Explicit Discretization Research Articles

Receding Horizon Control of Bilinear System with Hydraulic Delays

All flow systems are subject to transport delays, which are determined by the flow rates in the system. When the flow rates themselves are control inputs, the system becomes subject to input-dependent state delays, which poses significant theoretical problems. In this paper we propose a model predictive control scheme for a generic multi-variable heat transport system, where flows to the individual consumers can be manipulated by a centralized controller. The control design takes the transport delay into account by means of an explicit discretization of the transport equation, which is a partial differential equation. Different discretization methods are considered, giving rise to a high-order bilinear system model in discrete time. The control problem is formulated as a linear quadratic receding horizon problem with terminal cost. Simulation studies show that the control problem can be solved using standard software and that the Lax-Friedrichs discretization method appears to be the most suitable of the investigated methods.

The Impact of Frame Transformations on Power System EMT Simulation

This article investigates the impact of frame transformations on the accuracy of numerical discretization in power system transient and stability studies. As analysed, frame transformations influence the convergence of the numerical discretization. Specifically, for an explicit discretization method (e.g., forward Euler method), the stability of the original system is best preserved in the frame where the system eigenvalue is closer to the origin of the complex plane, e.g., in the stationary frame for inductors and capacitors, and in the synchronous frame for <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">dq-</i> frame controllers of inverters. Simulation results are given to validate the theoretical analysis.

Numerical Schemes and Stability Analysis for Fractional Models of Transient Uni- and Bidimensional flow in Petroleum Reservoirs

Objective: To propose a modeling for the flow in oil reservoirs using Caputo’s definition of fractional derivative applied to the spatial coordinate of the problem. Furthermore, delimit a region of numerical stability for the explicit method of solving the model equations. Methodology: Starting from the material balance in a differential control volume and the generalized Darcy's law, and coupling it with an equation of state, models describing flow in porous media were derived. A numerical solution scheme using an explicit discretization was proposed. For this purpose, the L1 method for approximating spatial fractional derivatives and finite difference approximation for temporal derivatives were employed.. Results and Discussion: Through the Von Neumann stability analysis, it was observed that the explicit solution schemes for the models are conditionally stable and dependent on the order of the fractional derivative. Additionally, numerical stability regions were constructed for one and two-dimensional transient schemes, revealing that the one-dimensional scheme is stable for , and the two-dimensional scheme is stable for for all derivative orders between 0 and 1. Implications of the research: Historically, flow models in petroleum reservoirs have been based on the application of Darcy's law; however, its use is limited due to some constraints. Recently, fractional calculus has played a significant role in generalizing and obtaining more accurate models in various application areas. In this context, Darcy's law has been generalized, allowing for the derivation of fractional models that are more suitable for describing flow in porous media. These models, in general, are solved using a numerical method, and it is crucial to understand the stability region of the method for their resolution.

An implicit–explicit time discretization for elastic wave propagation problems in plates

AbstractWe propose a new implicit–explicit scheme to address the challenge of modeling wave propagation within thin structures using the time‐domain finite element method. Compared to standard explicit schemes, our approach renders a time marching algorithm with a time step independent of the plate thickness and its associated discretization parameters (mesh step and order of approximation). Relying on the standard three dimensional elastodynamics equations, our strategy can be applied to any type of material, either isotropic or anisotropic, with or without discontinuities in the thickness direction. Upon the assumption of an extruded mesh of the plate‐like geometry, we show that the linear system to be solved at each time step is partially lumped thus efficiently treated. We provide numerical evidence of an adequate convergence behavior, similar to a reference solution obtained using the well‐known leapfrog scheme. Further numerical investigations show significant speed up factors compared to the same reference scheme, proving the efficiency of our approach for the configurations of interest.

