Accurate Numerical Scheme Research Articles

A Comparison on Solutions of Fifth-Order Boundary- Value Problems

A fast and accurate numerical scheme for the solution of fifth-order boundary-value problems has been investigated in this work. We apply the reproducing kernel method (RKM) for solving this problem. The analytical results of the equations have been acquired in terms of convergent series with easily computable components. We compare our results with the numerical methods: B-spline method, decomposition method, variational iteration method, Sinc-Galerkin method and homotopy perturbation method. The comparison of the results with exact ones is made to confirm the validity and efficiency.

Discretization of the divergence formulation of the bed slope term in the shallow-water equations and consequences in terms of energy balance

In this research, the influence on energy balance of the discretization scheme of the divergence formulation of the bed slope term in the shallow-water equations is analyzed theoretically (for a single topographic step) and based on two numerical tests. Different values of the main parameter controlling the discretization scheme of the divergence formulation are analyzed to identify the formulation which minimizes the energy variation resulting from the discretization. For a wide range of ambient Froude numbers and relative step heights, the theoretical value of the control parameter minimizing the energy variation falls within a very narrow range, which can reasonably be approximated by a single “optimal” value. This is a result of high practical relevance for the design of accurate numerical schemes, as confirmed by the results of the numerical tests.

A GPU-accelerated adaptive discontinuous Galerkin method for level set equation

ABSTRACTThis paper presents a GPU-accelerated nodal discontinuous Galerkin method for the solution of two- and three-dimensional level set (LS) equation on unstructured adaptive meshes. Using adaptive mesh refinement, computations are localised mostly near the interface location to reduce the computational cost. Small global time step size resulting from the local adaptivity is avoided by local time-stepping based on a multi-rate Adams–Bashforth scheme. Platform independence of the solver is achieved with an extensible multi-threading programming API that allows runtime selection of different computing devices (GPU and CPU) and different threading interfaces (CUDA, OpenCL and OpenMP). Overall, a highly scalable, accurate and mass conservative numerical scheme that preserves the simplicity of LS formulation is obtained. Efficiency, performance and local high-order accuracy of the method are demonstrated through distinct numerical test cases.

ADI-FDTD Method for Two-Dimensional Transient Electromagnetic Problems

AbstractAn efficient and accurate numerical scheme is proposed for solving the transverse electric (TE) mode electromagnetic (EM) propagation problem in two-dimensional earth. The scheme is based on the alternating direction finite-difference time-domain (ADI-FDTD) method. Unlike the conventional upward continuation approach for the earth-air interface, an integral formulation for the interface boundary is developed and it is effectively incorporated to the ADI solver. Stability and convergence analysis together with an error estimate are presented. Numerical simulations are carried out to validate the proposed method, and the advantage of the present method over the popular Du-Fort-Frankel scheme is clearly demonstrated. Examples of the electromagnetic field propagation in the ground with anomaly further verify the effectiveness of the proposed scheme.

Self-Similar Solutions for Reversing Interfaces in the Slow Diffusion Equation with Strong Absorption

We consider the slow nonlinear diffusion equation subject to a strong absorption rate and construct local self-similar solutions for reversing (and antireversing) interfaces, where an initially advancing (receding) interface gives way to a receding (advancing) one. We use an approach based on invariant manifolds, which allows us to determine the required asymptotic behavior for small and large values of the concentration. We then “connect” the requisite asymptotic behaviors using a robust and accurate numerical scheme. By doing so, we are able to furnish a rich set of self-similar solutions for both reversing and antireversing interfaces. The stability of these self-similar solutions is validated against direct numerical simulation in the case of constant absorption.

Summation-by-Parts Operators for Non-Simply Connected Domains

We construct fully discrete stable and accurate numerical schemes for solving partial differential equations posed on non-simply connected spatial domains. The schemes are constructed using summati ...

