First-order Finite Difference Research Articles

Numerical validation of pressure and flow characteristics across a control valve in a feed line

This work is intended to understand the variation of pressure and flow at the pump inlet of liquid rocket engine. The opening and closure of the valve upstream of the pump features complex phenomenon. The opening and closing of the valve cause pressure and flow variations at the pump inlet which may lead to combustion instabilities in combustion chamber of engine, hydraulic transients in feedlines, and off-design operation of turbo-pumps which are fundamental to the efficient testing and operation of engine. A numerical model to predict the pressure and flow transients across a control valve for different rate of opening in fluid feed systems has been developed using first-order finite difference technique. In case of flow in pipes, the velocity and pressure is governed by momentum and continuity equations. A computer code for the prediction of fluid transients is developed based on method of characteristics for one-dimensional fluid flow in pipelines and compared with test data for validation. The control valve is considered to be in-line with the feed line and modeled based on the valve coefficient vs. percent opening of valve. This model can subsequently be used to predict the effect of opening/closing time of the valve on pressure surges across the control valve and corresponding flow rate in the feedline for different opening of the valve.

Physics-Capturing Discretizations for Spectral Wind-Wave Models

This paper discusses the discretization methods that have been commonly employed to solve the wave action balance equation, and that have gained a renewed interest with the widespread use of unstructured grids for third-generation spectral wind-wave models. These methods are the first-order upwind finite difference and first-order vertex-centered upwind finite volume schemes for the transport of wave action in geographical space. The discussion addresses the derivation of these schemes from a different perspective. A mathematical framework for mimetic discretizations based on discrete calculus is utilized herein. A key feature of this algebraic approach is that the process of exact discretization is segregated from the process of interpolation, the latter typically involved in constitutive relations. This can help gain insight into the performance characteristics of the discretization method. On this basis, we conclude that the upwind finite difference scheme captures the wave action flux conservation exactly, which is a plus for wave shoaling. In addition, we provide a justification for the intrinsic low accuracy of the vertex-centred upwind finite volume scheme, due to the physically inaccurate but common flux constitutive relation, and we propose an improvement to overcome this drawback. Finally, by way of a comparative demonstration, a few test cases is introduced to establish the ability of the considered methods to capture the relevant physics on unstructured triangular meshes.

Optimal Dispatch of Integrated Energy Campus Microgrids Considering the Time-Delay of Pipelines

Energy transmission in cooling/heating and gas pipelines has time-delay, which impacts the power balance between various sources and loads in the optimal dispatch of an integrated energy campus microgrid (IECM). This article proposed an optimal dispatch model for IECMs, which considered the time-delay of energy transmission in cooling/heating and gas pipelines. In this model, the electricity and natural gas purchase cost of the IECM was made into an objective function. Partial differential equations (PDEs) were used to describe the time-delay in pipelines, and the discrete variables that described the switching number of absorption chillers and heat exchange units were included in the constraints. The proposed model was essentially a mixed-integer nonlinear programming (MINLP) model with PDE constraints. Orthogonal collocation on finite elements (OCFE) in the two-dimensional domain was used to transform PDEs into algebraic equations (AEs). Several linearization methods, including piecewise linearization and the big M method, were used to transform the initial MINLP model into a mixed-integer linear programming (MILP) model to reduce computational complexity. OCFE was compared with the first-order finite difference method, and simulation results were used to demonstrate the accuracy and efficiency of the proposed algorithm. The impact of the time-delay of pipelines on IECM was analyzed through comparison with a steady-state model.

Comparison and Analysis of Two Different Calculation Schemes of SLEVE- Hybrid Coordinate in GRAPES_MESO Model

The calculation scheme of the smoothed-level and hybrid (SLEVE-hybrid for short) coordinates in numerical forecasting model is not limited in number. It is divided into the semi-analytical scheme and the finite differential scheme in terms of the various differential methods of the coordinate deformation variables. Having compared the dynamic equation and the long-time batch simulation results of the two schemes, the present study draws the following conclusions. The first-order finite difference accuracy of the coordinate deformation variables in the finite differential scheme is theoretically lower than that in the semi-analytical scheme. The larger the vertical gradient of the layer thickness is, the larger the relative errors of the finite differential scheme are. The long-time batch simulation test in the GRAPES model dynamic core demonstrates that the bias of the temperature and the geopotential height in the semi-analytical scheme is smaller under the default layering, while the simulation difference of the two schemes is greatly reduced when the layering is more uniform.

