A high-order splitting scheme for the advection diffusion equation

A high-order splitting scheme for the advection-diffusion equation of pollutants is proposed in this paper. The multidimensional advection-diffusion equation is splitted into several one-dimensional equations that are solved by the scheme. Only three spatial grid points are needed in each direction and the scheme has fourth-order spatial accuracy. Several typically pure advection and advection-diffusion problems are simulated. Numerical results show that the accuracy of the scheme is much higher than that of the classical schemes and the scheme can he efficiently solved with little programming effort.

A High-Order Spatiotemporal Precision-Matching Taylor–Li Scheme for Time-Dependent Problems

Based on the Taylor series method and Li’s spatial differential method, a high-order hybrid Taylor–Li scheme is proposed.The results of a linear advection equation indicate that, using the initial values of the square-wave type, a result with thirdorder accuracy occurs. However, using initial values associated with the Gaussian function type, a result with very high precision appears. The study demonstrates that, when the order of the time integral is more than three, the corresponding optimal spatial difference order could be higher than six. The results indicate that the reason for why there is no improvement related to an order of spatial difference above six is the use of a time integral scheme that is not high enough. The author also proposes a recursive differential method to improve the Taylor–Li scheme’s computation speed. A more rapid and highprecision program than direct computation of the high-order space differential item is employed, and the computation speed is dramatically boosted. Based on a multiple-precision library, the ultrahigh-order Taylor–Li scheme can be used to solve the advection equation and Burgers’ equation.

High-order schemes for predicting computational aeroacoustic propagation with adaptive mesh refinement

High-order schemes based on block-structured adaptive mesh refinement method are prepared to solve computational aeroacoustic （CAA） problems with an aim at improving computational efficiency. A number of numerical issues associated with high-order schemes on an adaptively refined mesh, such as stability and accuracy are addressed. Several CAA benchmark problems are used to demonstrate the feasibility and efficiency of the approach.

A－HIGH－ORDER ACCURACY EXPLICIT DIFFERENCE SCHEME FOR SOLVING THE EQUATION OF TWO－DIMENSIONAL PARABOLIC TYPE

In this paper. a three explicit difference shcemes with high order accuracy for solving the equations of two-dimensional parabolic type is proposed. The stability condition is r=△t/△x￣ 2=△t/△y￣2≤1/4 and the truncation error is O (△t￣2 + △x￣4 ).

A FAMILY OF HIGH-ORDER ACCURACY EXPLICIT DIFFERENCE SCHEMES WITH BRANCHING STABILITY FOR SOLVING 3-D PARABOLIC PARTIAL DIFFERENTIAL EQUATION

A family of high-order accuracy explict difference schemes for solving 3-dimension parabolic P. D. E. is constructed. The stability condition is r = Deltat/Deltax(2) Deltat/Deltay(2) = Deltat/Deltaz(2) < 1/2 ,and the truncation error is 0(<Delta>t(2) + Deltax(4)).

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.

Stability of High-Order Staggered-Grid Schemes for 3D Elastic Wave Equation in Heterogeneous Media

In this paper, we firstly derive the stability conditions of high-order staggered-grid schemes for the three-dimensional (3D) elastic wave equation in heterogeneous media based on the energy method. Moreover, the plane wave analysis yields a sufficient and necessary stability condition by the von Neumann criterion in homogeneous case. Numerical computations for 3D wave simulation with point source excitation are given.

Perfect plane-wave source for a high-order symplectic finite-difference time-domain scheme

The method of splitting a plane-wave finite-difference time-domain （SP-FDTD） algorithm is presented for the initiation of plane-wave source in the total-field / scattered-field （TF/SF） formulation of high-order symplectic finite- difference time-domain （SFDTD） scheme for the first time. By splitting the fields on one-dimensional grid and using the nature of numerical plane-wave in finite-difference time-domain （FDTD）, the identical dispersion relation can be obtained and proved between the one-dimensional and three-dimensional grids. An efficient plane-wave source is simulated on one-dimensional grid and a perfect match can be achieved for a plane-wave propagating at any angle forming an integer grid cell ratio. Numerical simulations show that the method is valid for SFDTD and the residual field in SF region is shrinked down to -300 dB.

