An Arbitrarily High Order and Asymptotic Preserving Kinetic Scheme in Compressible Fluid Dynamic

We present a class of arbitrarily high order fully explicit kinetic numerical methods in compressible fluid dynamics,both in time and space,which include the relaxation schemes by Jin and Xin.These methods can use the CFL number larger or equal to unity on regular Cartesian meshes for the multi-dimensional case.These kinetic models depend on a small parameter that can be seen as a"Knudsen"number.The method is asymptotic preserving in this Knudsen number.Also,the computational costs of the method are of the same order of a fully explicit scheme.This work is the extension of Abgrall et al.(2022)[3]to multidimensional systems.We have assessed our method on several problems for two-dimensional scalar problems and Euler equations and the scheme has proven to be robust and to achieve the theoretically predicted high order of accuracy on smooth solutions.

High-Order Soliton Solutions and Hybrid Behavior for the (2 + 1)-Dimensional Konopelchenko-Dubrovsky Equations

In this paper, the evolutionary behavior of N-solitons for a (2 + 1)-dimensional Konopelchenko-Dubrovsky equations is studied by using the Hirota bilinear method and the long wave limit method. Based on the N-soliton solution, we first study the evolution from N-soliton to T-order (T=1,2) breather wave solutions via the paired-complexification of parameters, and then we get the N-order rational solutions, M-order (M=1,2) lump solutions, and the hybrid behavior between a variety of different types of solitons combined with the parameter limit technique and the paired-complexification of parameters. Meanwhile, we also provide a large number of three-dimensional figures in order to better show the degeneration of the N-soliton and the interaction behavior between different N-solitons.

Focused Wave Properties Based on A High Order Spectral Method with A Non-Periodic Boundary

In this paper, a numerical model is developed based on the High Order Spectral （HOS） method with a non-periodic boundary. A wave maker boundary condition is introduced to simulate wave generation at the incident boundary in the HOS method. Based on the numerical model, the effects of wave parameters, such as the assumed focused amplitude, the central frequency, the frequency bandwidth, the wave amplitude distribution and the directional spreading on the surface elevation of the focused wave, the maximum generated wave crest, and the shifting of the focusing point, are numerically investigated. Especially, the effects of the wave directionality on the focused wave properties are emphasized. The numerical results show that the shifting of the focusing point and the maximum crest of the wave group are dependent on the amplitude of the focused wave, the central frequency, and the wave amplitude distribution type. The wave directionality has a definite effect on multidirectional focused waves. Generally, it can even out the difference between the simulated wave amplitude and the amplitude expected from theory and reduce the shifting of the focusing points, implying that the higher order interaction has an influence on wave focusing, especially for 2D wave. In 3D wave groups, a broader directional spreading weakens the higher nonlinear interactions.

Numerical Simulation for Water Entry of a Wedge at Varying Speed by a High Order Boundary Element Method

A high order boundary element method was developed for the complex velocity potential problem. The method ensures not only the continuity of the potential at the nodes of each element but also the velocity. It can be applied to a variety of velocity potential problems. The present paper, however, focused on its application to the problem of water entry of a wedge with varying speed. The continuity of the velocity achieved herein is particularly important for this kind of nonlinear free surface flow problem, because when the time stepping method is used, the free surface is updated through the velocity obtained at each node and the accuracy of the velocity is therefore crucial. Calculation was made for a case when the distance S that the wedge has travelled and time t follow the relationship s=Dtα, where D and α are constants, which is found to lead to a self similar flow field when the effect due to gravity is ignored.

Re-study on Recurrence Period of Stokes Wave Train with High Order Spectral Method

Owing to the Benjamin-Feir instability, the Stokes wave train experiences a modulation-demodulation process, and presents a recurrence characteristics. Stiassnie and Shemer researched the unstable evolution process and provided a theoretical formulation for the recurrence period in 1985 on the basis of the nonlinear cubic Schrodinger equation （NLS）. However, NLS has limitations on the narrow band and the weak nonlinearity. The recurrence period is re-investigated in this paper by using a highly efficient High Order Spectral （HOS） method, which can be applied for the direct phase- resolved simulation of the nonlinear wave train evolution. It is found that the Stiassnie and Shemer＇s formula should be modified in the cases with most unstable initial conditions, which is important for such topics as the generation mechanisms of freak waves. A new recurrence period formula is presented and some new evolution characteristics of the Stokes wave train are also discussed in details.

High-order discontinuous Galerkin method for applications to multicomponent and chemically reacting flows

This article focuses on the development of a discontinuous Galerkin (DG) method for simulations of multicomponent and chemically reacting flows. Compared to aerodynamic flow applications, in which DG methods have been successfully employed, DG simulations of chemically reacting flows introduce challenges that arise from flow unsteadiness, combustion, heat release, compressibility effects, shocks, and variations in thermodynamic properties. To address these challenges, algorithms are developed, including an entropy-bounded DG method, an entropy-residual shock indicator, and a new formulation of artificial viscosity. The performance and capabilities of the resulting DG method are demonstrated in several relevant applications, including shock/bubble interaction, turbulent combustion, and detonation. It is concluded that the developed DG method shows promising performance in application to multicomponent reacting flows. The paper concludes with a discussion of further research needs to enable the application of DG methods to more complex reacting flows.