Reused LU factorization as a preconditioner for efficient solution of the parabolized stability equations

The parabolized stability equations (PSE) are a widely used, efficient method to calculate the evolution of streamwise traveling instabilities in spatially developing, weakly nonparallel flows. Although the PSE are very economical, computational time can be significant due to the repeated solution of linear systems of equations in each marching step. The linear systems of equations are typically solved by LU factorization because of the moderate size and the sparsity of the matrices. In this paper, instead of repeated calculation of the LU factorization, a single LU factorization is calculated in each marching step and used as a preconditioner for an iterative solver. Numerical experiments are conducted on the plane-marching (3D) incompressible PSE, nonlinear PSE (NLPSE) with explicit discretization of the nonlinear terms, and NLPSE with implicit treatment of the nonlinearities. It is shown that the time spent on the solution of the linear systems of equations was reduced by a factor of 2.5, 4.5–7.5, and 20–60 for the three cases, respectively. The only requirement of the proposed numerical techniques is that the solution varies slowly in the marching direction; therefore it is expected to be applicable to similar boundary layer stability problems or slowly varying parabolic partial differential equations. This is supported by the fact that the numerical method was applied to the Boundary Region Equations (BRE) and the line-marching (2D) PSE, the latter being a much smaller problem than its plane-marching counterpart. In these cases, a factor of 50% and 15–30% decrease in the net time of the linear equation solution is reported, respectively.

Partial Learning Using Partially Explicit Discretization for Heterogeneous Transport Problem Simulation

Partial Learning Using Partially Explicit Discretization for Heterogeneous Transport Problem Simulation

High-fidelity modeling of a vertical axis tidal turbine model under realistic flow conditions

In the context of the global energy crisis, developing renewable energies is of primal importance. Among the renewable resources available are tidal currents. Tidal currents can be exploited with tidal turbines. Different concepts of tidal turbine exist, we here investigate a vertical axis tidal turbine or VATT prototype made by HydroQuest and CMN. More precisely, we carry on numerical investigations on a Hydroquest/CMN tidal turbine model. Experimental studies are limited to turbine models and thus suffer from scale effects. In situ measurement are still expensive and difficult to acquire. Computational Fluid Dynamics (CFD) is an interesting alternative in order to go beyond what experiments can offer. One major drawback of CFD is that it requires validation of the models used. A validation of our CFD tool is included in this paper. Because the hydrodynamic of vertical axis tidal turbines is highly unsteady, Large Eddy Simulation (LES) are well adapted here. Blade-resolved Computational Fluid Dynamics (CFD) approaches are appropriate to study vertical axis tidal turbines, but their cost is still prohibitive for real size tidal turbine. Simplified approaches are preferred solutions. Literature showed that Actuator Line Model (ALM) LES is well suited to model vertical axis turbine. Most tidal sites are characterized with strong turbulence intensities. The presence of turbulence in most tidal sites has led companies and academics to study the effects of ambient turbulence over tidal turbines. It has been shown that turbulence can have a non-negligible influence on the turbine performances and wake. To account for the effects of turbulence, synthetic turbulence is introduced at the inlet of the simulation. The LBM is an unsteady weakly-compressible CFD approach. It solves the Boltzmann equation using an explicit time discretization and a uniform Cartesian grid called lattice. It has been proven to be an efficient approach for modeling tidal turbines using LES and ALM, Grondeau et al. (2019). In this paper, we study a tidal turbine model of the Hydroquest/CMN VATT using an ALM-LBM-LES approach. This model was tested at the Ifremer Boulogne-sur-Mer testing facilities. We first compare our results with the experimental ones in a scenario without turbulence and for flood and ebb tide configurations. A numerical investigation of the influence of upstream turbulence on the turbine performances and wake is then realized. Prospects of this study include among others the effects of turbines interactions and waves.

Bayesian decision making using partial data for fractured poroelastic media

Partial data is a common challenge in many applied problems, where we can measure only the part of the solution. This paper considers the poroelasticity problem in fractured media, where only partial measurements of fracture pressures are available. The poroelasticity models have applications in many engineering and scientific fields, such as geomechanics, hydrology, and reservoir simulation. This paper aims to develop a Bayesian decision making approach using partial data and consider its application to the poroelasticity problem in fractured media. The methodology of the proposed approach combines Proper Orthogonal Decomposition, Discrete Empirical Interpolation Method, Bayesian Inversion Approach, and Bayesian Decision Theory. It allows us to compute posterior probabilities and decide according to the optimal decision rule based on the partial data. Moreover, we apply partially explicit temporal discretization to solve forward problems efficiently. For testing the proposed approach, we consider a two-dimensional model problem. We simulate the cases when we need to make a single decision about the reservoir or decisions about the elements of the set of reservoirs. The results show that the proposed approach can efficiently compute the posterior probabilities and make a decision in the presence of only partial data for both cases.

