High-order Numerical Algorithm Research Articles

A new class of high-order supplementary variable methods for the Klein–Gordon–Zakharov system

In this paper, based on the paradigm of supplementary variable method (SVM), we reformulate the Klein–Gordon–Zakharov system into equivalent optimization problem subject to PDE constraints, and then present a novel class of high-order energy-preserving numerical algorithms to solve it numerically. The optimization model is discretized by applying the Gauss collocation method as well as the prediction–correction technique in time and the sine pseudo-spectral method in space. Experimental results and some comparisons between this method and other reported ones are provided to favor the suggested method in the overall performance.

A study of the occurrence of DDT in an array of obstacles: Combined effects of longitudinal and transverse obstacle spacings

Combustion wave propagation in a channel containing a stoichiometric hydrogen-air mixture and an array of obstacles was investigated, aiming to explore the effects of longitudinal (ls) and transverse (ts) spacings of the obstacles on the occurrence of deflagration-to-detonation transition (DDT). A high-order numerical algorithm and adaptive mesh refinement were adopted to solve reactive Navier-Stokes equations. The spacings, ls and ts, are varied from 1.2 to 6 times the obstacle diameter (do = 12.7 mm). The results show that the occurrence of DDT is largely dependent on ts. The DDT is likely to occur when the ratio of transverse obstacle gap to detonation cell size d/λ ≥ 0.64. The DDT run-up distance and time are closely related to ls and ts. For large ts, the DDT occurrence time increases with ls. However, the DDT distance may decrease significantly as ls increases when ls is relatively small, due to frequent local explosions in the isolated pockets of unreacted material. A requirement for the generation of these isolated explosions can be quantified as (ls/ts)2(12–4br) < 1, where br is the blockage ratio of the channel. For small ts, the increase of DDT distance with ls/ts slows down notably when ls/ts exceeds a threshold, because there is a mechanism characterized by large fluctuations in flame surface area that promotes the occurrence of DDT in the case with a large enough ls/ts.

High-order Lagrange multiplier method for the coupled Klein-Gordon-Schrödinger system

In this work, a novel class of high-order energy-preserving algorithms are developed for simulating the coupled Klein-Gordon-Schrödinger equations. We introduce a Lagrange multiplier approach and derive a new system which warrants exactly the original total energy, instead of the modified quadratic energy in the previous energy quadratization approach. The Lagrange system is then reformulated equivalently into an optimization problem subject to PDE constraints. We discretize the optimization system by applying the Gauss collocation method together with the prediction-correction technique in time and the sine pseudo-spectral method in space, which leads to a family of fully discrete high-order energy-preserving schemes. In the numerical experiments, we compare this method with other previous methods and verify the high accuracy as well as its ability to maintain the original energy at each time layer. The feasibility and validity of new method are also witnessed for the KGS model in the nonrelativistic limit regime. Additionally, enlightened by this seminal idea, another type of high-order numerical algorithms conserving two invariants of mass and energy are also presented and relevant experimental results are listed to verify the power of the suggested method.

Effect of wall roughness on flame acceleration and deflagration-to-detonation transition in a narrow channel

Numerical simulations are conducted to investigate the effect of wall roughness on flame acceleration (FA), deflagration-to-detonation transition (DDT), and detonation propagation in a narrow channel filled with stoichiometric hydrogen-air mixture. The wall roughness is determined by the element height h relative to the pipe diameter d and can be described using a dimensionless number Ra = 2h/d. A high-order numerical algorithm is employed to solve the Navier–Stokes equations on an adaptive mesh. The results show that the roughness enhances the effect of boundary layer and promotes FA and DDT. In channels with small roughness (Ra<0.1), detonation is not observed. Flame instabilities are caused by the interaction between the flame surfaces and reflected waves from the sidewalls, wrinkling of the flame front, leading to additional flame acceleration, and the production of intense pressure waves. In comparison, in channels with large roughness (Ra>0.1), vortices, shears, and even turbulence are produced in the cavity-like regions as leading shock passes over the elements. The mechanisms of the final detonation transition in the rough narrow channel are thought to be the formation of local hot spots arising from the multiple interactions of shocks with the roughness elements and viscous heating of unburned gas in the highly turbulent boundary layer.

