Crank-Nicolson Quasi-Compact Scheme for the Nonlinear Two-Sided Spatial Fractional Advection-Diffusion Equations

The higher-order numerical scheme of nonlinear advection-diffusion equations is studied in this article, where the space fractional derivatives are evaluated by using weighted and shifted Grünwald difference operators and combining the compact technique, in the time direction is discretized by the Crank-Nicolson method. Through the energy method, the stability and convergence of the numerical scheme in the sense of L2-norm are proved, and the convergence order is . Some examples are given to show that our numerical scheme is effective.

THE THREE-POINT FIFTH-ORDER ACCURATE GENERALIZED COMPACT SCHEME AND ITS APPLICATIONS

A three-point fifth-order accurate generalized compact scheme (GC scheme) with a spectral-like resolution is constructed in a general way. The scheme satisfies the principle of stability and the principle about suppression of the oscillations, therefore numerical errors can decay automatically and no spurious oscillations are generated around shocks. The third-order TVD type Runge-Kutta method is employed for the time integration, thus making the GC scheme best suited for unsteady problems. Numerical results show that the GC scheme is shock-capturing. The time-dependent boundary conditions proposed by Thompson are well employed when the algorithm is applied to the Euler equations of gas dynamics.

Stabilized seventh-order dissipative compact scheme for two-dimensional Euler equations

We derive in this paper a time stable seventh-order dissipative compact finite difference scheme with simultaneous approximation terms(SATs) for solving two-dimensional Euler equations. To stabilize the scheme, the choice of penalty coefficients for SATs is studied in detail. It is demonstrated that the derived scheme is quite suitable for multi-block problems with different spacial steps. The implementation of the scheme for the case with curvilinear grids is also discussed.Numerical experiments show that the proposed scheme is stable and achieves the design seventh-order convergence rate.

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.

WEIGHTED COMPACT SCHEME FOR SHOCK CAPTURING

A new class of finite difference schemes--the weighted compact schemes are proposed. According to the idea of the WENO schemes, the weighted compact scheme is constructed by a combination of the approximations of derivatives on candidate stencils with properly assigned weights so that the non oscillatory property is achieved when discontinuities appear. The primitive function reconstruction method of ENO schemes is applied to obtain the conservative form of the weighted compact scheme. This new scheme not only preserves the characteristic of standard compact schemes and achieves high order accuracy and high resolution using a compact stencil, but also can accurately capture shock waves and discontinuities without oscillation. Numerical examples show that the new scheme is very promising and successful.

Stability Analysis of Inverse Lax-Wendroff Procedure for a High order Compact Finite Difference Schemes

This paper considers the finite difference(FD)approximations of diffusion operators and the boundary treatments for different boundary conditions.The proposed schemes have the compact form and could achieve arbitrary even order of accuracy.The main idea is to make use of the lower order compact schemes recursively,so as to obtain the high order compact schemes formally.Moreover,the schemes can be implemented efficiently by solving a series of tridiagonal systems recursively or the fast Fourier transform(FFT).With mathematical induction,the eigenvalues of the proposed differencing operators are shown to be bounded away from zero,which indicates the positive definiteness of the operators.To obtain numerical boundary conditions for the high order schemes,the simplified inverse Lax-Wendroff(SILW)procedure is adopted and the stability analysis is performed by the Godunov-Ryabenkii method and the eigenvalue spectrum visualization method.Various numerical experiments are provided to demonstrate the effectiveness and robustness of our algorithms.

