Accurate Solution Algorithm Research Articles

Assessment of the cubic Fokker–Planck–DSMC hybrid method for hypersonic rarefied flows past a cylinder

Hypersonic vehicles experience a wide range of Knudsen number regimes due to changes in atmospheric density. The Direct Simulation Monte Carlo (DSMC) method is physically accurate for all flow regimes, however it is relatively computationally expensive in high density, and low Knudsen number regions. Recent advances in the Fokker–Planck (FP) kinetic models have addressed this issue by approximating the particle collisions involved in the Boltzmann collision integral with continuous stochastic processes. Furthermore, a coupled FP–DSMC solution method has been devised aiming at a universally efficient yet accurate solution algorithm for rarefied gas flows. Well known Lofthouse case of a generic hypersonic flow about a cylinder (Mach 10, Kn 0.01, Argon) is selected to investigate the performance of a hybrid FP–DSMC implementation. The effect of molecular potential on the accuracy of the scheme is mainly analyzed. Furthermore, spatial resolution of cubic FP scheme is studied. Finally, detailed study of accuracy and efficiency of FP–DSMC hybrid scheme is discussed. It is found that the presented adaptive grid together with the FP–DSMC method results in a factor of six speed up for considered hypersonic flow about a cylinder.

A continuous stochastic model for non-equilibrium dense gases

While accurate simulations of dense gas flows far from the equilibrium can be achieved by direct simulation adapted to the Enskog equation, the significant computational demand required for collisions appears as a major constraint. In order to cope with that, an efficient yet accurate solution algorithm based on the Fokker-Planck approximation of the Enskog equation is devised in this paper; the approximation is very much associated with the Fokker-Planck model derived from the Boltzmann equation by Jenny et al. [“A solution algorithm for the fluid dynamic equations based on a stochastic model for molecular motion,” J. Comput. Phys. 229, 1077–1098 (2010)] and Gorji et al. [“Fokker–Planck model for computational studies of monatomic rarefied gas flows,” J. Fluid Mech. 680, 574–601 (2011)]. The idea behind these Fokker-Planck descriptions is to project the dynamics of discrete collisions implied by the molecular encounters into a set of continuous Markovian processes subject to the drift and diffusion. Thereby, the evolution of particles representing the governing stochastic process becomes independent from each other and thus very efficient numerical schemes can be constructed. By close inspection of the Enskog operator, it is observed that the dense gas effects contribute further to the advection of molecular quantities. That motivates a modelling approach where the dense gas corrections can be cast in the extra advection of particles. Therefore, the corresponding Fokker-Planck approximation is derived such that the evolution in the physical space accounts for the dense effects present in the pressure, stress tensor, and heat fluxes. Hence the consistency between the devised Fokker-Planck approximation and the Enskog operator is shown for the velocity moments up to the heat fluxes. For validation studies, a homogeneous gas inside a box besides Fourier, Couette, and lid-driven cavity flow setups is considered. The results based on the Fokker-Planck model are compared with respect to benchmark simulations, where good agreement is found for the flow field along with the transport properties.

A fast sweeping algorithm for accurate solution of the tilted transversely isotropic eikonal equation using factorization

Traveltime computation is essential for many seismic data processing applications and velocity analysis tools. High-resolution seismic imaging requires eikonal solvers to account for anisotropy whenever it significantly affects the seismic wave kinematics. Moreover, computation of auxiliary quantities, such as amplitude and take-off angle, relies on highly accurate traveltime solutions. However, the finite-difference-based eikonal solution for a point-source initial condition has upwind source singularity at the source position because the wavefront curvature is large near the source point. Therefore, all finite-difference solvers, even the high-order ones, show inaccuracies because the errors due to source-singularity spread from the source point to the whole computational domain. We address the source-singularity problem for tilted transversely isotropic (TTI) eikonal solvers using factorization. We solve a sequence of factored tilted elliptically anisotropic (TEA) eikonal equations iteratively, each time by updating the right-hand-side function. At each iteration, we factor the unknown TEA traveltime into two factors. One of the factors is specified analytically, such that the other factor is smooth in the source neighborhood. Through this iterative procedure, we obtain an accurate solution to the TTI eikonal equation. Numerical tests show significant improvement in accuracy due to factorization. The idea can be easily extended to compute accurate traveltimes for models with lower anisotropic symmetries, such as orthorhombic, monoclinic, or even triclinic media.

