Second-order Numerical Scheme Research Articles

RAMMS: Numerical simulation of dense snow avalanches in three-dimensional terrain

Abstract Numerical avalanche dynamics models have become an essential part of snow engineering. Coupled with field observations and historical records, they are especially helpful in understanding avalanche flow in complex terrain. However, their application poses several new challenges to avalanche engineers. A detailed understanding of the avalanche phenomena is required to construct hazard scenarios which involve the careful specification of initial conditions (release zone location and dimensions) and definition of appropriate friction parameters. The interpretation of simulation results requires an understanding of the numerical solution schemes and easy to use visualization tools. We discuss these problems by presenting the computer model RAMMS, which was specially designed by the SLF as a practical tool for avalanche engineers. RAMMS solves the depth-averaged equations governing avalanche flow with accurate second-order numerical solution schemes. The model allows the specification of multiple release zones in three-dimensional terrain. Snow cover entrainment is considered. Furthermore, two different flow rheologies can be applied: the standard Voellmy–Salm (VS) approach or a random kinetic energy (RKE) model, which accounts for the random motion and inelastic interaction between snow granules. We present the governing differential equations, highlight some of the input and output features of RAMMS and then apply the models with entrainment to simulate two well-documented avalanche events recorded at the Vallee de la Sionne test site.

Fertigation in Furrows and Level Furrow Systems. I: Model Description and Numerical Tests

The simulation of fertigation in furrows and level furrow systems faces a number of problems resulting in relevant restrictions to its widespread application. In this paper, a simulation model is proposed that addresses some of these problems by: (1) implementing an infiltration model that adjusts to the variations in wetted perimeter; (2) using a friction model that adjusts to different flows and which uses an absolute roughness parameter; (3) adopting an equation for the estimation of the longitudinal diffusion coefficient; and (4) implementing a second-order TVD numerical scheme and specific treatments for the boundary conditions and the junctions. The properties of the proposed model were demonstrated using three numerical tests focusing on the numerical scheme and the treatments. The application of the model to the simulation of furrows and furrow systems is presented in a companion paper, in which the usefulness of the innovative aspects of the proposed model is demonstrated.

A Second-Order Algorithm for Solving Dynamic Cell Membrane Equations

This paper describes an extension of the so-called Rush-Larsen scheme, which is a widely used numerical method for solving dynamic models of cardiac cell electrophysiology. The proposed method applies a local linearization of nonlinear terms in combination with the analytical solution of linear ordinary differential equations to obtain a second-order accurate numerical scheme. We compare the error and computational load of the second-order scheme to the original Rush-Larsen method and a second-order Runge-Kutta (RK) method. The numerical results indicate that the new method outperforms the original Rush-Larsen scheme for all the test cases. The comparison with the RK solver reveals that the new method is more efficient for stiff problems.

Accuracy of Kinematic Wave and Diffusion Wave Approximations for Flood Routing. I: Steady Analysis

The applicability of the kinematic wave (KW) and diffusive wave (DW) approximations was investigated for steady flow in prismatic channels by using a second-order two-step Lax-Wendroff numerical scheme coupled with the characteristic method at the boundaries. Two types of downstream boundary conditions, critical-flow depth and zero-flow depth gradient, were considered together with the condition of discharge hydrograph reaching the steady state at the upstream end. The role of inertial, pressure, friction, and gravity forces was investigated for 16 test cases defined through the kinematic wave number K and the Froude number F0 , whose range was assumed to be (3–30) and (0.1–1), respectively. Errors were computed by comparing the steady dimensionless profiles of the flow depth with those estimated by the dynamic wave solution. The accuracy of the two approximations was assessed through the mean value e of the magnitudes of errors computed for the channel region where the solution was not significantly infl...

A second-order accurate non-linear difference scheme for the N -component Cahn–Hilliard system

We consider a second-order conservative nonlinear numerical scheme for the N -component Cahn–Hilliard system modeling the phase separation of a N -component mixture. The scheme is based on a Crank–Nicolson finite-difference method and is solved by an efficient and accurate nonlinear multigrid method. We numerically demonstrate the second-order accuracy of the numerical scheme. We observe that our numerical solutions are consistent with the exact solutions of linear stability analysis results. We also describe numerical experiments such as the evolution of triple junctions and the spinodal decomposition in a quaternary mixture. We investigate the effects of a concentration dependent mobility on phase separation.