Novel advances in high-order numerical algorithm for evaluation of the shallow water wave equations

In this paper, we propose a high-order nonlinear algorithm based on a finite difference method modification to the regularized long wave equation and the Benjamin–Bona–Mahony–Burgers equation subject to the homogeneous boundary. The consequence system of nonlinear equations typically trades with high computation burden. This dilemma can be overcome by establishing a fast numerical algorithm procedure without a reduction of numerical accuracy. The proposed algorithm forms a linear system with constant coefficient matrix at each time step and produces numerical solutions, which remarkably gains many computational advantages. In terms of analysis, a priori estimation for the numerical solution is derived to obtain the convergence and stability analysis. Additionally, the algorithm is globally mass preserving to avoid nonphysical behavior. Two benchmarks, including a single solitary wave to both equations, are given to validate the applicability and accuracy of the proposed method. Numerical results are obtained and compared to other approaches available in the literature. From the comparisons it is clear that the proposed approach produces accurate and precise results at low computational cost. Besides, the proposed scheme is applied to study the effect of the viscous term on a single solitary wave. It is shown that the viscous term results in the amplitude and width of the initial condition but not in its velocities in the case of a single solitary wave. As a consequence, theoretical and numerical findings provide a new area to investigate and expand the high-order algorithm for the family of wave equations.

High-order numerical algorithm and error analysis for the two-dimensional nonlinear spatial fractional complex Ginzburg–Landau equation

In this paper, we first construct an appropriate new generating function, and then based on this function, we establish a fourth-order numerical differential formula approximating the Riesz derivative with order γ∈(1,2]. Subsequently, we apply the formula to numerically study the two-dimensional nonlinear spatial fractional complex Ginzburg–Landau equation and obtain a difference scheme with convergence order Oτ2+hx4+hy4, where τ denotes the time step size, hx and hy denote the space step sizes, respectively. Furthermore, with the help of some newly derived discrete fractional Sobolev embedding inequalities, the unique solvability, the unconditional stability, and the convergence of the constructed numerical algorithm under different norms are proved by using the discrete energy method. Finally, some numerical results are presented to confirm the correctness of the theoretical results and verify the effectiveness of the proposed scheme.

Influence of longitudinal obstacle spacing on the deflagration-to-detonation transition through a bank of obstacles in a hydrogen-air mixture

Flame acceleration and deflagration-to-detonation transition (DDT) in a channel containing an array of staggered cylindrical obstacles and a stoichiometric hydrogen-air mixture were studied by solving the fully-compressible reactive Navier-Stokes equations using a high-order numerical algorithm and adaptive mesh refinement. Four different longitudinal spacings (ls) of the neighboring obstacle rows (i.e., ls = 15.28, 19.1, 25.4, and 38.2 mm, corresponding to 1.2, 1.5, 2 and 3 times of obstacle diameter, respectively) were used to examine the effect of obstacle spacing on flame acceleration and DDT. The results show that the main mechanisms of flame acceleration and transition to detonation in all the cases studied are consistent. While the flame acceleration is caused by the growth of flame surface area in the initial stage, it is governed by shock-flame interactions in the later stage when shock waves are generated. The focusing of strong shocks at flame front is responsible for the initiation of detonation. It was found that the flame propagation speed and the DDT run-up distance and time are highly dependent on ls. Specifically, the flame acceleration declines as ls increases, since a larger ls leads to less disturbance of flow by obstacles per unit channel length. For detonation initiation, both the run-up distance and time increase with the increase of ls. It is interesting to note that the DDT distance and time increase significantly as ls increases from 19.1 mm to 25.4 mm. This is related to the slowdown of the increase rate of energy release over a period before DDT occurs under large ls condition, because every time the flame passes over an obstacle row the shock-flame interaction is delayed and numerous isolated pockets of unburned gas material are formed.