A Direct Implicit Discretization Approach for Arbitrary Non-conformal Interface by Using a Virtual Cell

A direct implicit discretization approach is presented for an arbitrary non-conformal interface by using a virtual cell on unstructured meshes. In this study, the virtual cell is supposed to be located at the neighbour of a target cell adjacent to the non-conformal interface. Any computational value on a virtual cell is formulated as a linear combination of values on contribution cells, which are determined by the nearest neighbouring cells or surface intersecting cells. The linear formulation simplifies the process of implementing implicit discretization for the governing equations. The approach is simple and efficient for information transfer through the interface because it avoids mesh stitching or iterations based on explicit discretization for treating topologically inconsistent meshes. Several numerical tests verify that the proposed method is capable and efficient for dealing with the arbitrary non-conformal interface.

An adaptive space and time method in partially explicit splitting scheme for multiscale flow problems

We propose a space and time adaptive framework for a partially explicit splitting scheme of flow problems in multiscale media. We consider the parabolic equation with high-contrast multiscale coefficients. It is known that sufficient number of multiscale basis are needed to capture non-local features and information across all scales. Moreover, the time step size in an explicit discretization of the problem is affected by the magnitude of the coefficient. To improve computational efficiency, in this work, we introduce an adaptive algorithm to refine temporal size and enrich multiscale spaces. We first utilize a stable temporal splitting scheme with an appropriate construction of multiscale spaces to ensure that the time step size is independent of the contrast. The multiscale subspaces are then constructed to handle the fast-flow and slow-flow parts separately, and implicit and explicit discretization is adopted for the degrees of freedom with respect to the two multiscale subspaces, respectively. A multirate time stepping is introduced for the two parts considering that the flow rates vary significantly in different regions due to the high-contrast features of the multiscale coefficients. We derive both temporal and spatial error estimators to identify local regions for two components of the solutions. Start with coarser time step sizes and a few important multiscale basis functions in each subspace, we can adaptively refine time stepping and add additional basis with constraint energy minimizing properties based on the derived error indicators. The stability and convergence of the method are studied, and several numerical tests are presented to demonstrate the performance of the proposed method.

Explicit spectral element collocation method for nonlinear transient heat transfer

In this paper, we proposed a novel collocation method to solve nonlinear transient heat transfer problems in typical intelligent materials. The new method is derived from the weighted residual method in which the weight of Jacobian is proven to be the best choice. In the paper, the physical variables and their fluxes are respectively approximated by two series of Lagrange interpolation polynomials at Chebyshev-Gauss-Lobatto nodes. In the proposed method, the mass matrices of the final linear system of equations are diagonal and the explicit time discretization method can be employed. Using the proposed method, we solved the heat transfer in intelligent materials and it is proven to be quite efficient and accurate, which benefits from that no large sparse matrix is generated and solved. By the proposed method, one can easily solve the problems with complex geometries.

Numerical analysis of a time relaxation finite difference method for the heat equation

In this study, we first consider the time-relaxation model, which consists of adding the term $\kappa \left( u-\overline{u}\right) $ to the heat equation. Then, an explicit discretization scheme for the model is introduced to find the finite difference solutions. We first obtain the solutions by using the scheme and then investigate the method’s consistency, stability, and convergence properties. We prove that the method is consistent and unconditionally stable for any given value of $r$ and appropriate values of $\kappa$ and $\delta$. As a result, the method obtained by adding the time relaxation term to the first-order finite-difference explicit method behaves like the second-order implicit method. Finally, we apply the method to some test examples.