A reduced order finite difference method for solving space-fractional reaction-diffusion systems: The Gray-Scott model

In this paper, we want to present a fast, efficient, and robust numerical procedure for solving a system of PDEs with regard to the fractional Laplacian equation. In the developed approximate scheme, the spatial direction is discretized by a second-order finite difference formula and the temporal direction is discretized by a first-order finite difference approximation. In order to improve the accuracy and to decrease the used CPU time an alternative direction implicit (ADI) method is employed. Furthermore, we use a reduced order model (ROM) based on the proper orthogonal decomposition (POD) technique to reduce the elapsed computational time. The developed numerical scheme is well known as the reduced order finite difference scheme. To emphasize the fast and efficiency of the proposed algorithm, we apply it for the two-dimensional case.

Completed repeated Richardson extrapolation for compressible fluid flows

Abstract Richardson extrapolation is a powerful approach for reducing the spatial discretization errors and increasing, in this way, the accuracy of the computed solution obtained by using many numerical methods for solving different scientific and engineering problems. This approach has been used in a variety of computational fluid dynamics problems to reduce the numerical error, but it has been mainly restricted to the computation of incompressible fluid flows and on grids with coincident nodes. The purpose of this work is to present a completed repeated Richardson extrapolation (CRRE) procedure for a more generic type of grids not necessarily with coincident nodes, and test it on compressible fluid flows. Three tests are performed for one- and quasi-one-dimensional Euler equations, i.e. (i) the Rayleigh flow, (ii) the isentropic flow, and (iii) the adiabatic flow through a nozzle. The last test presents a normal shock wave. In order to build a simple CRRE solver, these problems are solved using a first-order upwind-type finite difference method as the base scheme. The normal shock wave problem is also solved using a high-order weighted essentially non-oscillatory (WENO) scheme to compare it with the CRRE procedure. The procedure we have proposed can increase the achieved accuracy and significantly decrease the magnitude of the spatial error in all three tests. Its performance can be best demonstrated in the Rayleigh flow test, where the spatial discretization error is reduced by seven orders of magnitude and the achieved accuracy is increased from 0.998 to 6.62 on the grid with 10240 nodes. Similar performance can be observed for the isentropic flow, for which the spatial discretization error is reduced by nine orders of magnitude and the achieved accuracy is again increased from 0.996 to 6.73 on the grid with 10240 nodes. Finally, in the adiabatic flow with a normal shock wave, the procedure can reduce the spatial discretization error both upstream and downstream of the shock. We do remark, however, that the more expensive high-order WENO scheme has errors of lower magnitude upstream of the shock and has a sharper shock transition for this shocked test case.

Parabolic Equation Modeling of a Seismic Airgun Array

The numerical modeling of seismic airgun sources, with a specific emphasis on the impact of sound on the marine ecosystem, is a challenging problem. In this paper, the simplified range-independent isovelocity problem, set up as part of the International Airgun Modelling Workshop, is solved using the parabolic equation (PE). The acoustic pressure and particle acceleration are computed including the arrival time series and source energy level in decidecade bands for ranges spanning from 30 m to 30 km. The particle acceleration is computed by taking the spatial first-order finite difference of the pressure on the PE computational grid (~1/16th wavelength). The two significant challenges this problem poses to the PE solution are the very large bandwidth (5-4500 Hz) and the computational accuracy required for time-series and particle acceleration computations. Each of the metrics outlined in the statement of the problem is computed and presented. In addition to computing the required fields, the 17-gun volumetric array is placed in a 3-D ocean environment in deep water in the Gulf of Mexico, with peak sound pressure computed out to 30 km and the band integrated sound exposure level (L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E,p</sub> ) computed to 400 km.