Numerical Study of the Temperature Separation in the Ranque-Hilsch Vortex Tube

This research has been carried out in order to provide an understanding of the physical behaviors of the flow variation of pressure and temperature in a vortex tube. A computational fluid dynamics model is used to predict the flow fields and the associated temperature separation within a Ranque-Hilsch vortex tube. The CFD model is a steady axisymmetric model (with swirl) that utilizes the standard k-e turbulence model. The second-order numerical schemes, was used to carry out all the computations. Vortex tube with a circumferential inlet stream and an axial (cold) outlet stream and a axial (hot) outlet stream was considered. Performance curves (temperature separation versus cold outlet mass fraction) were obtained for a specific vortex tube with a given inlet mass flow rate. Simulations have been carried out for varying amounts of cold outlet mass flow rates. The model results has reasonable agreement with experimental data.

Parameter-Free Second Order Numerical Scheme for Constrained Multibody Dynamical Systems

This paper describes a second-order numerical scheme for constrained multibody dynamics that is simple to implement and does not require the selection of parameters for constraint satisfaction. The method is a predictor-corrector type of method, and both the predictor and the corrector steps require solution of a symmetric positive-definite linear system with no iteration of nonlinear equations. The basic philosophy of the method is to satisfy the constraint equations to the same order as the discretization of the ordinary differential equations.

Numerical simulation of 3D turbulent isothermal flow in a vortex combustor

Numerical simulations of strongly swirling turbulent flows in a vortex combustor (VC) are conducted. A comprehensive investigation of a three-dimensional isothermal VC flow using three first-order turbulence models: the standard k–ε turbulence model, Renormalized Group (RNG) k–ε model and shear stress transport (SST) k–ω model; and a second-order turbulence model, Reynolds stress model (RSM) together with a second-order numerical differencing scheme is conducted in the present work. The computation indicates that the RSM is superior to the other turbulence models in capturing the swirl flow effect in comparison with measurements. The numerical results for the VC flow provide the characteristics of the flow in terms of relevant parameters for the VC design and operation, composed of axial and tangential velocities, pressure fields, and turbulence kinetic energy.

Second-order accurate integration algorithms for von-Mises plasticity with a nonlinear kinematic hardening mechanism

Two second-order numerical schemes for von-Mises plasticity with a combination of linear isotropic and nonlinear kinematic hardening are presented. The first scheme is the generalized midpoint integration procedure, originally introduced by Ortiz and Popov in 1985, detailed and applied here to the case of Armstrong–Frederick nonlinear kinematic hardening. The second algorithm is based on the constitutive model exponential-based reformulation and on the integration procedure previously introduced by the authors in the context of linearly hardening materials. There are two main targets to the work. Firstly, we wish to extensively test the generalized midpoint procedure since in the scientific literature no thorough numerical testing campaign has been carried out on this second-order algorithm. Secondly, we wish to extend the exponential-based integration technique also to nonlinear hardening materials. A wide numerical investigation is carried out in order to compare the performance of the two methods.

Large eddy simulation of the generation and breakdown of a tumbling flow

Large eddy simulations (LES) are performed in order to reproduce the generation and the breakdown of a tumbling motion in the simplified model engine [Borée, J., Maurel, S., Bazile, R., 2002. Disruption of a compressed vortex. Phys. Fluids, 14 (7) 2543–2556]. A second-order accurate numerical scheme is applied in conjunction with a mixed finite volume/finite element formulation adapted for unstructured deforming meshes. Subgrid terms are kept as simple as possible with a Smagorinsky model in order to build a methodology devoted to engine-like flows. The main statistical quantities, such as mean velocity and turbulent kinetic energy, are obtained from a set of independent cycles and compared to experiments. Important experimental features, such as oscillations of the intake jet, vortex precession and a turbulent kinetic energy peak near the vortex core, are well reproduced.