An efficient and long-time accurate third-order algorithm for the Stokes–Darcy system

A third-order in time numerical IMEX-type algorithm for the Stokes---Darcy system for flows in fluid saturated karst aquifers is proposed and analyzed. A novel third-order Adams---Moulton scheme is used for the discretization of the dissipative term whereas a third-order explicit Adams---Bashforth scheme is used for the time discretization of the interface term that couples the Stokes and Darcy components. The scheme is efficient in the sense that one needs to solve, at each time step, decoupled Stokes and Darcy problems. Therefore, legacy Stokes and Darcy solvers can be applied in parallel. The scheme is also unconditionally stable and, with a mild time-step restriction, long-time accurate in the sense that the error is bounded uniformly in time. Numerical experiments are used to illustrate the theoretical results. To the authors' knowledge, the novel algorithm is the first third-order accurate numerical scheme for the Stokes---Darcy system possessing its favorable efficiency, stability, and accuracy properties.

Simulation of underresolved turbulent flows by adaptive filtering using the high order discontinuous Galerkin spectral element method

Scale-resolving simulations of turbulent flows in complex domains demand accurate and efficient numerical schemes, as well as geometrical flexibility. For underresolved situations, the avoidance of aliasing errors is a strong demand for stability. For continuous and discontinuous Galerkin schemes, an effective way to prevent aliasing errors is to increase the quadrature precision of the projection operator to account for the non-linearity of the operands (polynomial dealiasing, overintegration). But this increases the computational costs extensively. In this work, we present a novel spatially and temporally adaptive dealiasing strategy by projection filtering. We show this to be more efficient for underresolved turbulence than the classical overintegration strategy. For this novel approach, we discuss the implementation strategy and the indicator details, show its accuracy and efficiency for a decaying homogeneous isotropic turbulence and the transitional Taylor-Green vortex and compare it to the original overintegration approach and a state of the art variational multi-scale eddy viscosity formulation.

Local maximum principle satisfying high‐order non‐oscillatory schemes

SummaryThe main contribution of this work is to classify the solution region including data extrema for which high‐order non‐oscillatory approximation can be achieved. It is performed in the framework of local maximum principle (LMP) and non‐conservative formulation. The representative uniformly second‐order accurate schemes are converted in to their non‐conservative form using the ratio of consecutive gradients. Using the local maximum principle, these non‐conservative schemes are analyzed for their non‐linear LMP/total variation diminishing stability bounds which classify the solution region where high‐order accuracy can be achieved. Based on the bounds, second‐order accurate hybrid numerical schemes are constructed using a shock detector. The presented numerical results show that these hybrid schemes preserve high accuracy at non‐sonic extrema without exhibiting any induced local oscillations or clipping error. Copyright © 2015 John Wiley & Sons, Ltd.

Fundamental kernel-based method for backward space–time fractional diffusion problem

Based on kernel-based approximation technique, we devise in this paper an efficient and accurate numerical scheme for solving a backward space–time fractional diffusion problem (BSTFDP). The kernels used in the approximation are the fundamental solutions of the space–time fractional diffusion equation expressed in terms of inverse Fourier transform of Mittag-Leffler functions. The use of Inverse fast Fourier transform (IFFT) technique enables an accurate and efficient evaluation of the fundamental solutions and gives a robust numerical algorithm for the solution of the BSTFDP. Since the BSTFDP is intrinsic ill-posed, we apply the standard Tikhonov regularization technique to obtain a stable solution to the highly ill-conditioned resultant system of linear equations. For choosing optimal regularization parameter, we combine the regularization technique with the generalized cross validation (GCV) method for an optimal placement of the source points in the use of fundamental solutions. Meanwhile, the proposed algorithm also speeds up the previous method given in Dou and Hon (2014). Several numerical examples are constructed to verify the accuracy and efficiency of the proposed method.