A NEW HIGH-ORDER ACCURACY EXPLICIT DIFFERENCE SCHEME FOR SOLVING THREE-DIMENSIONAL PARABOLIC EQUATIONS

In this paper, a new three-level explicit difference scheme with high-order accuracy is proposed for solving three-dimensional parabolic equations. The stability condition is r = Delta t/Delta x(2) = Delta t/Delta gamma(2) = Delta t/Delta z(2) less than or equal to 1/4, and the truncation error is O(Delta t(2) + Delta x(4)).

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.

Implicit high-order discontinuous Galerkin method with HWENO type limiters for steady viscous flow simulations

Two types of implicit algorithms have been im- proved for high order discontinuous Galerkin （DG） method to solve compressible Navier-Stokes （NS） equations on tri- angular grids. A block lower-upper symmetric Gauss- Seidel （BLU-SGS） approach is implemented as a nonlin- ear iterative scheme. And a modified LU-SGS （LLU-SGS） approach is suggested to reduce the memory requirements while retain the good convergence performance of the origi- nal LU-SGS approach. Both implicit schemes have the sig- nificant advantage that only the diagonal block matrix is stored. The resulting implicit high-order DG methods are applied, in combination with Hermite weighted essentially non-oscillatory （HWENO） limiters, to solve viscous flow problems. Numerical results demonstrate that the present implicit methods are able to achieve significant efficiency improvements over explicit counterparts and for viscous flows with shocks, and the HWENO limiters can be used to achieve the desired essentially non-oscillatory shock tran- sition and the designed high-order accuracy simultaneously.

Stabilized seventh-order dissipative compact scheme using simultaneous approximation terms

To ensure time stability of a seventh-order dissipative compact finite difference scheme, fourth-order boundary closures are used near domain boundaries previously. However, this would reduce the global convergence rate to fifth-order only. In this paper, we elevate the boundary closures to sixth-order to achieve seventh-order global accuracy. To keep the improved scheme time stable, the simultaneous approximation terms (SATs) are used to impose boundary conditions weakly. Eigenvalue analysis shows that the improved scheme is time stable. Numerical experiments for linear advection equations and one-dimensional Euler equations are implemented to validate the new scheme.

CONSTRUCTION OF THIRD-ORDER WNND SCHEMEAND ITS APPLICATION IN COMPLEX FLOW

According to the Liu's weighted idea, a space third-order WNND (weighted non-oscillatory, containing no free parameters, and dissipative scheme) scheme was constructed based on the stencils of second-order NND (non-oscillatory, containing no free parameters, and dissipative scheme) scheme. It was applied in solving linear-wave equation, 1D Euler equations and 3D Navier-Stokes equations. The numerical results indicate that the WNND scheme which does not increase interpolated point(compared to NND scheme) has more advantages in simulating discontinues and convergence than NND scheme. Appling WNND scheme to simulating the hypersonic flow around lift-body shows:With the AoA(angle of attack) increasing from 0° to 50°, the structure of limiting streamline of leeward surface changes from unseparating,open-separating to separating, which occurs from the combined-point (which consists of saddle and node points). The separating area of upper wing surface is increasing with the (AoA's) increasing. The topological structures of hypersonic flowfield based on the sectional flow patterns perpendicular to the body axis agree well with Zhang Hanxin's theory. Additionally, the unstable-structure phenomenon which is showed by two saddles connection along leeward symmetry line occurs at some sections when the AoA is bigger than 20°.

Prediction of Better Flow Control Parameters in MHD Flows Using a High Accuracy Finite Difference Scheme

We have successfully attempted to solve the equations of full-MHD model within the framework of &Psi;- &omega;formulation with an objective to evaluate the performance of a new higher order scheme to predict better values of control parameters of the flow. In particular for MHD flows, magnetic field and electrical conductivity are the control parameters. In this work, the results from our efficient high order accurate scheme are compared with the results of second order method and significant discrepancies are noted in separation length, drag coefficient and mean Nusselt number. The governing Navier-Stokes equation is fully nonlinear due to its coupling with Maxwell’s equations. The momentum equation has several highly nonlinear body-force terms due to full-MHD model in cylindrical polar system. Our high accuracy results predict that a relatively lower magnetic field is sufficient to achieve full suppression of boundary layer and this is a favorable result for practical applications. The present computational scheme predicts that a drag-coefficient minimum can be achieved when &beta;=0.4 which is much lower when compared to the value &beta;=1 as given by second order method. For a special value of &beta;=0.65, it is found that the heat transfer rate is independent of electrical conductivity of the fluid. From the numerical values of physical quantities, we establish that the order of accuracy of the computed numerical results is fourth order accurate by using the method of divided differences.