A Super-High-Accuracy Attitude Measurement Method of SINS Based on PWQHN Algorithm in the High-Dynamic Maneuver Environment

To reduce noncommutative error (NCE) and improve the attitude-update accuracy in the attitude algorithm of the strapdown inertial navigation system (SINS), we assume that angular rate is represented as a polynomial to deduce the relationship between the angular increment sampling sequence and the derivatives of the quaternion. Subsequently, a high-order numerical solution of the quaternion differential equation is obtained by substituting the derivatives of the quaternion into its Taylor expansion. Generally, the quaternion will be constructed with a series of angular increment samples in a constant updating interval. Furthermore, the piecewise quaternion high-order numerical (PWQHN) algorithm is deduced by reconstructing quaternion in each sampling interval to ulteriorly improve attitude accuracy. The simulation results indicate that the PWQHN algorithm is stable in two kinds of simulation conditions. In the coning motion condition, PWQHN algorithm has the highest accuracy. In the high-dynamic condition, the extreme accuracy of PWQHN is not significantly higher than other algorithms. However, PWQHN can still output accurate attitude in the situation without many angular increment subsamples.

High-order numerical algorithm for fractional-order nonlinear diffusion equations with a time delay effect

&lt;abstract&gt;&lt;p&gt;In this paper, we examine and provide numerical solutions to the nonlinear fractional order time-space diffusion equations with the influence of temporal delay. An effective high-order numerical scheme that mixes the so-called Alikhanov $ L2-1_\sigma $ formula side by side to the power of the Galerkin method is presented. Specifically, the time-fractional component is estimated using the uniform $ L2-1_{\sigma} $ difference formula, while the spatial fractional operator is approximated using the Legendre-Galerkin spectral approximation. In addition, Taylor's approximations are used to discretize the term of the nonlinear source function. It has been shown theoretically that the suggested scheme's numerical solution is unconditionally stable, with a second-order time-convergence and a space-convergent order of exponential rate. Furthermore, a suitable discrete fractional Grönwall inequality is then utilized to quantify error estimates for the derived solution. Finally, we provide a numerical test that closely matches the theoretical investigation to assess the efficacy of the suggested method.&lt;/p&gt;&lt;/abstract&gt;

Flame acceleration and DDT in a channel with fence-type obstacles: Effect of obstacle shape and arrangement

Experiments and numerical simulations were conducted to study the effects of obstacle shape and arrangement on the flame acceleration and deflagration-to-detonation transition (DDT) in an obstructed channel filled with a stoichiometric hydrogen–oxygen mixture. Two different shapes and arrangements of fence-type obstacles with the same blockage area were considered: continuous triangular and discrete rectangular obstacles. In the experiments, high-speed schlieren photography was used to record the flame acceleration and DDT process with a blockage ratio of 0.5. In the calculations with the different blockage ratios, a high-order numerical algorithm was used to solve the multidimensional, fully compressible, reactive Navier–Stokes equations coupled to a calibrated single-step chemical-diffusive model. The quantitative agreement between the calculations and experiments allows a detailed analysis of the DDT mechanism and the influence of obstacle shape/arrangement using the numerical results. The results show that the differences in the flame acceleration and distance to DDT between the triangular and rectangular obstacle arrays are closely related to the blockage ratio. For a low blockage ratio (br<0.5), flame accelerates faster and DDT occurs in a shorter time in the channel with the triangular obstacles because the inclined surfaces of the triangular obstacles allow stronger flame-vortex interactions and are more favorable to detonation survival once a detonation is initiated in the gap between obstacles. Nevertheless, for a high blockage ratio (br>0.5), the lateral triangle surfaces become steeper, and the advantages of triangular obstacles for promoting DDT become less effective. In addition, the triangles have a more significant weakening effect on the leading shock, while the rectangular obstacles allow the formation of Mach stems. These effects make the rectangular obstacles with a high blockage ratio more conducive to the occurrence of DDT than their triangular counterparts.

Structure preserving fourth-order difference scheme for the nonlinear spatial fractional Schrödinger equation in two dimensions

In this paper, we focus on develop high-order and structure-preserving numerical algorithm for the two-dimensional nonlinear space fractional Schrödinger equations. By constructing a new generating function, we obtain a fourth-order numerical differential formula and use it to approximate the spatial Riesz derivative, while the Crank–Nicolson method is applied for the time derivative. Based on the energy method, the conservation, solvability and convergence of the numerical algorithm are proved. Finally, some numerical examples are used to verify the correctness of the theoretical analysis and the validity of the numerical algorithm.