Adaptive time stepping for vesicle suspensions

We present an adaptive high-order accurate time-stepping numerical scheme for the flow of vesicles suspended in Stokesian fluids. Our scheme can be summarized as an approximate implicit spectral deferred correction (SDC) method. Applying a fully-implicit SDC scheme to vesicle flows is prohibitively expensive. For this reason we introduce several approximations. Our scheme is based on a semi-implicit linearized low-order time stepping method. (Our discretization is spectrally accurate in space.) We expect that the accuracy can be arbitrary-order, but our examples suffer from order reduction which limits the observed accuracy to third-order. We also use invariant properties of vesicle flows, constant area and boundary length in two dimensions, to reduce the computational cost of error estimation for adaptive time stepping. We present results in two dimensions for single-vesicle flows, constricted geometry flows, converging flows, and flows in a Couette apparatus. We experimentally demonstrate that the proposed scheme enables automatic selection of the time step size and high-order accuracy.

Numerical simulations of cryogenic cavitating flows

The present study deals with a numerical method for cryogenic cavitating flows. Recently, we have developed an accurate and efficient baseline numerical scheme for all-speed water-gas two-phase flows. By extending such progress, we modify the numerical dissipations to be properly scaled so that it does not show any deficiencies in low Mach number regions. For dealing with cryogenic two-phase flows, previous EOS-dependent shock discontinuity sensing term is replaced with a newly designed EOS-free one. To validate the proposed numerical method, cryogenic cavitating flows around hydrofoil are computed and the pressure and temperature depression effect in cryogenic cavitation are demonstrated. Compared with Hord's experimental data, computed results are turned out to be satisfactory. Afterwards, numerical simulations of flow around KARI turbopump inducer in liquid rocket are carried out under various flow conditions with water and cryogenic fluids, and the difference in inducer flow physics depending on the working fluids are examined.

Theoretical study of steam condensation induced water hammer phenomena in horizontal pipelines

AbstractSteam condensation induced water hammer (CIWH) phenomena are investigated and new theoretical results are presented. We use the WAHA3 model based on two-phase flow six first-order partial differential equations that present one dimensional, surface averaged mass, momentum and energy balances. A second order accurate high-resolution shock-capturing numerical scheme was applied with different kind of limiters in the numerical calculations. The applied two-fluid model shows some similarities to RELAP5 which is widely used in the nuclear industry to simulate nuclear power plant accidents. This model was validated with different CIWH experiments which were performed in the PMK-2 facility, which is a full-pressure thermohydraulic model of the nuclear power plant of VVER-440/312 type in the Energy Research Center of the Hungarian Academy of Sciences and in the Rosa facility of the Japan Atomic Energy Agency. In our present study we show the first part of a planned large database which will give us the upper and lower flooding mass flow rates for various pipe geometries where CIWH can happen. Such a reliable database would be a great help for future reactor constructions and scheming.

A comparison of higher-order finite-difference shock capturing schemes

The efficiency of computational fluid dynamics simulations can be greatly enhanced by employing higher-order accurate numerical schemes which provide superior accuracy for a given cost. For unsteady turbulent flow simulations involving shocks, contacts, and/or material discontinuities, various higher-order shock capturing schemes are available in the literature. The desired numerical scheme should be free of spurious numerical oscillations across discontinuities and it should obtain higher-order accuracy in smooth flow regions in an efficient manner. Sufficient robustness is necessary for effectively utilizing these numerical methods in engineering and science applications. Three classes of higher-order shock capturing schemes are compared in this paper: (1) central finite-difference schemes with explicit artificial dissipation, (2) a compact centered finite-difference scheme with localized artificial diffusivity and (3) weighted essentially non-oscillatory schemes in both explicit and compact finite difference forms. Multiple variations of these methods were implemented and tested using a block-structured Cartesian mesh solver. The current paper assesses shock capturing capabilities as well as effects on the accuracy in smooth flow regions using a variety of test cases that range from canonical shock problems to homogeneous isotropic turbulence at a turbulent Mach number of 0.5 where shocklets are formed. Finally, a computational cost breakdown for each scheme is provided and the overall computational efficiency of the different schemes are compared to each other.

