibilities, mean depths and pore volumes related to the degrees of freedom associated with fractures and rock matrix. An important aspect of the work with respect to others presented in the literature is the enhancement of the upscaling process by splitting the degrees of freedom associated with unconnected portions of the rock matrix. Numerical examples confirm the effectiveness of the proposed approach. We also compare two ways of setting up the boundary conditions of the local problems, that is a well known open issue, used to compute the upscaled transmissibility between cells. Though the experiments lead to a similar result, at the best of our knowledge, it is the first time that such a comparison is made for fractured reservoir.

Driving on Point Clouds: Motion Planning, Trajectory Optimization, and Terrain Assessment in Generic Nonplanar Environments

We present a practical approach to global motion planning and terrain assessment for ground robots in generic three‐dimensional (3D) environments, including rough outdoor terrain, multilevel facilities, and more complex geometries. Our method computes optimized six‐dimensional trajectories compliant with curvature and continuity constraints directly on unordered point cloud maps, omitting any kind of explicit surface reconstruction, discretization, or topology extraction. We assess terrain geometry and traversability on demand during motion planning, by fitting robot‐sized planar patches to the map and analyzing the local distribution of map points. Our motion planning approach consists of sampling‐based initial trajectory generation, followed by precise local optimization according to a custom cost measure, using a novel, constraint‐aware trajectory optimization paradigm. We embed these methods in a complete autonomous navigation system based on localization and mapping by means of a 3D laser scanner and iterative closest point matching, suitable for both static and dynamic environments. The performance of the planning and terrain assessment algorithms is evaluated in offline experiments using recorded and simulated sensor data. Finally, we present the results of navigation experiments in three different environments—rough outdoor terrain, a two‐level parking garage, and a dynamic environment, demonstrating how the proposed methods enable autonomous navigation in complex 3D terrain.

Von Neumann stability analysis of the u–p reproducing kernel formulation for saturated porous media

This paper introduces the von Neumann method to investigate the temporal stability of the displacement---pressure $$(u{-}p)$$(u-p) reproducing kernel formulation for saturated porous media. Both dynamic and quasi-static formulations are considered and the critical time steps are derived. The effect of lumped and consistent matrices on temporal stability is analyzed under explicit temporal discretization. It is shown that lumped matrices have better temporal stability than consistent matrices. The study also shows that nodal support size greatly affects the critical time step size of the formulations. For consistent matrices, larger support size results in smaller critical time step size; however, opposite relation occurs if lumped scheme is used. The numerical study shows that stabilization parameter of the stabilized nodal integration methods reduces the critical time step size. Transient analyses are performed to verify the results from von Neumann analysis.

A compact exponential method for the efficient numerical simulation of the dewetting process of viscous thin films

In this work, we propose a discrete mathematical system to model the evolution of the thickness of two-dimensional viscous thin films subject to a dewetting process. The continuous model under consideration is a degenerate partial differential equation that generalizes the classical thin film equation, and considers the inclusion of a singular potential. The analytical model is discretized using an exponential method that is capable of preserving the positive character of the approximations. In addition, the explicit nature of our approach results in an economic computer implementation which produces fast simulations. We provide some illustrative examples on the dynamics of the growth of thin films in the presence/absence of a dewetting process. The qualitative results exhibit the appearance of typical patterns obtained in experimental settings. The technique was validated against Bhattacharya’s method and a standard explicit discretization of the mathematical model.

A modified Bhattacharya exponential method to approximate positive and bounded solutions of the Burgers–Fisher equation

In this work, we investigate numerically the classical Burgers–Fisher equation using a modified Bhattacharya method. The partial differential equation under investigation possesses nonnegative and bounded solutions and, under some suitable parameter conditions, these solutions are traveling waves. It is well known that the use of the Bhattacharya approach leads to the design of numerical techniques that are sensitive to zero solutions. However, in this manuscript, we provide a correction of that technique in order to approximate solutions of the Burgers–Fisher equation that are bounded in [0,1]. The proposed methodology is explicit, and we establish thoroughly the capability of the technique to preserve the non-negativity, the boundedness and the monotonicity of the numerical approximations, as well as the constant solutions of the continuous model. The new class of methods introduced in this work considers the presence of a free parameter, and we show that this family tends to an explicit and standard discretization of the Burgers–Fisher equation when the free parameter tends to infinity. Some simulations illustrate the main features of the method.