Solution of the Problem of Smoothing of the Signals at the Preprocessing of Thermal Images

Smoothing two-dimensional digital signals is important for a number of applications. The paper presents a mathematical method and an algorithm for smoothing two-dimensional digital signals. The method is based on minimizing the objective function using criteria of the first-order finite difference between the rows and columns of the image as a measure of distance. To estimate the parameters of the developed method, a non-iterative algorithm is used. The present study shows results of changing the smoothing filter core depending on variations in the method parameters.

Alternating-direction implicit finite difference methods for a new two-dimensional two-sided space-fractional diffusion equation

According to the principle of conservation of mass and the fractional Fick’s law, a new two-sided space-fractional diffusion equation was obtained. In this paper, we present two accurate and efficient numerical methods to solve this equation. First we discuss the alternating-direction finite difference method with an implicit Euler method (ADI–implicit Euler method) to obtain an unconditionally stable first-order accurate finite difference method. Second, the other numerical method combines the ADI with a Crank–Nicolson method (ADI–CN method) and a Richardson extrapolation to obtain an unconditionally stable second-order accurate finite difference method. Finally, numerical solutions of two examples demonstrate the effectiveness of the theoretical analysis.

Parabolic equation modeling of a seismic airgun array

The numerical modelling of seismic airgun sources, with a specific emphasis on the impact of sound on the marine ecosystem, is a challenging problem. In this paper, the simplified range-independent iso-velocity problem set up as part if the International Airgun Modelling Workshop (IAMW), is solved using the Parabolic Equation (PE). The acoustic pressure and particle acceleration are computed including the arrival time series and source energy level in deci-decade bands for ranges spanning from 30m to 30 km. The particle acceleration is computed by taking the spatial gradient of the pressure on the PE computational grid. The two significant challenges this problem poses to the PE solution are the very large bandwidth (5-4500 Hz) and the computational accuracy required for time-series and particle acceleration computations. Each of the metrics outlined in the statement of the problem are computed and presented. In addition to computing the required fields, the 17-gun volumetric array is placed in a 3D ocean environment in deep water in the Gulf of Mexico, with peak pressure computed out to 30km and the band integrated Signal Energy Level (SEL) computed to 400 km.

Solving Stiff Reaction-Diffusion Equations Using Exponential Time Differences Methods

Reaction-diffusion equations modeling Predator-Prey interaction are of current interest. Standard approaches such as first-order (in time) finite difference schemes for approximating the solution are widely spread. Though, this paper shows that recent advance methods can be more favored. In this work, we have incorporated, throughout numerical comparison experiments, spectral methods, for the space discretization, in conjunction with second and fourth-order time integrating methods for approximating the solution of the reaction-diffusion differential equations. The results have revealed that these methods have advantages over the conventional methods, some of which to mention are: the ease of implementation, accuracy and CPU time.

Calculation of the filtration problem by finite differences methods

The filtration problem of a suspension in a porous medium is relevant for the construction industry. In the design of hydraulic structures, construction of waterproof walls in the ground, grouting the loose soil, it is necessary to calculate the transfer and deposition of solid particles by the fluid flow. A one-dimensional filtration problem of a monodisperse suspension in a porous medium with a size-exclusion capture mechanism is considered. It is assumed that as the deposit grows, the porosity and admissible flow of particles through the porous medium change. The solution of the initial filtration model and the equivalent equations are calculated. For the numerical calculation of the problem, both standard first-order finite difference formulas and more accurate second-order schemes were used. The obtained solutions are compared with the results given by the TVD-scheme.

Reconstruction of Cell Focal Adhesions using Physical Constraints and Compressive Regularization

We develop a method to reconstruct, from measured displacements of an underlying elastic substrate, the spatially dependent forces that cells or tissues impart on it. Given newly available high-resolution images of substrate displacements, it is desirable to be able to reconstruct small-scale, compactly supported focal adhesions that are often localized and exist only within the footprint of a cell. In addition to the standard quadratic data mismatch terms that define least-squares fitting, we motivate a regularization term in the objective function that penalizes vectorial invariants of the reconstructed surface stress while preserving boundaries. We solve this inverse problem by providing a numerical method for setting up a discretized inverse problem that is solvable by standard convex optimization techniques. By minimizing the objective function subject to a number of important physically motivated constraints, we are able to efficiently reconstruct stress fields with localized structure from simulated and experimental substrate displacements. Our method incorporates the exact solution for a stress tensor accurate to first-order finite differences and motivates the use of distance-based cutoffs for data inclusion and problem sparsification.