High-Order Multivariate Spectral Algorithms for High-Dimensional Nonlinear Weakly Singular Integral Equations with Delay

One of the open problems in the numerical analysis of solutions to high-dimensional nonlinear integral equations with memory kernel and proportional delay is how to preserve the high-order accuracy for nonsmooth solutions. It is well-known that the solutions to these equations display a typical weak singularity at the initial time, which causes challenges in developing high-order and efficient numerical algorithms. The key idea of the proposed approach is to adopt a smoothing transformation for the multivariate spectral collocation method to circumvent the curse of singularity at the beginning of time. Therefore, the singularity of the approximate solution can be tailored to that of the exact one, resulting in high-order spectral collocation algorithms. Moreover, we provide a framework for studying the rate of convergence of the proposed algorithm. Finally, we give a numerical test example to show that the approach can preserve the nonsmooth solution to the underlying problems.

A high-order adaptive numerical algorithm for fractional diffusion wave equation on non-uniform meshes

A high-order adaptive numerical algorithm for fractional diffusion wave equation on non-uniform meshes

High-order numerical algorithms for the time-fractional convection–diffusion equation

In this paper, efficient methods are derived for seeking a numerical solution to the time-fractional convection–diffusion equation whose solution very likely exhibits a weak regularity at the starting time. Here, the time-fractional derivative in the Caputo sense with order in is discretized by the - methods with uniform and non-uniform meshes and the spatial derivative is approximated by the local discontinuous Galerkin (LDG) finite element methods. The fully discrete schemes for both situations are established and analyzed. It is shown that the derived schemes are numerically stable and convergent. Finally, some numerical examples are performed to testify the effectiveness of the obtained algorithm.

A Coupled LES-Synthetic Turbulence Method for Jet Noise Prediction

Large-eddy simulation (LES)-based jet noise predictions do not resolve the entire broadband noise spectra, often under-predicting high frequencies that correspond to un-resolved small-scale turbulence. The coupled LES-synthetic turbulence (CLST) model is presented which aims to model the missing high frequencies. The CLST method resolves large-scale turbulent fluctuations from coarse-grid large-eddy simulations (CLES) and models small-scale fluctuations generated by a synthetic eddy method (SEM). Noise is predicted using a formulation of the linearized Euler equations (LEE), where the acoustic waves are generated by source terms from the combined fluctuations of the CLES and the stochastic fields. Sweeping and straining of the synthetic eddies are accounted for by convecting eddies with the large turbulent scales from the CLES flow field. The near-field noise of a Mach 0.9 jet at a Reynolds number of 100,000 is predicted with LES. A high-order numerical algorithm, involving a dispersion relation preserving scheme for spatial discretization and an Adams–Bashforth scheme for time marching, is used for both LES and LEE solvers. Near-field noise spectra from the LES solver are compared to published results. Filtering is applied to the LES flow field to produce an under-resolved CLES flow field, and a comparison to the un-filtered LES spectra reveals the missing noise for this case. The CLST method recovers the filtered high-frequency content, agreeing well with the spectra from LES and showing promise at modeling the high-frequency range in the acoustic noise spectrum at a reasonable expense.

Verification of depletion capability of OpenMC using VERA depletion benchmark

Depletion capability has been implemented in the open-source state-of-the-art Monte Carlo particle transport code, OpenMC, via the Python application programming interface (API) and C API. The Python API provides the main depletion functionality including the depletion matrix solver based on the Chebyshev rational approximation method, and high-fidelity numerical integration algorithms for flux and nuclide densities updates. OpenMC serves as the data provider for problem dependent transmutation reactions needed to form the depletion matrix. The data exchange between the OpenMC solver and the depletion solver is controlled by the C API to facilitate in-memory transfers. This paper firstly demonstrates the accuracy of this implementation by comparing results of VERA depletion benchmark against the well-verified Monte Carlo depletion code, Serpent2, including both multiplication factor and nuclide densities of interest. Additionally, a simplified depletion chain based on that of the CASL project is evaluated by comparing its accuracy and efficiency with a full depletion chain based on ENDF/B-VII.1 library. The benefits of high-order numerical integration algorithms have also been verified on selected benchmarks with and without burnable absorbers. Finally, the impact of the ENDF/B-VIII.0 data library on select benchmarks was also evaluated.