New Sixth-Order Compact Schemes for Poisson/Helmholtz Equations

Some new sixth-order compact finite difference schemes for Poisson/Helmholtz equations on rectangular domains in both two-and three-dimensions are developed and analyzed.Different from a few sixth-order compact finite difference schemes in the literature,the finite difference and weight coefficients of the new methods have analytic simple expressions.One of the new ideas is to use a weighted combination of the source term at staggered grid points which is important for grid points near the boundary and avoids partial derivatives of the source term.Furthermore,the new compact schemes are exact for 2D and 3D Poisson equations if the solution is a polynomial less than or equal to 6.The coefficient matrices of the new schemes are M-matrices for Helmholtz equations with wave number K≤0,which guarantee the discrete maximum principle and lead to the convergence of the new sixth-order compact schemes.Numerical examples in both 2D and 3D are presented to verify the effectiveness of the proposed schemes.

Developing Hybrid cell-edge and cell-node Dissipative Compact Scheme for Complex Geometry Flows

Developing high resolution finite difference scheme and enabling the use of this scheme on complex geometry are the aims of this study.High resolution has been achieved by Dissipative Compact Schemes(DCS),however,according to the recent research,applications of DCS on complex geometry may have serious problem for that the Geometric Conservation Law(GCL)is not satisfied,and this may cause numerical instability.To cope with this problem,a new scheme named Hybrid cell-edge and cell-node Dissipative Compact Scheme(HDCS)has been formulated.The formulation of the HDCS contains two steps.First,a new central compact scheme is formulated for the purpose of conveniently fulfilling the GCL,and then dissipation is added on the central scheme by high-order dissipative interpolation of cell-edge variables.The solutions of Euler and Navier-Stokes equations show that the HDCS can be applied successfully on complex geometry,while the DCS may suffer numerical instabilities.Moreover,high resolution of the HDCS may be observed in the test of scattering of acoustic waves by multiple cylinders.

AN ALE METHOD AND DDM WITH HIGH ACCURATE COMPACT SCHEMES FOR VORTEX-INDUCED VIBRATIONS OF AN ELASTIC CIRCULAR CYLINDER

A numerical study was conducted for the vortex-induced vibrations of anelastic circular cylinder at low Reynolds numbers. An Arbitrary Lagrangian-Eulerian (ALE) method wasemployed to deal with the fluid-structure interaction with an H-O type of non-staggered gridsincorporating the domain decomposition method (DDM), which could save the computational CPU time dueto re-meshing. The computational domain was divided into nine sub-domains including one ALEsub-domain and eight Eulerian sub-domains. The convection term and dissipation term in the N-Sequations were discretized using the third-order upwind compact scheme and the fourth-order centralcompact scheme, respectively. The motion of the cylinder was modeled by a spring-damper-mass systemand solved using the Runge-Kutta method. By simulating the non-linear fluid-structure interaction,the ''lock-in'', ''beating'' and ''phase switch'' phenomena were successfully captured, and the resultsagree with experimental data Furthermore, the vortex structure, the unsteady lift and drag on thecylinder, and the cylinder displacement at various natural frequency of the cylinder for Re = 200were discussed in detail, by which a jump transition of the wake structure was captured.

Fourth order accurate compact scheme with group velocity control (GVC)

For solving complex flow field with multi-scale structure higher order accurate schemes are preferred. Among high order schemes the compact schemes have higher resolving efficiency. When the compact and upwind compact schemes are used to solve aerodynamic problems there are numerical oscillations near the shocks. The reason of oscillation production is because of non-uniform group velocity of wave packets in numerical solutions. For improvement of resolution of the shock a parameter function is introduced in compact scheme to control the group velocity. The newly developed method is simple. It has higher accuracy and less stencil of grid points.