Review:Recent Development of High⁃Order⁃Spectral Method Combined with Computational Fluid Dynamics Method for Wave⁃Structure Interactions

The present paper reviews the recent developments of a high⁃order⁃spectral method(HOS)and the combination with computational fluid dynamics(CFD)method for wave⁃structure interactions.As the numerical simulations of wave⁃structure interaction require efficiency and accuracy,as well as the ability in calculating in open sea states,the HOS method has its strength in both generating extreme waves in open seas and fast convergence in simulations,while computational fluid dynamics(CFD)method has its advantages in simulating violent wave⁃structure interactions.This paper provides the new thoughts for fast and accurate simulations,as well as the future work on innovations in fine fluid field of numerical simulations.

Hermite WENO-based limiters for high order discontinuous Galerkin method on unstructured grids

A novel class of weighted essentially nonoscillatory （WENO） schemes based on Hermite polynomi- als, termed as HWENO schemes, is developed and applied as limiters for high order discontinuous Galerkin （DG） method on triangular grids. The developed HWENO methodology utilizes high-order derivative information to keep WENO re- construction stencils in the von Neumann neighborhood. A simple and efficient technique is also proposed to enhance the smoothness of the existing stencils, making higher-order scheme stable and simplifying the reconstruction process at the same time. The resulting HWENO-based limiters are as compact as the underlying DG schemes and therefore easy to implement. Numerical results for a wide range of flow conditions demonstrate that for DG schemes of up to fourth order of accuracy, the designed HWENO limiters can simul- taneously obtain uniform high order accuracy and sharp, es- sentially non-oscillatory shock transition.

3-D simulations of freak waves based on high-order spectral method

Three-dimensional ( 3-D) directional wave focusing is one of the mechanisms that contribute to the generation of freak waves. To simulate and analyze this phenomenon,a 3-D wave focusing model is proposed based on the enhanced high-order spectral method,which solves the fully nonlinear potential flow equations with a free surface within periodic unbounded 3-D domains. The numerical model is validated against a fifth-order Stokes solution for regular waves. Laboratory-scale freak waves are observed with wave components having equal amplitudes. Investigations of the appearance and propagation of freak-wave events in a 3-D open wavefield defined by a directional wave spectrum are then realized.

Coupling of high order multiplication perturbation method and reduction method for variable coefcient singular perturbation problems

Based on the precise integration method （PIM）, a coupling technique of the high order multiplication perturbation method （HOMPM） and the reduction method is proposed to solve variable coefficient singularly perturbed two-point boundary value prob lems （TPBVPs） with one boundary layer. First, the inhomogeneous ordinary differential equations （ODEs） are transformed into the homogeneous ODEs by variable coefficient dimensional expansion. Then, the whole interval is divided evenly, and the transfer ma trix in each sub-interval is worked out through the HOMPM. Finally, a group of algebraic equations are given based on the relationship between the neighboring sub-intervals, which are solved by the reduction method. Numerical results show that the present method is highly efficient.

High order multiplication perturbation method for singular perturbation problems

This paper presents a high order multiplication perturbation method for sin- gularly perturbed two-point boundary value problems with the boundary layer at one end. By the theory of singular perturbations, the singularly perturbed two-point boundary value problems are first transformed into the singularly perturbed initial value problems. With the variable coefficient dimensional expanding, the non-homogeneous ordinary dif- ferential equations （ODEs） are transformed into the homogeneous ODEs, which are then solved by the high order multiplication perturbation method. Some linear and nonlinear numerical examples show that the proposed method has high precision.

High-Order Discontinuous Galerkin Method for Hovering Rotor Simulations Based on a Rotating Reference Frame

An implicit higher ? order discontinuous Galerkin(DG) spatial discretization for the compressible Euler equations in a rotating frame of reference is presented and applied to a rotor in hover using hexahedral grids. Instead of auxiliary methods like grid adaptation,higher ? order simulations(fourth ? and fifth ? order accuracy) are adopted.Rigorous numerical experiments are carefully designed,conducted and analyzed. The results show generally excellent consistence with references and vigorously demonstrate the higher?order DG method's better performance in loading distribution computations and tip vortex capturing, with much fewer degrees of freedom(DoF). Detailed investigations on the outer boundary conditions for hovering rotors are presented as well. A simple but effective speed smooth procedure is developed specially for the DG method. Further results reveal that the rarely used pressure restriction for outlet speed has a considerable advantage over the extensively adopted vertical speed restriction.