An efficient particle Fokker–Planck algorithm for rarefied gas flows

This paper is devoted to the algorithmic improvement and careful analysis of the Fokker–Planck kinetic model derived by Jenny et al. [1] and Gorji et al. [2]. The motivation behind the Fokker–Planck based particle methods is to gain efficiency in low Knudsen rarefied gas flow simulations, where conventional direct simulation Monte Carlo (DSMC) becomes expensive. This can be achieved due to the fact that the resulting model equations are continuous stochastic differential equations in velocity space. Accordingly, the computational particles evolve along independent stochastic paths and thus no collision needs to be calculated. Therefore the computational cost of the solution algorithm becomes independent of the Knudsen number. In the present study, different computational improvements were persuaded in order to augment the method, including an accurate time integration scheme, local time stepping and noise reduction. For assessment of the performance, gas flow around a cylinder and lid driven cavity flow were studied. Convergence rates, accuracy and computational costs were compared with respect to DSMC for a range of Knudsen numbers (from hydrodynamic regime up to above one). In all the considered cases, the model together with the proposed scheme give rise to very efficient yet accurate solution algorithms.

A Discontinuous Galerkin Finite Element Method for Dynamic of Fully Saturated Soil / Rzwiazanie Zadania Dynamiki Całkowicie Nawodnionego Gruntu Przy Zastosowaniu Mes Z Nieciagłym Sformułowaniem Galerkina W Czasie

AbstractThe fully coupled, porous solid-fluid dynamic field equations with u-p formulation are used in this paper to simulate pore fluid and solid skeleton responses. The present formulation uses physical damping, which dissipates energy by velocity proportional damping. The proposed damping model takes into account the interaction of pore fluid and solid skeleton. The paper focuses on formulation and implementation of Time Discontinuous Galerkin (TDG) methods for soil dynamics in the case of fully saturated soil. This method uses both the displacements and velocities as basic unknowns and approximates them through piecewise linear functions which are continuous in space and discontinuous in time. This leads to stable and third-order accurate solution algorithms for ordinary differential equations. Numerical results using the time-discontinuous Galerkin FEM are compared with results using a conventional central difference, Houbolt, Wilson θ, HHT- α, and Newmark methods. This comparison reveals that the time-discontinuous Galerkin FEM is more stable and more accurate than these traditional methods.

Finite Element Method For Capturing Hydraulic Jump In Open Channel

An efficient and accurate solution algorithm is developed for 1D unsteady flow problems in open channels. The method was the finite element solution of the Saint-Venant equations using the bi-tridiagonal matrix algorithm. For numerical stability, the time step was a function of depth and flow velocity, to avoid restriction due to the wave celerity variation in the computational analysis when using the method of characteristics. Validity of numerical models is obvious compared with the experimental data. Two examples were solved with the finite element method (FEM). The first involves routing a discharge hydrograph down a rectangular channel. The second example consists of routing a sudden shut off of all flow at the downstream end of the channel. These examples are taken from the text book of Streeter and Wylie [Hydraulic transients (Ann Arbor: F.E.B. Press, 1983)]. Numerical results agree well with those obtained by these authors and show that the proposed method is consistent, accurate and highly stable in capturing discontinuities propagation in free surface flows.

Demand and generation cost uncertainty modelling in power system optimization studies

This paper describes the formulations and the solution algorithms developed to include uncertainties in the generation cost function and in the demand on DC OPF studies. The uncertainties are modelled by trapezoidal fuzzy numbers and the solution algorithms are based on multiparametric linear programming techniques. These models are a development of an initial formulation detailed in several publications co-authored by the second author of this paper. Now, we developed a more complete model and a more accurate solution algorithm in the sense that it is now possible to capture the widest possible range of values of the output variables reflecting both demand and generation cost uncertainties. On the other hand, when modelling simultaneously demand and generation cost uncertainties, we are representing in a more realistic way the volatility that is currently inherent to power systems. Finally, the paper includes a case study to illustrate the application of these models based on the IEEE 24 bus test system.