Aerodynamic Noise Propagation Simulation using Immersed Boundary Method and Finite Volume Optimized Prefactored Compact Scheme

Based on the immersed boundary method （IBM） and the finite volume optimized pre-factored compact （FVOPC） scheme, a numerical simulation of noise propagation inside and outside the casing of a cross flow fan is estab- lished. The unsteady linearized Euler equations are solved to directly simulate the aero-acoustic field. In order to validate the FVOPC scheme, a simulation case： one dimensional linear wave propagation problem is carried out using FVOPC scheme, DRP scheme and HOC scheme. The result of FVOPC is in good agreement with the ana- lytic solution and it is better than the results of DRP and HOC schemes, the FVOPC is less dispersion and dissi- pation than DRP and HOC schemes. Then, numerical simulation of noise propagation problems is performed. The noise field of 36 compact rotating noise sources is obtained with the rotating velocity of 1000r/min. The PML absorbing boundary condition is applied to the sound far field boundary condition for depressing the numerical reflection. Wall boundary condition is applied to the casing. The results show that there are reflections on the casing wall and sound wave interference in the field. The FVOPC with the IBM is suitable for noise propagation problems under the complex geometries for depressing the dispersion and dissipation, and also keeping the high order precision.

A CLASS OF COMPACT UPWIND TVD DIFFERENCE SCHEMES

A new method was proposed for constructing total variation diminishing （TVD） upwind schemes in conservation forms. Two limiters were used to prevent nonphysical oscillations across discontinuity. Both limiters can ensure the nonlinear compact schemes TVD property. Two compact TVD （CTVD） schemes were tested, one is thirdorder accuracy, and the other is fifth-order. The performance of the numerical algorithms was assessed by one-dimensional complex waves and Riemann problems, as well as a twodimensional shock-vortex interaction and a shock-boundary flow interaction. Numerical results show their high-order accuracy and high resolution, and low oscillations across discontinuities.

CONSERVATION OF THREE-POINT COMPACT SCHEMES ON SINGLE AND MULTIBLOCK PATCHED GRIDS FOR HYPERBOLIC PROBLEMS

For nonlinear hyperbolic problems, conservation of the numerical scheme is important for convergence to the correct weak solutions. In this paper the conservation of the well-known compact scheme up to fourth order of accuracy on a single and uniform grid is studied, and a conservative interface treatment is derived for compact schemes on patched grids. For a pure initial value problem, the compact scheme is shown to be equivalent to a scheme in the usual conservative form. For the case of a mixed initial boundary value problem, the compact scheme is conservative only if the rounding errors are small enough. For a patched grid interface, a conservative interface condition useful for mesh refinement and for parallel computation is derived and its order of local accuracy is analyzed.

ON THE EXPLICIT COMPACT SCHEMES II: EXTENSION OF THE STCE/CE METHOD ON NONSTAGGERED GRIDS ON THE EXPLICIT COMPACT SCHEMES II: EXTENSION OF THE STCE/CE METHOD ON NONSTAGGERED GRIDS

This paper continues to construct and study the explicit compact (EC) schemes for conservation laws. First, we axtend STCE/SE method on non-staggered grid, which has same well resolution as one in [1], and just requires half of the computational works. Then, we consider some constructions of the EC schemes for two-dimensional conservation laws, and some 1D and 2D numerical experiments are also given.

Multiderivative Combined Dissipative Compact Scheme Satisfying Geometric Conservation Law III: Characteristic-Wise Hybrid Method

Based on newly developed weight-based smoothness detectors and non-linear interpolations designed to capture discontinuities for the multiderivative com-bined dissipative compact scheme(MDCS),hybrid linear and nonlinear interpolations are proposed to form hybrid MDCS.These detectors are derived from the weights used for the nonlinear interpolations and can provide suitable switches between the linear and the nonlinear schemes to realize the characteristics for the hybrid MDCS of capturing discontinuities and maintaining high resolution in the region without large discontinuities.To save computational cost,the nonlinear scheme with characteris-tic decomposition is only applied in the detected discontinuities region by specially designed hybrid strategy.Typical tests show that the hybrid MDCS is capable of cap-turing discontinuities and maintaining high resolution power for the smooth region at the same time.With the satisfaction of the geometric conservative law(GCL),the MDCS is further applied on curvilinear mesh to present its promising capability of handling pragmatic simulations.