A stable loosely-coupled scheme for cardiac electro-fluid-structure interaction

We present a loosely coupled scheme for the numerical simulation of the cardiac electro-fluid-structure interaction problem, whose solution is typically computationally intensive due to the need to suitably treat the coupling of the different submodels. Our scheme relies on a segregated treatment of the subproblems, in particular on an explicit Robin-Neumann algorithm for the fluid-structure interaction, aiming at reducing the computational burden of numerical simulations. The results, both in an ideal and a realistic cardiac setting, show that the proposed scheme is stable at the regimes typical of cardiac simulations. From a comparison with a scheme with implicit fluid-structure interaction, it emerges that, while conservation properties are not fully preserved, computational times significantly benefit from the explicit scheme. Overall, the explicit discretization represents a good trade-off between accuracy and cost, and is a valuable alternative to implicit schemes for fast large-scale simulations.

Optimal convergence rate of the explicit Euler method for convection–diffusion equations II: High dimensional cases

AbstractThis is the second part of study on the optimal convergence rate of the explicit Euler discretization in time for the convection–diffusion equations [Zhang et al. Appl. Math. Lett. 131 (2022), 108048] which focuses on high‐dimensional linear/nonlinear cases under Dirichlet/Neumann boundary conditions. Several new difference schemes are proposed based on the explicit Euler discretization in temporal derivative and centered difference discretization in spatial derivatives. The priori estimate of the improved difference scheme with application to the constant convection coefficients is performed under the maximum norm and the optimal convergence rate four is achieved when the step‐ratios along each direction equal to . Also we give partial results for the three‐dimensional case. The improved difference schemes have essentially improved the CFL condition and the numerical accuracy comparing with the classical difference schemes. Numerical examples involving two‐/three‐dimensional linear/nonlinear problems under Dirichlet/Neumann boundary conditions such as the Fisher equation, the Chafee–Infante equation and the Burgers' equation substantiate the good properties claimed for the improved difference scheme.

Stability of Finite Difference Solution of Time-Dependent Schrodinger Equations

In this paper, the stability of finite difference methods for time-dependent Schrodinger equation with Dirichlet boundary conditions on a staggered mesh was considered with explicit and implicit discretization. Using the matrix representation for the numerical algorithm, it is shown that for the explicit finite difference method, the solution is conditionally stable while it becomes unconditionally stable for implicit finite difference methods. A 1D Harmonic Oscillator problem shall be used to illustrate this behaviour.

Efficient Multigrid Reduction-in-Time for Method-of-Lines Discretizations of Linear Advection

Parallel-in-time methods for partial differential equations (PDEs) have been the subject of intense development over recent decades, particularly for diffusion-dominated problems. It has been widely reported in the literature, however, that many of these methods perform quite poorly for advection-dominated problems. Here we analyze the particular iterative parallel-in-time algorithm of multigrid reduction-in-time (MGRIT) for discretizations of constant-wave-speed linear advection problems. We focus on common method-of-lines discretizations that employ upwind finite differences in space and Runge-Kutta methods in time. Using a convergence framework we developed in previous work, we prove for a subclass of these discretizations that, if using the standard approach of rediscretizing the fine-grid problem on the coarse grid, robust MGRIT convergence with respect to CFL number and coarsening factor is not possible. This poor convergence and non-robustness is caused, at least in part, by an inadequate coarse-grid correction for smooth Fourier modes in space-time known as characteristic components. We propose an alternative coarse-grid operator that provides a better correction of these modes. This coarse-grid operator is related to previous work and uses a semi-Lagrangian discretization combined with an implicitly treated truncation error correction. Theory and numerical experiments show the proposed coarse-grid operator yields fast MGRIT convergence for many of the method-of-lines discretizations considered, including for both implicit and explicit discretizations of high order. Parallel results demonstrate speed-up over sequential time-stepping.

