A simple method of depressing numerical dissipation effects during wave simulation within the Euler model

Numerical wave tanks are widely-acknowledged tools in studying waves and wave-structure interactions. They can generate waves under realistic scales and offers more information on the fluid field. However, most numerical wave tanks suffer from issues known as the numerical dissipation and numerical dispersion. The former causes wave energy to be slowly dissipated and the latter shifts wave frequencies during wave propagation. This paper proposes a simple method of depressing numerical dissipation effects on the basis of solving Euler equations using the finite difference method(FDM). The wave propagation solutions are solved analytically taking into account the influence of the damping terms. The main idea of the method is to append a source term to the momentum equation, whose strength is determined by how strong the numerical damping effect is. The method is verified by successfully depressing numerical effects during the simulation of regular linear waves, Stokes waves and irregular waves. By applying the method, wave energy is able to be close to its initial value after long distance of travel.

On the Reduction of Numerical Dissipation in Central-Upwind Schemes

We study central-upwind schemes for systems of hyperbolic conservation laws,recently introduced in[13].Similarly to staggered non-oscillatory central schemes,these schemes are central Godunov-type projection-evolution methods that enjoy the advantages of high resolution,simplicity,universality and robustness.At the same time,the central-upwind framework allows one to decrease a relatively large amount of numerical dissipation present at the staggered central schemes.In this paper,we present a modification of the one-dimensional fully-and semi-discrete central-upwind schemes,in which the numerical dissipation is reduced even further.The goal is achieved by a more accurate projection of the evolved quantities onto the original grid.In the semi-discrete case,the reduction of dissipation procedure leads to a new,less dissipative numerical flux.We also extend the new semi-discrete scheme to the twodimensional case via the rigorous,genuinely multidimensional derivation.The new semi-discrete schemes are tested on a number of numerical examples,where one can observe an improved resolution,especially of the contact waves.

NUMERICAL DISSIPATION FOR THREE-POINT DIFFERENCE SCHEMES TO HYPERBOLIC EQUATIONS WITH UNEVEN MESHES

The widely used locally adaptive Cartesian grid methods involve a series of abruptly refined interfaces. The numerical dissipation due to these interfaces is studied here for three-point difference approximations of a hyperbolic equation. It will be shown that if the wave moves in the fine-to-coarse direction then the dissipation is positive (stabilizing), and if the wave moves in the coarse-to-fine direction then the dissipation is negative (destabilizing).

Modifying and Reducing Numerical Dissipation in A Two-Dimensional Central-Upwind Scheme

This study presents a modification of the central-upwind Kurganov scheme for approximating the solution of the 2D Euler equation.The prototype,extended from a 1D model,reduces substantially less dissipation than expected.The problem arises from over-restriction of some slope limiters,which keep slopes between interfaces of cells to be Total-Variation-Diminishing.This study reports the defect and presents a re-derived optimal formula.Numerical experiments highlight the significance of this formula,especially in long-time,large-scale simulations.

Energetics of lateral eddy diffusion/advection: Part II. Numerical diffusion/diffusivity and gravitational potential energy change due to isopycnal diffusion

Study of oceanic circulation and climate requires models which can simulate tracer eddy diffusion and ad vection accurately. It is shown that the traditional Eulerian coordinates can introduce large artificial hori zontal diffusivity/viscosity due to the incorrect alignment of the axis. Therefore, such models can smear sharp fronts and introduce other numerical artifacts. For simulation with relatively low resolution, large lateral diffusion was explicitly used in models; therefore, such numerical diffusion may not be a problem. However, with the increase of horizontal resolution, the artificial diffusivity/viscosity associated with hori zontal advection in the commonly used Eulerian coordinates may become one of the most challenging ob stacles for modeling the ocean circulation accurately. Isopycnal eddy diffusion （mixing） has been widely used in numerical models. The common wisdom is that mixing along isopycnal is energy free. However, a careful examination reveals that this is not the case. In fact, eddy diffusion can be conceptually separated into two steps： stirring and subscale diffusion. Due to the thermobaric effect, stirring, or exchanging water masses, along isopycnal surface is associated with the change of GPE in the mean state. This is a new type of instability, called the thermobaric instability. In addition, due to cabbeling subscale diffusion of water parcels always leads to the release of GPE. The release of GPE due to isopycnal stirring and subscale diffusion may lead to the thermobaric instability.

NUMERICAL ANALYSIS AND CONSTRUCTION OF LIMITER OF HIGH RESOLUTION DIFFERENCE SCHEME

In the paper, based on the theory of the remainder effects of difference schemes, some typical limiters are analysed and compared. For different limiters, the different strength of numerical dissipation and dispersion of schemes is the reason why the schemes show obvious different characteristics. After analysing and comparing the numerical dissipation and dispersion of various schemes, a new kind of limiter is proposed. The new scheme has high resolution in sharp discontinuities, and avoids the 'distortion' due to the stronger numerical dispersion in the relatively more smooth region. Numerical experiments show that the scheme has good properties.

Fifth-Order A-WENO Schemes Based on the Adaptive Diffusion Central-Upwind Rankine-Hugoniot Fluxes

We construct new fifth-order alternative WENO(A-WENO)schemes for the Euler equations of gas dynamics.The new scheme is based on a new adaptive diffusion centralupwind Rankine-Hugoniot(CURH)numerical flux.The CURH numerical fluxes have been recently proposed in[Garg et al.J Comput Phys 428,2021]in the context of secondorder semi-discrete finite-volume methods.The proposed adaptive diffusion CURH flux contains a smaller amount of numerical dissipation compared with the adaptive diffusion central numerical flux,which was also developed with the help of the discrete RankineHugoniot conditions and used in the fifth-order A-WENO scheme recently introduced in[Wang et al.SIAM J Sci Comput 42,2020].As in that work,we here use the fifth-order characteristic-wise WENO-Z interpolations to evaluate the fifth-order point values required by the numerical fluxes.The resulting one-and two-dimensional schemes are tested on a number of numerical examples,which clearly demonstrate that the new schemes outperform the existing fifth-order A-WENO schemes without compromising the robustness.