Effects of obstacle layout and blockage ratio on flame acceleration and DDT in hydrogen-air mixture in a channel with an array of obstacles

Numerical simulations were performed to study flame acceleration and deflagration-to-detonation transition (DDT) in hydrogen-air mixture in a channel with a two-dimensional array of cylindrical obstacles. A high-order numerical algorithm with adaptive mesh refinement was applied to solve reactive Navier-Stokes equations. The effect of obstacle layout was examined by considering three layouts at a fixed blockage ratio of 0.5: staggered, inline-concentrated, and inline-scattered. Three blockage ratios, 0.33, 0.5, and 0.67, were used for the case of staggered obstacles to explore the influence of blockage ratio. The results show that both obstacle layout and blockage ratio have significant effects on flame acceleration and DDT occurrence, although the basic mechanism of detonation initiation is consistent for all the cases involving shock focusing. In the staggered case, the head-on collisions of flame and pressure waves with obstacles greatly promote the growth of flame surface area and thus lead to the fastest flame acceleration and shortest detonation onset time. In the inline-concentrated case, flame propagates slower than that in the staggered case due to smaller flame surface area. However, compared to the zigzag path in the staggered case, the straight passages parallel to flame propagation direction in the inline-concentrated case are more conducive to producing strong shock focusing and thus result in the shortest DDT distance. In the inline-scattered case, the straight passage along the centerline of channel facilitates the early acceleration of flame, but it has the slowest flame propagation in lateral directions and thereby the longest DDT time and distance. For the staggered obstacles at different blockage ratios, flame acceleration rate increases with increasing blockage ratio. The occurrence of DDT is hindered by the most congested obstacles, because the shock focusing is insufficiently strong to initiate detonation after passing through the excessively narrow gaps.

Prediction of multiphase flows with sharp interfaces using anisotropic mesh optimisation

We propose an integrated, parallelised modelling approach to solve complex multiphase flow problems with sharp interfaces. This approach is based on a finite-element, double control-volume methodology, and employs highly-anisotropic mesh optimisation within a framework of high-order numerical methods and algorithms, which include adaptive time-stepping, metric advection, flux limiting, compressive advection of interfaces, multi-grid solvers and preconditioners. Each method is integral to increasing the fidelity of representing the underlying physics while maximising computational efficiency, and, only in combination, do these methods result in the accurate, reliable, and efficient simulation of complex multiphase flows and associated regime transitions. These methods are applied simultaneously for the first time in this paper, although some of the individual methods have been presented previously. We validate our numerical predictions against standard benchmark results from the literature and demonstrate capabilities of our modelling framework through the simulation of laminar and turbulent two-phase pipe flows. These complex interfacial flows involve the creation of bubbles and slugs, which involve multi-scale physics and arise due to a delicate interplay amongst inertia, viscous, gravitational, and capillary forces. We also comment on the potential use of our integrated approach to simulate large, industrial-scale multiphase pipe flow problems that feature complex topological transitions.

A high-order numerical algorithm for solving Lane–Emden equations with various types of boundary conditions

A high-order numerical algorithm for solving Lane–Emden equations with various types of boundary conditions

High order algorithms for numerical solution of fractional differential equations

In this paper, two novel high order numerical algorithms are proposed for solving fractional differential equations where the fractional derivative is considered in the Caputo sense. The total domain is discretized into a set of small subdomains and then the unknown functions are approximated using the piecewise Lagrange interpolation polynomial of degree three and degree four. The detailed error analysis is presented, and it is analytically proven that the proposed algorithms are of orders 4 and 5. The stability of the algorithms is rigorously established and the stability region is also achieved. Numerical examples are provided to check the theoretical results and illustrate the efficiency and applicability of the novel algorithms.