A split-characteristic finite element model for 1-D unsteady flows

An efficient and accurate solution algorithm was proposed for 1-D unsteady flow problems widely existing in hydraulic engineering. Based on the split-characteristic finite element method, the numerical model with the Saint-Venant equations of 1-D unsteady flows was established. The assembled finite element equations were solved with the tri-diagonal matrix algorithm. In the semi-implicit and explicit scheme, the critical time step of the method was dependent on the space step and flow velocity, not on the wave celerity. The method was used to eliminate the restriction due to the wave celerity for the computational analysis of unsteady open-channel flows. The model was verified by the experimental data and theoretical solution and also applied to the simulation of the flow in practical river networks. It shows that the numerical method has high efficiency and accuracy and can be used to simulate 1-D steady flows, and unsteady flows with shock waves or flood waves. Compared with other numerical methods, the algorithm of this method is simpler with higher accuracy, less dissipation, higher computation efficiency and less computer storage.

Electron emission during combined attosecond pulses

When a strong attosecond laser pulse acts simultaneously with a fast ionic projectile, the electronic response of an atom changes and leads to a drastic change in the final momenta, which may be observed in recoil-ion momentum experiments. The nonlinear response to the combined fields leads in certain cases to ionization probabilities more than three times larger than the probabilities by either of the two fields individually. The results are based on an algorithm for accurate solution of the time-dependent Schr\"odinger equation in three dimensions, which has the property that it requires CPU time comparable with that of two-dimensional methods.

Inspecting the Mechanism: closed‐form Solutions for Asset Prices in Real Business Cycle Models

We derive closed-form solutions for asset prices in an RBC economy. The equations are based on a log-linear solution of the RBC model and allow a clearer understanding of the determination of risk premia in models with production. We demonstrate not only why the premium of equity over the risk-free rate is small but also why the premium of equity over a real long-term bond is small and often negative. In particular, risk premia for equity and long real bonds are negative when technology shocks are permanent. Recently, a growing literature has explored the asset pricing implications of real business cycle (RBC) models. Examples are Rouwenhorst (1995), Jermann (1998) and Boldrin et al. (1995). 1 From a methodological point of view, models with production allow a more realistic modelling of consumption and dividends than do the pure exchange economies of Lucas (1978). However, explaining the behaviour of asset prices in production economies is also more challenging. For example, in an exchange economy, an increase in risk premia can be obtained by increasing risk aversion. This is not necessarily true in production economies, since agents can choose a smoother consumption path by substituting between work and leisure in response to productivity shocks. The papers just cited find that RBC models tend to generate counterfactual asset pricing implications unless some extreme form of rigidities is introduced. This paper attempts to provide a clearer understanding of the challenges posed by asset pricing behaviour for RBC models. Instead of solving a standard model numerically and simulating it over a large number of parameter permutations, we argue here that this understanding can be better developed by analysing approximate closed-form solutions for prices of a variety of financial assets. Although there are now many accurate solution algorithms available for solving RBC models (e.g., see the overview in Taylor and Uhlig (1990)), the approximate closed-form expressions obtained here make the relationship between asset prices and exogenous technology shocks particularly transparent by providing simple analytical expressions that decompose the effect of technology shocks on asset prices into (i) a direct effect due to the shock itself and (ii) an indirect effect stemming from capital accumulation.

