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.

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.

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.

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.

Fourier Continuation Discontinuous Galerkin Methods for Linear Hyperbolic Problems

Fourier continuation(FC)is an approach used to create periodic extensions of non-periodic functions to obtain highly-accurate Fourier expansions.These methods have been used in partial differential equation(PDE)-solvers and have demonstrated high-order convergence and spectrally accurate dispersion relations in numerical experiments.Discontinuous Galerkin(DG)methods are increasingly used for solving PDEs and,as all Galerkin formulations,come with a strong framework for proving the stability and the convergence.Here we propose the use of FC in forming a new basis for the DG framework.

High Order Deep Domain Decomposition Method for Solving High Frequency Interface Problems

This paper proposes a high order deep domain decomposition method(HOrderDeepDDM)for solving high-frequency interface problems,which combines high order deep neural network(HOrderDNN)with domain decomposition method(DDM).The main idea of HOrderDeepDDM is to divide the computational domain into some sub-domains by DDM,and apply HOrderDNNs to solve the high-frequency problem on each sub-domain.Besides,we consider an adaptive learning rate annealing method to balance the errors inside the sub-domains,on the interface and the boundary during the optimization process.The performance of HOrderDeepDDM is evaluated on high-frequency elliptic and Helmholtz interface problems.The results indicate that:HOrderDeepDDM inherits the ability of DeepDDM to handle discontinuous interface problems and the power of HOrderDNN to approximate high-frequency problems.In detail,HOrderDeepDDMs(p>1)could capture the high-frequency information very well.When compared to the deep domain decomposition method(DeepDDM),HOrderDeepDDMs(p>1)converge faster and achieve much smaller relative errors with the same number of trainable parameters.For example,when solving the high-frequency interface elliptic problems in Section 3.3.1,the minimum relative errors obtained by HOrderDeepDDMs(p=9)are one order of magnitude smaller than that obtained by DeepDDMs when the number of the parameters keeps the same,as shown in Fig.4.

Semi-Eulerian and High Order Gaussian Beam Methods for the Schrödinger Equation in the Semiclassical Regime

A novel Eulerian Gaussian beam method was developed in[8]to compute the Schrödinger equation efficiently in the semiclassical regime.In this paper,we introduce an efficient semi-Eulerian implementation of this method.The new algorithm inherits the essence of the Eulerian Gaussian beam method where the Hessian is computed through the derivatives of the complexified level set functions instead of solving the dynamic ray tracing equation.The difference lies in that,we solve the ray tracing equations to determine the centers of the beams and then compute quantities of interests only around these centers.This yields effectively a local level set implementation,and the beam summation can be carried out on the initial physical space instead of the phase plane.As a consequence,it reduces the computational cost and also avoids the delicate issue of beam summation around the caustics in the Eulerian Gaussian beam method.Moreover,the semi-Eulerian Gaussian beam method can be easily generalized to higher order Gaussian beam methods,which is the topic of the second part of this paper.Several numerical examples are provided to verify the accuracy and efficiency of both the first order and higher order semi-Eulerian methods.

CONSTRUCTION OF HIGH ORDER SYMPLECTIC PRK METHODS

In this paper three classes of high order symplectic partitioned Runge-Kutta (PRK) methods, which are based on the W-transformation of Hairer and Wanner, and a particular class of diagonally implicit symplectic PRK methods of arbitrarily high order, which are based on the composite algorithm, are constructed. In particular, the symplectically composite PRK methods of arbitrarily high order become explicit when applied to separable Hamiltonian systems.

HIGH ORDER NUMERICAL METHODS TO TWO DIMENSIONAL HEAVISIDE FUNCTION INTEGRALS

In this paper we design and analyze a class of high order numerical methods to two dimensional Heaviside function integrals. Inspired by our high order numerical methods to two dimensional delta function integrals [19], the methods comprise approximating the mesh cell restrictions of the Heaviside function integral. In each mesh cell the two dimen- sional Heaviside function integral can be rewritten as a one dimensional ordinary integral with the integrand being a one dimensional Heaviside function integral which is smooth on several subsets of the integral interval. Thus the two dimensional Heaviside function inte- gral is approximated by applying standard one dimensional high order numerical quadra- tures and high order numerical methods to one dimensional Heaviside function integrals. We establish error estimates for the method which show that the method can achieve any desired accuracy by assigning the corresponding accuracy to the sub-algorithms. Numerical examples are presented showing that the in this paper achieve or exceed the expected second to fourth-order methods implemented accuracy.

Variable High Order Multiblock Overlapping Grid Methods for Mixed Steady and Unsteady Multiscale Viscous Flows

Flows containing steady or nearly steady strong shocks on parts of the flow field,and unsteady turbulence with shocklets on other parts of the flow field are difficult to capture accurately and efficiently employing the same numerical scheme,even under the multiblock grid or adaptive grid refinement framework.While sixthorder or higher-order shock-capturing methods are appropriate for unsteady turbulence with shocklets,third-order or lower shock-capturing methods are more effective for strong steady or nearly steady shocks in terms of convergence.In order to minimize the short comings of low order and high order shock-capturing schemes for the subject flows,a multiblock overlapping grid with different types of spatial schemes and orders of accuracy on different blocks is proposed.The recently developed single block high order filter scheme in generalized geometries for Navier Stokes and magnetohydrodynamics systems is extended to multiblock overlapping grid geometries.The first stage in validating the high order overlapping approach with several test cases is included.