A computationally efficient spectral method for modeling core dynamics

AbstractAn efficient, spectral numerical method is presented for solving problems in a spherical shell geometry that employs spherical harmonics in the angular dimensions and Chebyshev polynomials in the radial direction. We exploit the three‐term recurrence relation for Chebyshev polynomials that renders all matrices sparse in spectral space. This approach is significantly more efficient than the collocation approach and is generalizable to both the Galerkin and tau methodologies for enforcing boundary conditions. The sparsity of the matrices reduces the computational complexity of the linear solution of implicit‐explicit time stepping schemes to O(N) operations, compared to operations for a collocation method. The method is illustrated by considering several example problems of important dynamical processes in the Earth's liquid outer core. Results are presented from both fully nonlinear, time‐dependent numerical simulations and eigenvalue problems arising from the investigation of the onset of convection and the inertial wave spectrum. We compare the explicit and implicit temporal discretization of the Coriolis force; the latter becomes computationally feasible given the sparsity of the differential operators. We find that implicit treatment of the Coriolis force allows for significantly larger time step sizes compared to explicit algorithms; for hydrodynamic and dynamo problems at an Ekman number of , time step sizes can be increased by a factor of 3 to 16 times that of the explicit algorithm, depending on the order of the time stepping scheme. The implementation with explicit Coriolis force scales well to at least 2048 cores, while the implicit implementation scales to 512 cores.

An integrated workflow for characterization and simulation of complex fracture networks utilizing microseismic and horizontal core data

Hydraulic fracture treatments may induce complex network geometries that are challenging to incorporate in numerical flow simulators. To capture this complexity, we propose a workflow that integrates a semi-stochastic Discrete Fracture Network (DFN) generator constrained by Microseismic events and core data and an efficient Perpendicular Bisector (PEBI) grid generator for the explicit discretization of the network. We applied this workflow to evaluate the effect of DFN related uncertainties on production performance. We developed a DFN model that constrains the location of natural fractures by microseismic events and samples fracture characteristics from core-data-based Probability Density Functions (PDFs). The model also interconnects natural and hydraulic fractures through a geomechanics-based algorithm. For an efficient fracture discretization, we developed a PEBI meshing technique capable to conform to low-angle intersections of extensively clustered network, incorporating optimization algorithms that reduce highly skewed cells, and ensure good mesh quality. Finally, to evaluate the impact of DFN related uncertainties on production, we implemented an efficient Monte Carlo (MC) methodology that minimizes flow simulations. Our integrated workflow provides a methodology to efficiently model and mesh explicitly complex DFN systems for numerical fluid simulations, incorporating microseismic and core data. Furthermore, this methodology, together with the implementation of an efficient MC technique, allows to evaluate quantitatively the impact on production forecast of DFN related uncertainties, which come from the inherent randomness of the DFN modeling and the lack of accurate knowledge of PDF parameters due to incomplete fracture characterization.

Discontinuity preserving method for noise removal of multi-carrier signals

A nonlinear method based on anisotropic diffusion notion is proposed in this paper to remove noise from noisy signals modulated with multiple carrier signals by preserving carrier signals as well as discontinuities present in the original noiseless signals. Gaussian and correlated noise contaminating signals with up to four carriers are considered here. Our algorithm proposed here is implemented with both explicit and semi-implicit discretization schemes. Experiments presented here demonstrate promising results indicating a better performance for our nonlinear noise removal method in comparison with the state-of- art in the literature.

Efficient Eigen-Analysis for Large Delayed Cyber-Physical Power System Using Explicit Infinitesimal Generator Discretization

Time delays significantly compromise the performance of wide-area measurement and control system and thus may jeopardize the stability of cyber-physical power systems (CPPS). A delayed CPPS (DCPPS) has a transcendental characteristic equation, leading to an infinite number of eigenvalues basically unsolvable by traditional eigen-analysis methods. In this paper, an explicit infinitesimal generator discretization (EIGD) approach is presented to tackle the traditionally intractable problem. First, the delayed differential equation of DCPPS is transformed to an ordinary differential equation by using an operator called infinitesimal generator. The operator is then optimally discretized, resulting in a highly structured, sparse and explicit approximant matrix. By exploiting the sparsity of the matrix and that of system matrices, the rightmost eigenvalues of the original DCPPS can be accurately computed. The contributions of the EIGD approach lie in the following: 1) it forms a theoretical foundation for accurately obtaining the critical eigenvalues of a CPPS with multiple delays; 2) it constructs a highly structured approximant matrix that enables efficient eigen-analysis of a large DCPPS by making full use of sparsity techniques; and 3) it integrates the shift-invert transformation, Arnoldi algorithm, Newton correction and eigen-sensitivity to form a computational framework for the analysis of large DCPPS. The accuracy, efficiency and scalability of EIGD have been extensively studied and thoroughly validated on the two-area four-machine test system and a practical large transmission grid.