New insights into the differences between the dual node approach and the common node approach for coupling surface–subsurface flow

Abstract. The common node approach and the dual node approach are two widely applied approaches to coupling surface–subsurface flow. In this study both approaches are analyzed for cell-centered as well as vertex-centered finite difference schemes. It is shown that the dual node approach should be conceptualized and implemented as a one-sided first-order finite difference to approximate the vertical subsurface hydraulic gradient at the land surface. This results in a consistent dual node approach in which the coupling length is related to grid topology. In this coupling approach the coupling length is not to be interpreted as a nonphysical model parameter. Although this particular coupling approach is technically not new, the differences between this consistent dual node approach and the common node approach have not been studied in detail. In fact, this coupling scheme is often believed to be similar to the common node approach. In this study it is illustrated that in comparison to the common node approach, the head continuity at the surface–subsurface interface is formulated more correctly in the consistent dual node approach. Numerical experiments indicate that the consistent dual node approach is less sensitive to the vertical discretization when simulating excess infiltration. It is also found that the consistent dual node approach can be advantageous in terms of numerical efficiency.

Numerical solution of the optimal speed problem with a phase constraint for one parabolic equation

An algorithm for the numerical solution of the optimal speed problem with phase constraint for a parabolic equation describing the heat conduction processes in inhomogeneous media is proposed. To solve the problems with the use of first-order optimization methods and finite differences on non-uniform grids, analytical formulas are obtained for the components of the gradient of the functional with respect to controllable functions. A method is proposed for selecting initial approximations for optimal controls and a step in time at each iteration, which makes it possible to accelerate the computation process. To achieve the specified accuracy, the speed problem requires 6 iterations and . Based on the analysis of the results of numerical experiments, the influence of various parameters on the iterative process is investigated and recommendations are developed on the use of the proposed algorithm. In optimal control problems, the total number of iterations in option a by the conditional gradient method is 110 and the gradient projection method is 108. In option b, the total number of iterations is CGM – 81, GPM – 64, i. e., the total number of iterations in the optimal control problem in method b the choice of the initial approximation is much less than in variant a. The optimal speed control, obtained by both methods, is close enough to test controls. Numerical experiments are also carried out in the case when the control-optimal controls have two switching points. However, the nature of the results obtained does not change. The proposed algorithm can be used to determine the optimal regime and time of thermal conductivity processes in inhomogeneous media.

PyJac: Analytical Jacobian generator for chemical kinetics

Accurate simulations of combustion phenomena require the use of detailed chemical kinetics in order to capture limit phenomena such as ignition and extinction as well as predict pollutant formation. However, the chemical kinetic models for hydrocarbon fuels of practical interest typically have large numbers of species and reactions and exhibit high levels of mathematical stiffness in the governing differential equations, particularly for larger fuel molecules. In order to integrate the stiff equations governing chemical kinetics, generally reactive-flow simulations rely on implicit algorithms that require frequent Jacobian matrix evaluations. Some in situ and a posteriori computational diagnostics methods also require accurate Jacobian matrices, including computational singular perturbation and chemical explosive mode analysis. Typically, finite differences numerically approximate these, but for larger chemical kinetic models this poses significant computational demands since the number of chemical source term evaluations scales with the square of species count. Furthermore, existing analytical Jacobian tools do not optimize evaluations or support emerging SIMD processors such as GPUs. Here we introduce pyJac, a Python-based open-source program that generates analytical Jacobian matrices for use in chemical kinetics modeling and analysis. In addition to producing the necessary customized source code for evaluating reaction rates (including all modern reaction rate formulations), the chemical source terms, and the Jacobian matrix, pyJac uses an optimized evaluation order to minimize computational and memory operations. As a demonstration, we first establish the correctness of the Jacobian matrices for kinetic models of hydrogen, methane, ethylene, and isopentanol oxidation (number of species ranging 13–360) by showing agreement within 0.001% of matrices obtained via automatic differentiation. We then demonstrate the performance achievable on CPUs and GPUs using pyJac via matrix evaluation timing comparisons; the routines produced by pyJac outperformed first-order finite differences by 3–7.5 times and the existing analytical Jacobian software TChem by 1.1–2.2 times on a single-threaded basis. It is noted that TChem is not thread-safe, while pyJac is easily parallelized, and hence can greatly outperform TChem on multicore CPUs. The Jacobian matrix generator we describe here will be useful for reducing the cost of integrating chemical source terms with implicit algorithms in particular and algorithms that require an accurate Jacobian matrix in general. Furthermore, the open-source release of the program and Python-based implementation will enable wide adoption. Program summaryProgram Title: pyJacProgram Files doi:http://dx.doi.org/10.17632/mmr3z8j2m5.1Licensing provisions: MIT LicenseProgramming language: PythonExternal routines/libraries: required: NumPy; optional: Cython, Cantera, PyYAML, AdeptNature of problem: Automatic generation of source code to evaluate Jacobian matrices for chemical kinetic modelsSolution method: Chemical kinetic model interpreted from input file(s), and partial derivatives and necessary supporting functions are written to file based on a theoretical derivation.Additional comments: This paper describes pyJac v1.0.2, available at http://dx.doi.org/10.5281/zenodo.251144. Current version and support available at https://github.com/SLACKHA/pyjac/