Analysis of the local truncation error in the pressure‐free projection method for incompressible flows: a new accurate expression of the intermediate boundary conditions

AbstractThe numerical integration of the Navier–Stokes equations for incompressible flows demands efficient and accurate solution algorithms for pressure–velocity splitting. Such decoupling was traditionally performed by adopting the Fractional Time‐Step Method that is based on a formal separation between convective–diffusive momentum terms from the pressure gradient term. This idea is strictly related to the fundamental theorem on the Helmholtz–Hodge orthogonal decomposition of a vector field in a finite domain, from which the name projection methods originates. The aim of this paper is to provide an original evaluation of the local truncation error (LTE) for analysing the actual accuracy achieved by solving the de‐coupled system. The LTE sources are formally subdivided in two categories: errors intrinsically due to the splitting of the original system and errors due to the assignment of the boundary conditions. The main goal of the present paper consists in both providing the LTE analysis and proposing a remedy for the inaccuracy of some types of intermediate boundary conditions associated with the prediction equation. Such evaluations will be directly performed in the physical space for both the time continuous formulation and the finite volume discretization along with the discrete Adams–Bashforth/Crank–Nicolson time integration. A new proposal for a boundary condition expression, congruent with the discrete prediction equation is herein derived, fulfilling the goal of accomplishing the closure of the problem with fully second order accuracy. In our knowledge, this procedure is new in the literature and can be easily implemented for confined flows. The LTE is clearly highlighted and many computations demonstrate that our proposal is efficient and accurate and the goal of adopting the pressure‐free method in a finite domain with fully second order accuracy is reached. Copyright © 2003 John Wiley & Sons, Ltd.

A particular integral BEM/time-discontinuous FEM methodology for solving 2-D elastodynamic problems

This study proposes a time-discontinuous Galerkin finite element method (FEM) for solving second-order ordinary differential equations in the time domain. The equations are formulated using a particular integral boundary element method (BEM) in the space domain for elastodynamic problems. The particular integral BEM technique depends only on elastostatic displacement and traction fundamental solutions, without resorting to commonly used complex fundamental solutions for elastodynamic problems. Based on the time-discontinuous Galerkin FEM, the unknown displacements and velocities are approximated as piecewise linear functions in the time domain, and are permitted to be discontinuous at the discrete time levels. This leads to stable and third-order accurate solution algorithms for ordinary differential equations. Numerical results using the time-discontinuous Galerkin FEM are compared with results using a conventional finite difference method (the Houbolt method). Both methods are employed for a particular integral BEM analysis in elastodynamics. This comparison reveals that the time-discontinuous Galerkin FEM is more stable and more accurate than the traditional finite difference methods.

Simplified non‐linear dynamic analysis of base isolated buildings subjected to general plane motion

Considers hysteretic non‐linear models for modelling horizontal force‐displacement characteristics of an isolation system subjected to general plane motion. Simple close form solution of the stiff differential equation of hysteretic model for forces mobilized in the non‐linear elements of the base isolation system are obtained. Simulates experimental shear force‐displacement loops obtained from the bi‐axial tests by different investigators. Both the experimental and the simulated hysteresis loops are found to be in good agreement. Develops a unified solution algorithm for computation of response of different types of base isolated buildings, considering non‐linear behaviour of the isolation systems, subjected to multi‐directional motion. The solution algorithm is based on the implicit‐implicit partitioned Newmark’s method in predictor‐corrector form. Response of the base isolated symmetrical building as obtained from the solution algorithm and the computer programs developed are in good agreement with that obtained from the more complex numerical studies reported in the literature. Response of a three storeyed symmetrical building isolated by pure friction isolators and laminated rubber bearings has been obtained using this simple yet accurate solution algorithm.

Adaptive refinement and nonlinear fluid problems

Abstract Two principal and related topics are considered: (1) adaptive mesh refinement for finite element computations, and (2) mesh refinement specifically for nonlinear flow problems. In the first instance the residual and associated trace theorems for variational problems are introduced to relate the solution error to a computable residual. This provides a theoretical basis for a mesh refinement strategy. A corresponding adaptive refinement procedure that automatically and selectively refines the mesh is formulated and implemented for a class of nonlinear transport problems in chemical engineering. In particular, a nonlinear problem with a boundary-layer solution is investigated. The strategy of interweaving Newton solution and mesh refinement proves particularly efficient. Two-dimensional compressible and transonic flows are next examined. Mesh refinement of subrogions of the flow field is applied to yield high solution accuracy near a singularity for both the linear and nonlinear flows. Refinement and Newton iteration are combined, together with Mach number parameterization, to determine an efficient and accurate solution algorithm. Similar points for nonlinear viscous problems are also reviewed.