A numerical method based on polynomial sextic spline functions for the solution of special fifth-order boundary-value problems

A second-order accurate numerical scheme is presented for the solution of special fifth-order boundary-value problems with two-point-boundary conditions. The polynomial sextic spline functions are applied to construct the numerical algorithm. Convergence of the method is discussed through standard convergence analysis. A numerical illustration is given to show the pertinent features of the technique.

A staggered-grid numerical algorithm for the extended Boussinesq equations

A second-order accurate numerical scheme is developed to solve Nwogu’s extended Boussinesq equations. A staggered-grid system is introduced with the first-order spatial derivatives being discretized by the fourth-order accurate finite-difference scheme. For the time derivatives, the fourth-order accurate Adams predictor–corrector method is used. The numerical method is validated against available analytical solutions, other numerical results of Navier–Stokes equations, and experimental data for both 1D and 2D nonlinear wave transformation problems. It is shown that the new algorithm has very good conservative characteristics for mass calculation. As a result, the model can provide accurate and stable results for long-term simulation. The model has proven to be a useful modeling tool for a wide range of water wave problems.

A parallel implicit method for the direct numerical simulation of wall-bounded compressible turbulence

A new second-order accurate implicit temporal numerical scheme for the direct numerical simulation of turbulent flows is presented. The formulation of the implicit method and the corresponding tunable parameters are introduced. The numerical simulation results are compared with the results given by explicit Runge–Kutta schemes, theoretical results, and published experimental and numerical data. An assessment of the accuracy and performance of the method to simulate turbulent flows is made for temporally decaying isotropic turbulence and subsonic and supersonic turbulent boundary layers. Whereas no significant advantage over typical explicit time integration methods are found for the incompressible flows; it is shown that the implicit scheme yields significant reduction in computer cost while assuring time-accurate solutions for compressible turbulence simulations.

A class of methods based on non-polynomial sextic spline functions for the solution of a special fifth-order boundary-value problems

A family of fourth and second-order accurate numerical schemes is presented for the solution of fifth-order boundary-value problems with two-point-boundary conditions. The non-polynomial sextic spline functions are applied to construct the numerical algorithms. This approach generalizes polynomial spline algorithms, and provides solution at every point of range of integration. Convergence of the methods is discussed through standard convergence analysis. A numerical illustration is given to show the pertinent features of the technique.

A second-order multi-fluid model for evaporating sprays

The aim of this paper is to present a method using both the ideas of sectional approach and moment methods in order to accurately simulate evaporation phenomena in gas-droplets flows. Using the underlying kinetic interpretation of the sectional method [Y. Tambour, Combust. Flame 60 (1985) 15–28] exposed in [F. Laurent and M. Massot, Combust. Theory Model. 5 (2001) 537–572], we propose an extension of this approach based on a more accurate representation of the droplet size number density in each section ensuring the exact conservation of two moments (as opposed to only one moment used in the classical approach). A corresponding second-order numerical scheme, with respect to space and droplet size variables, is also introduced and can be proved to be positive and to satisfy a maximum principle on the velocity and the mean droplet mass under a suitable CFL-like condition. Numerical simulations have been performed and the results confirm the accuracy of this new method even when a very coarse mesh for the droplet size variable (i.e. : a low number of sections) is used.

Balanced finite volume WENO and central WENO schemes for the shallow water and the open-channel flow equations

The goal of this work is to extend finite volume WENO and central WENO schemes to the hyperbolic balance laws with geometrical source term and spatially variable flux function. In particular, we apply proposed schemes to the shallow water and the open-channel flow equations where the source term depends on the channel geometry. For obtaining stable numerical schemes that are free of spurious oscillations, it becomes crucial to use the decomposed source term evaluation, which maintains the balancing between the flux gradient and the source term. In addition, the open-channel flow equations contain spatially variable flux function. The appropriate definitions of the terms that arise in the source term decomposition, in combination with the Roe approximate Riemann solver that includes the spatial derivative of the flux function, lead to the finite volume WENO scheme that satisfies the exact conservation property – the property of preserving the quiescent flow exactly. When the central WENO schemes are applied, additional reformulations are introduced for the transition from the staggered values to the nonstaggered ones and vice versa by using the WENO reconstruction procedure. The proposed central WENO schemes also preserve the quiescent flow, but only in prismatic channels. In various test problems the obtained balanced schemes show improvements in comparison with the standard versions of the proposed type schemes, as well as with some other first- and second-order numerical schemes.