Local convergency rate of MSE in density estimation using the second-order modulus of smoothness

ABSTRACTIn this article, the local convergence rate of the mean square error (MSE) corresponding to a delta sequence-based density estimators is investigated by using second-order modulus of continuity type majorants. We look at the rate of convergency of the MSE of estimator for densities belonging to the class of functions which are defined by the second-order finite differences. The main contribution of this study is to obtain stronger convergence rate of a MSE by relaxing the second-order differentiation condition when compared with the class of density functions defined by the first-order finite differences.

Universal Raising and Lowering Operators for a Discrete Energy Spectrum

We consider the first-order finite-difference expression of the commutator between d / dx and x. This is the appropriate setting in which to propose commutators and time operators for a quantum system with an arbitrary potential function and a discrete energy spectrum. The resulting commutators are identified as universal lowering and raising operators. We also find time operators which are finite-difference derivations with respect to the energy. The matrix elements of the commutator in the energy representation are analyzed, and we find consistency with the equality \([\hat{T},\hat{H}]=i\hbar \). We apply the theory to the particle in an infinite well and for the Harmonic oscillator as examples.

Fast multipole accelerated dual reciprocity boundary element method for diffusion problems

A fast multipole boundary element method (FMBEM) is presented for diffusion problems based on a dual reciprocity formulation. In the dual reciprocity formulation, domain integrals that arise from solving the time-dependent boundary value problems are transformed into boundary integrals by constructing particular solutions. The time-derivatives in the governing differential equation are approximated with a first-order finite difference time-stepping scheme. Discontinuous linear elements, which are known to give more accurate results than constant or linear elements, are used in the implementation to discretize the boundary integral equations, in combination with the fast multipole method for speeding up the solution. Three numerical examples of diffusion are presented. The performance of the developed FMBEM is compared with that of a conventional BEM and a commercial finite element program. The results show that the developed FMBEM can be a reliable and efficient tool for solving diffusion problems.

Hencky Bar-Chain Model for Buckling and Vibration of Beams with Elastic End Restraints

This paper presents the Hencky bar-chain model (HBM) for buckling and vibration analyses of Euler–Bernoulli beams with elastic end restraints. The Hencky bar-chain comprises rigid beam segments (of length a = L/n where L is the total length of beam and n the number of beam segments) connected by frictionless hinges with elastic rotational springs of stiffness EI/a where EI is the flexural rigidity of the beam. The elasticity and the mass of the beam are concentrated at the hinges with rotational springs. The key contribution of this paper lies in the modeling of the elastic end restraints of the Hencky bar-chain that will simulate the same buckling and vibration results as that furnished by the first-order central finite difference beam model (FDM) which was earlier shown to be analogous to the HBM. The establishment of such a physical discrete beam model allows one to obtain solutions for beam-like structure with repetitive cells (or elements) as well as to calibrate the Eringen's coefficient e0 in the nonlocal beam theory that captures the small length scale effect.