On implementing the implicit discrete‐time super‐twisting observer on mechanical systems

AbstractIn this paper, an extension of an algorithm for the implicit discretization of a super‐twisting sliding mode observer is presented. Implicit and explicit discretization algorithms for homogeneous differentiators, where no physical model information is considered, are investigated in literature. This article studies the behavior when considering models of a rather general class of nonlinear systems. The discrete equations of the super‐twisting observer are reformulated as generalized equation and an algorithm for the step‐by‐step solution is given. The uniqueness of the derived algorithm is investigated with an equivalent variational inequality formulation which is derived for a class of nonlinear systems. Furthermore, a semi‐implicit predictor‐corrector discretization is presented which is an approximation method for the presented algorithms and allows an explicit implementation in practical applications. Accuracy properties under noise and sampling are given. The algorithm is applied on two mechanical example systems taken from practice.

Tikhonov regularization in general form with Chebfun

AbstractLinear ill‐posed problems often are analyzed in function spaces using tools from functional analysis, while their numerical solution typically is computed by first discretizing the problem and then applying tools from finite‐dimensional linear algebra. The Chebfun package makes it possible to solve linear ill‐posed problems without explicit discretization. This work discusses the use of Tikhonov regularization with a fairly general linear regularization operator within the Chebfun framework.

HODG:high-order discontinuous Galerkin methods for solving compressible Euler and Navier-Stokes equations - an open-source component-based development framework

In this work, we present HODG, an open-source component-based development framework based on high order Discontinuous Galerkin (DG) methods for solving compressible Euler and Navier-Stokes equations. This framework is written in pure C++11, and proposes “component” as the basic function unit, which is the key to the Interface-Oriented Programming principle and Aspect-Oriented Programming (AOP) technology. Built on the top-level design of components, HODG is a flexible yet pragmatic development framework that works right out of the box and is easy to use for starters and developers.The current release of HODG supports structured, unstructured, hybrid, and second-order meshes available. It is capable of solving Euler and Navier-Stokes (N-S) equations in two dimensions and three dimensions. In this development framework, spatial accuracy is easy to be extended to higher orders, and DG-P0, DG-P1, DG-P2 (up to third order) are implemented in the current version. Moreover, implicit temporal discretization and explicit discretization are available. Especially for viscous fluxes, the direct Discontinuous Galerkin (DDG) formulation and Bassi and Rebay II (BR2) scheme are implemented. In the case of strong discontinuities or shock waves, artificial viscosity is applied to capture shock waves. Various benchmark numerical examples are provided to demonstrate the full capabilities of HODG. The software is freely available under an MIT license. Program summaryProgram Title: HODG v2.0.0CPC Library link to program files:https://doi.org/10.17632/835732yz8s.1Code Ocean capsule:https://codeocean.com/capsule/4788938Licensing provisions: MITProgramming language: C++11Nature of problem: Compressible Euler and Navier–Stokes equations of fluid dynamics; potential for any advection–diffusion type problem.Solution method: DG-P0, DG-P1, DG-P2 spatial discretization, combining with DDG and BR2 schemes for viscous fluxes, explicit and implicit time schemes, and supporting structured, unstructured, hybrid, and high-order meshes.unusual features: developed based on modern C++11 features; propose the “component” as basic function unit, which follows the practice of Interface-Oriented Programming principle; Special component named “action” is used, and is the key to AOP technology in this framework; Implicitly implements the Dependency Injection (DI) technology; Flexible yet pragmatic, and works right out of the box.

THE ROLE OF STIFFNESS IN TRAINING AND GENERALIZATION OF RESNETS

Neural ordinary differential equations (NODEs) have recently regained popularity as large-depth limits of a large class of neural networks. In particular, residual neural networks (ResNets) are equivalent to an explicit Euler discretization of an underlying NODE, where the transition from one layer to the next is one time step of the discretization. The relationship between continuous and discrete neural networks has been of particular interest. Notably, analysis from the ordinary differential equation viewpoint can potentially lead to new insights for understanding the behavior of neural networks in general. In this work, we take inspiration from differential equations to define the concept of stiffness for a ResNet via the interpretation of a ResNet as the discretization of a NODE. We then examine the effects of stiffness on the ability of a ResNet to generalize, via computational studies on example problems coming from climate and chemistry models. We find that penalizing stiffness does have a unique regularizing effect, but we see no benefit to penalizing stiffness over L&lt;sup&gt;2&lt;/sup&gt; regularization (penalization of network parameter norms) in terms of predictive performance.