Numerical Issues in the Implementation of High Order Polynomial Multi-Domain Penalty Spectral Galerkin Methods for Hyperbolic Conservation Laws

In this paper,we consider high order multi-domain penalty spectral Galerkin methods for the approximation of hyperbolic conservation laws.This formulation has a penalty parameter which can vary in space and time,allowing for flexibility in the penalty formulation.This flexibility is particularly advantageous for problems with an inhomogeneous mesh.We show that the discontinuous Galerkin method is equivalent to the multi-domain spectral penalty Galerkin method with a particular value of the penalty parameter.The penalty parameter has an effect on both the accuracy and stability of the method.We examine the numerical issues which arise in the implementation of high order multi-domain penalty spectral Galerkin methods.The coefficient truncation method is proposed to prevent the rapid error growth due to round-off errors when high order polynomials are used.Finally,we show that an inconsistent evaluation of the integrals in the penalty method may lead to growth of errors.Numerical examples for linear and nonlinear problems are presented.

High Order Deep Neural Network for Solving High Frequency Partial Differential Equations

This paper proposes a high order deep neural network(HOrderDNN)for solving high frequency partial differential equations(PDEs),which incorporates the idea of“high order”from finite element methods(FEMs)into commonly-used deep neural networks(DNNs)to obtain greater approximation ability.The main idea of HOrderDNN is introducing a nonlinear transformation layer between the input layer and the first hidden layer to form a high order polynomial space with the degree not exceeding p,followed by a normal DNN.The order p can be guided by the regularity of solutions of PDEs.The performance of HOrderDNNis evaluated on high frequency function fitting problems and high frequency Poisson and Helmholtz equations.The results demonstrate that:HOrderDNNs(p>1)can efficiently capture the high frequency information in target functions;and when compared to physics-informed neural network(PINN),HOrderDNNs(p>1)converge faster and achieve much smaller relative errors with same number of trainable parameters.In particular,when solving the high frequency Helmholtz equation in Section 3.5,the relative error of PINN stays around 1 with its depth and width increase,while the relative error can be reduced to around 0.02 as p increases(see Table 5).

NUMERICAL SIMULATIONS OF EVOLUTIONS FROM WALL PULSE TO TURBULENT COHE RENT STRUCTURES

The high order compact d if ference method is developed for solving the perturbation equations based on Navi er Stokes equations, and is used in studying complex evolution processes from w all negative pulse to the turbulent coherent structure in the channel flow. Th is method contains three dimensional coupling difference scheme with high accur acy and high resolution, and the high order time splitting methods. Compared with the general spectral method, the method can be used to research turbule nt coherent structure under more general boundary conditions and in flow domains . In this paper, the generation and evolution of the turbulent coherent structur es ind uced by wall pulse in the channel flow are simulated, and the basic characterist ics and rules of the turbulent coherent structure are shown. Computational r esults indicate that a wall negative pulse is more convenient than the resonant three wave model.

A THREE-DIMENSIONAL DESINGULARIZED HIGH ORDER PANEL METHOD BASED ON NURBS

A desingularized high order panel method based on Non-Uniform Rational B-Spline （NURBS） was developed to deal with three-dimensional potential flow problems. A NURBS surface was used to precisely represent the body geometry. Velocity potential on the body surface was described by the B-spline after the source density distribution on the body surface had been solved. The collocation approach was employed to satisfy the Neurnann boundary condition and Gaussian quadrature points were chosen as both the collocation points and the source points. The singularity was removed by a combined method, so the process of the numerical computation was non-singular. In order to verify the method proposed, the unbounded flow problems of sphere and ellipsoid, the wave-making problem of a submerged ellipsoid were chosen as computational examples. It is shown that the numerical results are in good agreement with analytical solutions and other numerical results in all cases, and sufficient accuracy of numerical solution can be reached with a small number of panels.

Numerical Research on Wave-making Resistance of Trimaran

A 3D rankine panel method was developed for calculating the linear wave-making resistance of a tri-maran with Wigley hulls. In order to calculate the normal vector and derivative of the body surface accurately, non-uniform rational B-spline (NURBS) was adopted to represent body surface and rankine source density. The radiation condition is satisfied using the numerical technology of staggered grids. Numerical results show that the linear wave-making resistance of the trimaran can be calculated effectively using this method.

Fully Nonlinear Simulations of Wave Resonance by An Array of Cylinders in Vertical Motions

The finite element method （FEM） is employed to analyze the resonant oscillations of the liquid confined within multiple or an array of floating bodies with fully nonlinear boundary conditions on the free surface and the body surface in two dimensions. The velocity potentials at each time step are obtained through the FEM with 8-node quadratic shape functions. The finite element linear system is solved by the conjugate gradient （CG） method with a symmetric successive overelaxlation （SSOR） preconditioner. The waves at the open boundary are absorbed by the combination of the damping zone method and the Sommerfeld-Orlanski equation. Numerical examples are given by an array of floating wedge- shaped cylinders and rectangular cylinders. Results are provided for heave motions including wave elevations, profiles and hydrodynamic forces. Comparisons are made in several cases with the results obtained from the second order solution in the time domain. It is found that the wave amplitude in the middle region of the array is larger than those in other places, and the hydrodynamic force on a cylinder increases with the cylinder closing to the middle of the array.