Investigation of the characteristic properties of high-order harmonic spectrum in atoms using Bohmian trajectories

We study the electron quantum dynamics of high harmonic generation (HHG) processes of atomic hydrogen under intense near infrared (NIR) laser fields by means of the De Broglie–Bohm’s framework of Bohmian mechanics. The proposed accurate 3D numerical scheme is utilized to discuss the mechanism of the multiple plateau generation and the cut-off extension, as the main characteristic features of HHG spectrum. One-color (1600 nm) and two-color (1600 nm + 800 nm) laser fields with different time delays are used to investigate the effect of pulse shape on electron dynamics and HHG process. The presented results on Bohmian trajectories and their energy content, along with the analysis of the emission time period of different groups of trajectories, provide a comprehensive and fresh electron dynamical picture and uncover novel mechanisms of the HHG processes and power spectra.

Effects of confinement on bubble dynamics in a square duct

In this paper, we simulate the effects of confinement on three-dimensional bubble dynamics in a square duct. A systematic study of three confinement ratios, four Bond numbers and two Morton numbers has been conducted. A GPU implemented VOF numerical algorithm on a Cartesian collocated grid with an improved method to handle the surface tension force has been developed. A three-dimensional geometry construction method with a piecewise linear interpolation scheme for approximating the interface segments is used to compute the interface shape and liquid volume flux from the cell faces. The Navier–Stokes equations are solved with a space–time second-order accurate fractional step numerical scheme. The surface tension force is treated as a pressure gradient and an additional Poisson equation is solved. Transient and steady state bubble deformations, rise velocities and aspect ratios are presented.

Efficient methods for highly oscillatory integrals with weak and Cauchy singularities

In this paper, we present two fast and accurate numerical schemes for the approximation of highly oscillatory integrals with weak and Cauchy singularities. For analytical kernel functions, by using the Cauchy theorem in complex analysis, we transform the integral into two line integrals in complex plane, which can be calculated by some proper Gauss quadrature rules. For general kernel functions, the non-oscillatory and nonsingular part of the integrand is replaced by a polynomial interpolation in Chebyshev points, and the integral is then evaluated by using recurrence relations. Furthermore, several numerical experiments are shown to verify the validity of such methods.

The elliptical power law profile lens

The deflection, potential, shear and magnification of a gravitational lens following an elliptical power law mass model are investigated. This mass model is derived from the circular power law profile through a rescaling of the axes, similar to the case of a singular isothermal ellipsoid. The resulting deflection can be calculated explicitly and given in terms of the Gaussian hypergeometric function. Analytic expressions for the remaining lensing properties are found as well. Because the power law profile lens contains a number of well-known lens models as special cases, the equivalence of the new expressions with known results is checked. Finally, it is shown how these results naturally lead to a fast and accurate numerical scheme for computing the deflection and other lens quantities, making this method a useful tool for realistically modelling observed lenses.

The walk on moving spheres: A new tool for simulating Brownian motion’s exit time from a domain

In this paper we introduce a new method for the simulation of the exit time and exit position of a δ-dimensional Brownian motion from a domain. The main interest of our method is that it avoids splitting time schemes as well as inversion of complicated series. The method, called walk on moving spheres algorithm, was first introduced for hitting times of Bessel processes. In this study this method is adapted and developed for the first time for the Brownian motion hitting times. The idea is to use the connexion between the δ-dimensional Bessel process and the δ-dimensional Brownian motion thanks to an explicit Bessel hitting time distribution associated with a particular curved boundary. This allows to build a fast and accurate numerical scheme for approximating the hitting time. We introduce also an overview of existing methods for the simulation of the Brownian hitting time and perform numerical comparisons with existing methods.

Pricing derivatives with counterparty risk and collateralization: A fixed point approach

This paper studies a valuation framework for financial contracts subject to reference and counterparty default risks with collateralization requirement. We propose a fixed point approach to analyze the mark-to-market contract value with counterparty risk provision, and show that it is a unique bounded and continuous fixed point via contraction mapping. This leads us to develop an accurate iterative numerical scheme for valuation. Specifically, we solve a sequence of linear inhomogeneous PDEs, whose solutions converge to the fixed point price function. We apply our methodology to compute the bid and ask prices for both defaultable equity and fixed-income derivatives, and illustrate the non-trivial effects of counterparty risk, collateralization ratio and liquidation convention on the bid-ask spreads.