A UNIFORM HIGH-ORDER METHOD FOR A SINGULAR PERTURBATION PROBLEM IN CONSERVATIVE FORM

A uniform high-order method is. presented for the numerical solution of a singular perturbation problem in conservative form. We firest replace the original second-order problem (1.1) by two equivalent first-order problems ( 1.4), i.e., the solution of (1.1) is a linear combination of the solutions of (1.4). Then we derive a uniformly O (hm+1) accurate scheme for the first-order problems (1.4), where m is an arbitrary nonnegative integer, so we can get a uniformly O (hm+1) accurate solution of the original problem (1.1) by relation (1.3). Some illustrative numerical results are also given.

The Hirota bilinear method for the coupled Burgers equation and the high-order Boussinesq-Burgers equation

This paper studies the coupled Burgers equation and the high-order Boussinesq-Burgers equation. The Hirota bilinear method is applied to show that the two equations are completely integrable. Multiple-kink （soliton） solutions and multiple-singular-kink （soliton） solutions are derived for the two equations.

A force control high-order single-step-β method (HSM) for substructure pseudo-dynamic testing

This paper proposes a new technique which introduces the high-order single-step-β method(HSM)into the experimental study on the substructure pseudo-dynamic testing(SPDT).The technique is based on the proposed concept of equivalent shear stiffness which can meet the requirement of the HSM algorithm.A study is done to theoretically validate the technique by the numerical analysis of two-storey shear building structure,in comparison of the proposed substructure pseudo-dynamic testing algorithm with the central difference method(CDM).Then,a full-scale SPDT model,the three-storey frame-supported reinforced concrete short-limb masonry shear wall structure,is designed and tested to simulate the seismic response of the corresponding six-storey structure and verify the proposed force control HSM technique.Meanwhile,the techniques of both stiffness correction and force control are suggested to control algorithmic error,control error and measurement error.The results indicate that the force control HSM can be used in the full-scale multi-degree-of-freedom(MDOF)substructure pseudo-dynamic testing before descent segment of structure restoring force properties.

A mass-conserved multiphase lattice Boltzmann method based on high-order difference

The Z–S–C multiphase lattice Boltzmann model [Zheng, Shu, and Chew(ZSC), J. Comput. Phys. 218, 353(2006)]is favored due to its good stability, high efficiency, and large density ratio. However, in terms of mass conservation, this model is not satisfactory during the simulation computations. In this paper, a mass correction is introduced into the ZSC model to make up the mass leakage, while a high-order difference is used to calculate the gradient of the order parameter to improve the accuracy. To verify the improved model, several three-dimensional multiphase flow simulations are carried out,including a bubble in a stationary flow, the merging of two bubbles, and the bubble rising under buoyancy. The numerical simulations show that the results from the present model are in good agreement with those from previous experiments and simulations. The present model not only retains the good properties of the original ZSC model, but also achieves the mass conservation and higher accuracy.

A high-order accurate wavelet method for solving Schrdinger equations with general nonlinearity

A sampling approximation for a function defined on a bounded interval is proposed by combining the Coiflet-type wavelet expansion and the boundary extension technique. Based on such a wavelet approximation scheme, a Galerkin procedure is developed for the spatial discretization of the generalized nonlinear Schr6dinger （NLS） equa- tions, and a system of ordinary differential equations for the time dependent unknowns is obtained. Then, the classical fourth-order explicit Runge-Kutta method is used to solve this semi-discretization system. To justify the present method, several widely considered problems are solved as the test examples, and the results demonstrate that the proposed wavelet algorithm has much better accuracy and a faster convergence rate in space than many existing numerical methods.

High-Order Iterative Methods Repeating Roots a Constructive Recapitulation

This paper considers practical, high-order methods for the iterative location of the roots of nonlinear equations, one at a time. Special attention is being paid to algorithms also applicable to multiple roots of initially known and unknown multiplicity. Efficient methods are presented in this note for the evaluation of the multiplicity index of the root being sought. Also reviewed here are super-linear and super-cubic methods that converge contrarily or alternatingly, enabling us, not only to approach the root briskly and confidently but also to actually bound and bracket it as we progress.

Finding Periodic Solutions of High Order Duffing Equations via Homotopy Method

This paper presents a detailed analysis of finding the periodic solutions for the high order Duffing equation x^（2n） ＋ g（x） = e（t） （n ≥ 1）. Firstly, we give a constructive proof for the existence of periodic solutions via the homotopy method. Then we establish an efficient and global convergence method to find periodic solutions numerically.

High Order Compact Difference Scheme and Multigrid Method for 2D Elliptic Problems with Variable Coefficients and Interior/Boundary Layers on Nonuniform Grids

In this paper, a high order compact difference scheme and a multigrid method are proposed for solving two-dimensional (2D) elliptic problems with variable coefficients and interior/boundary layers on nonuniform grids. Firstly, the original equation is transformed from the physical domain (with a nonuniform mesh) to the computational domain (with a uniform mesh) by using a coordinate transformation. Then, a fourth order compact difference scheme is proposed to solve the transformed elliptic equation on uniform girds. After that, a multigrid method is employed to solve the linear algebraic system arising from the difference equation. At last, the numerical experiments on some elliptic problems with interior/boundary layers are conducted to show high accuracy and high efficiency of the present method.