Weighted Averaged Flux-Type Scheme for Shallow-Water Equations with Fractional Step Method

A numerical model describing two-dimensional fluid motions has been developed on an unstructured grid system. By using a fractional step method, a two-dimensional problem governed by the two-dimensional shallow-water equations is treated as two one-dimensional problems. Thus it is possible to simulate two-dimensional numerical problems with a higher computational efficiency. One-dimensional problems are solved by using an upwind total variation diminishing version of the second-order weighted averaged flux method with an approximate Riemann solver. Numerical oscillations commonly observed in second-order numerical schemes are controlled by exploiting a flux limiter. For the general purpose, the model can simulate on an arbitrary topography, treat a moving boundary, and resolve a shock. Five ideal and practical problems are tested. Very accurate results are observed.

WAF 기법을 이용한 천수방정식 해석

본 연구에서는 비구조격자 좌표계상에서 천수방정식 해석을 위한 수치모형을 개발하였다. Fractional step method를 이용하여 이차원의 천수방정식을 두개의 일차원 문제로 분리하여 계산 효율이 우수한 수치연산을 수행하였다. 분리된 일차원 문제는 이차정확도의 TVD 기법을 이용하여 해석하였다. TVD 기법은 HLLC 기법을 이용하여 Riemann 해를 구한후 WAF 기법을 기반으로 이용하였으며 WAF 흐름율 제한자를 이용하여 이차정확도 문제에 수반되는 비물리적인 수치진동을 제어하였다. 개발된 모형을 다양한 문제에 적용한 결과, 해석해와 계산된 결과가 매우 잘 일치하였으며, 본 모형이 불연속해나 상류사류의 혼합흐름 및 이동경계 문제 등에 이용될 수 있음을 보였다. A numerical model for the solution of two-dimensional free surface flow is developed on unstructured grid. By using fractional step method, the two-dimensional shallow water equations (SWE) are treated as two one-dimensional problems. Thus, it is possible to simulate computational hydraulic problems with higher computational efficiency. The one-dimensional problems are solved using upwind TVD version of second-order Weighted Averaged Flux (WAF) scheme with HLLC approximate Riemann solver. The numerical oscillations which are common with second-order numerical scheme are controlled by exploiting WAF flux limiter, Some idealized test problems are solved using this model and very accurate and stable solutions are obtained. It can be concluded as an efficient implement for the computation of SWE including dam break problems that concerning discontinuities, subcritical and supercritical flows and complex domain.

Weak first- and second-order numerical schemes for stochastic differential equations appearing in Lagrangian two-phase flow modeling

Weak first- and second-order numerical schemes are developed to integrate the stochastic differential equations that arise in mean-field - pdf methods (Lagrangian stochastic approach) for modeling polydispersed turbulent two-phase flows. These equations present several challenges, the foremost being that the problem is characterized by the presence of different time scales that can lead to stiff equations, when the smallest time-scale is significantly less than the time-step of the simulation. The numerical issues have been detailed by Minier [Monte Carlo Meth. and Appl. 7 295-310, (2000)] and the present paper proposes numerical schemes that satisfy these constraints. This point is really crucial for physical and engineering applications, where various limit cases can be present at the same time in different parts of the domain or at different times. In order to build up the algorithm, the analytical solutions to the equations are first carried out when the coefficients are constant. By freezing the coefficients in the analytical solutions, first and second order unconditionally stable weak schemes are developed. A prediction/ correction method, which is shown to be consistent for the present stochastic model, is used to devise the second-order scheme. A complete numerical investigation is carried out to validate the schemes, having included also a comprehensive study of the different error sources. The final method is demonstrated to have the required stability, accuracy and efficiency.

Grid Convergence Error Analysis for Mixed-Order Numerical Schemes

Grid Convergence Error Analysis for Mixed-Order Numerical Schemes