A Compact Scheme for Coupled Stochastic Nonlinear Schrodinger Equations

In this paper,we propose a compact scheme to numerically study the coupled stochastic nonlinear Schrodinger equations.We prove that the compact scheme preserves the discrete stochastic multi-symplectic conservation law,discrete charge conservation law and discrete energy evolution law almost surely.Numerical experiments confirm well the theoretical analysis results.Furthermore,we present a detailed numerical investigation of the optical phenomena based on the compact scheme.By numerical experiments for various amplitudes of noise,we find that the noise accelerates the oscillation of the soliton and leads to the decay of the solution amplitudes with respect to time.In particular,if the noise is relatively strong,the soliton will be totally destroyed.Meanwhile,we observe that the phase shift is sensibly modified by the noise.Moreover,the numerical results present inelastic interaction which is different from the deterministic case.

Three-point explicit compact difference scheme with arbitrary order of accuracy and its application in CFD

Based on the successive iteration in the Taylor series expansion method, a three-point explicit compact difference scheme with arbitrary order of accuracy is derived in this paper. Numerical characteristics of the scheme are studied by the Fourier analysisl Unlike the conventional compact difference schemes which need to solve the equation to obtain the unknown derivatives in each node, the proposed scheme is explicit and can achieve arbitrary order of accuracy in space. Application examples for the convectiondiffusion problem with a sharp front gradient and the typical lid-driven cavity flow are given. It is found that the proposed compact scheme is not only simple to implement and economical to use, but also is effective to simulate the convection-dominated problem and obtain high-order accurate solution in coarse grid systems.

A Fifth-Order Low-Dissipative Conservative Upwind Compact Scheme Using Centered Stencil

In this paper,a conservative fifth-order upwind compact scheme using centered stencil is introduced.This scheme uses asymmetric coefficients to achieve the upwind property since the stencil is symmetric.Theoretical analysis shows that the proposed scheme is low-dissipative and has a relatively large stability range.To maintain the convergence rate of the whole spatial discretization,a proper non-periodic boundary scheme is also proposed.A detailed analysis shows that the spatial discretization implemented with the boundary scheme proposed by Pirozzoli[J.Comput.Phys.,178(2001),pp.81–117]is approximately fourth-order.Furthermore,a hybridmethodology,coupling the compact scheme with WENO scheme,is adopted for problems with discontinuities.Numerical results demonstrate the effectiveness of the proposed scheme.

Modified Upwinding Compact Scheme for Shock and Shock Boundary Layer Interaction

Standard compact scheme and upwinding compact scheme have high order accuracy and high resolution,but cannot capture the shock which is a discontinuity.This work developed a modified upwinding compact scheme which uses an effective shock detector to block compact scheme to cross the shock and a control function to mix the flux with WENO scheme near the shock.The new scheme makes the original compact scheme able to capture the shock sharply and,more importantly,keep high order accuracy and high resolution in the smooth area which is particularly important for shock boundary layer and shock acoustic interactions.Numerical results show the scheme is successful for 2-D Euler and 2-D Navier-Stokes solvers.The examples include 2-D incident shock,2-D incident shock and boundary layer interaction.The scheme is robust,which does not involve case related parameters.

High Order Compact Schemes in Projection Methods for Incompressible Viscous Flows

Within the projection schemes for the incompressible Navier-Stokes equations(namely"pressure-correction"method),we consider the simplest method(of order one in time)which takes into account the pressure in both steps of the splitting scheme.For this scheme,we construct,analyze and implement a new high order compact spatial approximation on nonstaggered grids.This approach yields a fourth order accuracy in space with an optimal treatment of the boundary conditions(without error on the velocity)which could be extended to more general splitting.We prove the unconditional stability of the associated Cauchy problem via von Neumann analysis.Then we carry out a normal mode analysis so as to obtain more precise results about the behavior of the numerical solutions.Finally we present detailed numerical tests for the Stokes and the Navier-Stokes equations(including the driven cavity benchmark)to illustrate the theoretical results.