A Third-Order Scheme for Numerical Fluxes to Guarantee Non-Negative Coefficients for Advection-Diffusion Equations

According to Godunov theorem for numerical calculations of advection equations, there exist no high-er-order schemes with constant positive difference coefficients in a family of polynomial schemes with an accuracy exceeding the first-order. In case of advection-diffusion equations, so far there have been not found stable schemes with positive difference coefficients in a family of numerical schemes exceeding the second-order accuracy. We propose a third-order computational scheme for numerical fluxes to guarantee the non-negative difference coefficients of resulting finite difference equations for advection-diffusion equations. The present scheme is optimized so as to minimize truncation errors for the numerical fluxes while fulfilling the positivity condition of the difference coefficients which are variable depending on the local Courant number and diffusion number. The feature of the present optimized scheme consists in keeping the third-order accuracy anywhere without any numerical flux limiter by using the same stencil number as convemtional third-order shemes such as KAWAMURA and UTOPIA schemes. We extend the present method into multi-dimensional equations. Numerical experiments for linear and nonlinear advection-diffusion equations were performed and the present scheme’s applicability to nonlinear Burger’s equation was confirmed.

Efficient high-order immersed interface methods for heat equations with interfaces

An efficient high-order immersed interface method （IIM） is proposed to solve two-dimensional （2D） heat problems with fixed interfaces on Cartesian grids, which has the fourth-order accuracy in the maximum norm in both time and space directions. The space variable is discretized by a high-order compact （HOC） difference scheme with correction terms added at the irregular points. The time derivative is integrated by a Crank-Nicolson and alternative direction implicit （ADI） scheme. In this case, the time accuracy is just second-order. The Richardson extrapolation method is used to improve the time accuracy to fourth-order. The numerical results confirm the convergence order and the efficiency of the method.

Optimization of a global seventh-order dissipative compact finite-difference scheme by a genetic algorithm

A global seventh-order dissipative compact finite-difference scheme is optimized in terms of time stability. The dissipative parameters appearing in the boundary closures are assumed to be different, resulting in an optimization problem with several parameters determined by applying a generic algorithm. The optimized schemes are analyzed carefully from the aspects of the eigenvalue distribution, the ε-pseudospectra, the short time behavior, and the Fourier analysis. Numerical experiments for the Euler equations are used to show the effectiveness of the final recommended scheme.

高阶精度非线性加权格式权函数研究综述

在简述用于捕捉间断的非线性格式发展历程的基础上,依托五阶精度WENO格式,介绍了非线性加权格式在候选模板集选取、光滑度指标计算方法及其与非线性权的函数关系等方面做出的努力。对于非等宽候选模板集的情况,给出了保证非线性加权格式精度的非线性权的量阶关系,强调了光滑度指标计算方法对格式精度和效率的重要性,提出了进一步开展多宽度模板集非线性加权格式研究的建议。

2维薛定谔方程的一种高精度紧致差分格式

该文对2维薛定谔方程利用局部一维化方法,将2维方程分裂为x、y方向的2个1维薛定谔方程,然后采用6阶紧致格式的离散方法来处理空间变量的2阶导数项,将薛定谔方程转化为一个常微分方程组.通过L-稳定Simpson方法对上述空间离散化得到的常微分方程进行离散化,得到了一种具有空间6阶精度和时间3阶精度的格式,并证明了该格式无条件稳定性.并通过数值模拟和对比方法验证了格式的有效性.

Research of influence of reduced-order boundary on accuracy and solution of interior points

The flow field with a high order scheme is usually calculated so as to solve complex flow problems and describe the flow structure accurately. However, there are two problems, i.e., the reduced-order boundary is inevitable and the order of the scheme at the discontinuous shock wave contained in the flow field as the supersonic flow field is low. It is questionable whether the reduced-order boundary and the low-order scheme at the shock wave have an effect on the numerical solution and accuracy of the flow field inside. In this paper, according to the actual situation of the direct numerical simulation of the flow field, two model equations with the exact solutions are solved, which are steady and unsteady, respectively, to study the question with a high order scheme at the interior of the domain and the reduced-order method at the boundary and center of the domain. Comparing with the exact solutions, it is found that the effect of reduced-order exists and cannot be ignored. In addition, the other two model equations with the exact solutions, which are often used in fluid mechanics, are also studied with the same process for the reduced-order problem.