Analytical exploration of γ-function explicit method for pseudodynamic testing of nonlinear systems

It has been well studied that the γ-function explicit method can be effective in providing favorable numerical dissipation for linear elastic systems. However, its performance for nonlinear systems is unclear due to a lack of analytical evaluation techniques. Thus, a novel technique is proposed herein to evaluate its efficiency for application to nonlinear systems by introducing two parameters to describe the stiffness change. As a result, the numerical properties and error propagation characteristics of the γ-function explicit method for the pseudodynamic testing of a nonlinear system are analytically assessed. It is found that the upper stability limit decreases as the step degree of nonlinearity increases; and it increases as the current degree of nonlinearity increases. It is also shown that this integration method provides favorable numerical dissipation not only for linear elastic systems but also for nonlinear systems. Furthermore, error propagation analysis reveals that the numerical dissipation can effectively suppress the severe error propagation of high frequency modes while the low frequency responses are almost unaffected for both linear elastic and nonlinear systems.

Shedding vortex simulation method based on viscous compensation technology research

Owing to the influence of the viscosity of the flow field,the strength of the shedding vortex decreases gradually in the process of backward propagation.Large-scale vortexes constantly break up,forming smaller vortexes.In engineering,when numerical simulation of vortex evolution process is carried out,a large grid is needed to be arranged in the area of outflow field far from the boundary layer in order to ensure the calculation efficiency.As a result,small scale vortexes at the far end of the flow field cannot be captured by the sparse grid in this region,resulting in the dissipation or even disappearance of vortexes.In this paper,the effect of grid scale is quantified and compared with the viscous effect through theoretical derivation.The theoretical relationship between the mesh viscosity and the original viscosity of the flow field is established,and the viscosity term in the turbulence model is modified.This method proves to be able to effectively improve the intensity of small-scale shedding vortexes at the far end of the flow field under the condition of sparse grid.The error between the simulation results and the results obtained by using fine mesh is greatly reduced,the calculation time is shortened,and the high-precision and efficient simulation of the flow field is realized.

Improved quadratic isogeometric element simulation of one-dimensional elastic wave propagation with central difference method

Two improved isogeometric quadratic elements and the central difference scheme are used to formulate the solution procedures of transient wave propagation prob- lems. In the proposed procedures, the lumped matrices corresponding to the isogeomet- ric elements are obtained. The stability conditions of the solution procedures are also acquired. The dispersion analysis is conducted to obtain the optimal Courant-Friedrichs- Lewy （CFL） number or time-step sizes corresponding to the spatial isogeometric elements. The dispersion analysis shows that the isogeometric quadratic element of the fourth-order dispersion error （called the isogeometric analysis （IGA）-f quadratic element） provides far more desirable numerical dissipation/dispersion than the element of the second-order dis- persion error （called the IGA-s quadratic element） when appropriate time-step sizes are selected. The numerical simulations of one-dimensional （1D） transient wave propagation problems demonstrate the effectiveness of the proposed solution procedures.

Investigation and Improvement of the Staggered Labyrinth Seal

Recent studies on staggered labyrinth seals have focused on the effects of different parameters,such as the pressure ratio and rotational speed on the leakage flow rate.However,few investigations pay sufficient attention to flow details and the sealing mechanism,which would be of practical importance in designing seals having higher performance.This paper establishes a theoretical model to study the seal mechanism,thus revealing that leakage is determined by the pressure ratio and geometric structure.Numerical simulation is implemented to illustrate details of the flow field within the seal structure.Viscous dissipation is used to quantitatively investigate the contribution that each location makes to the seal performance,revealing that orifices and stagnation points are the most important positions in the seal structure,generating the most dissipation.The orifice is carefully studied by using the theoretical model.Experiments for different pressure ratios are conducted and the results match well with those of the theoretical model and numerical simulation,verifying the theoretical model and analysis of the seal mechanism.Three new designs,based on a good understanding of the seal mechanism,are presented,with one reducing leakage by 24.5%.

An improved CESE method and its application to steady-state coronal structure simulation

This paper presents an improved space-time conservation element and solution element(CESE)method by applying a non-staggered space-time mesh system and simply improving the calculation of flow variables and applies it to magnetohydrodynamics(MHD)equations.The improved CESE method can improve the solution quality even with a large disparity in the Courant number(CFL)when using a fixed global marching time.Moreover,for a small CFL(say<0.1),the method can significantly reduce the numerical dissipation and retain the solution quality,which are verified by two benchmark problems.And meanwhile,comparison with the original CESE scheme shows better resolution of the improved scheme results.Finally,we demonstrate its validation through the application of this method in three-dimensional coronal dynamical structure with dipole magnetic fields and measured solar surface magnetic fields as the initial input.

A high-order SPH method by introducing inverse kernels

The smoothed particle hydrodynamics（SPH） method is usually expected to be an efficient numerical tool for calculating the fluid-structure interactions in compressors; however, an endogenetic restriction is the problem of low-order consistency. A high-order SPH method by introducing inverse kernels, which is quite easy to be implemented but efficient, is proposed for solving this restriction. The basic inverse method and the special treatment near boundary are introduced with also the discussion of the combination of the Least-Square（LS） and Moving-Least-Square（MLS） methods. Then detailed analysis in spectral space is presented for people to better understand this method. Finally we show three test examples to verify the method behavior.