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 oper...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 L<sub>2</sub>-norm are proved, and the convergence order is . Some examples are given to show that our numerical scheme is effective.展开更多
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 suppre...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.展开更多
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 ch...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.展开更多
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 ra...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.展开更多
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 deri...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.展开更多
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 ...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.展开更多
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...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.展开更多
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,acco...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.展开更多
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 inte...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.展开更多
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...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.展开更多
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 est...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.展开更多
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 e...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.展开更多
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...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.展开更多
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 requir...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.展开更多
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 nonline...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.展开更多
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 la...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.展开更多
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 sch...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.展开更多
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 an...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.展开更多
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 use...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.展开更多
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 b...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.展开更多
文摘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 L<sub>2</sub>-norm are proved, and the convergence order is . Some examples are given to show that our numerical scheme is effective.
基金The project supported by the National Natural Science Foundation of China (19972038)Foundation of the National CFD Laboratory of China
文摘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.
基金Project supported by the National Natural Science Foundation of China(Grant No.11601517)the Basic Research Foundation of National University of Defense Technology(Grant No.ZDYYJ-CYJ20140101)
文摘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.
基金supported by the National Natural Science Foundation of China(No.11601517)the Basic Research Foundation of National University of Defense Technology(No.ZDYYJ-CYJ20140101)
文摘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.
文摘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.
基金supported by the NSFC grant 11801143J.Lu’s research is partially supported by the NSFC grant 11901213+3 种基金the National Key Research and Development Program of China grant 2021YFA1002900supported by the NSFC grant 11801140,12171177the Young Elite Scientists Sponsorship Program by Henan Association for Science and Technology of China grant 2022HYTP0009the Program for Young Key Teacher of Henan Province of China grant 2021GGJS067.
文摘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.
基金supported by the National Natural Science Foundation of China(Grant No.42274101)and the Excellent Youth Foundation of Hunan Province of China(Grant No.2018JJ1042)Hongling Hu was supported by the National Natural Science Foundation of China(Grant No.12071128)the Natural Science Foundation of Hunan Province(Grant No.2021JJ30434).Zhilin Li was partially supported by a Simons Grant No.633724.
文摘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.
基金supported by the National Basic Research Program of China(Grant no.2009CB723800)National Natural Science Foundation of China(Grand Nos.11072259 and 11202226)the Foundation of State Key Laboratory of Aerodynamics(Grand Nos.JBKY11030902 and JBKY11010100)
文摘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.
文摘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.
基金the computers of the State Kay Laboratory of Scientific and Engineering Computing, the Chinese Academy of Sciences This work was supporded by the National Natural Science Foundation of China (Grant No. 19972070) "95" Project and 973 Project (Grant N
文摘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.
基金the university doctorate fund of China(Grant No.20060487036)the National Natural Science Foundation of China (Grant No.50676035)
文摘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.
基金Project supported by the National Natural Science Foundation of China (Nos. 10172015 and 90205010)
文摘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.
基金This work was supported by Chinese NSF(Contract No.10025210).Running head:Conservation of Compact Schemes.
文摘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.
基金This work was supported in part by National Natural Science Foundation of China, the StateMajor Key Project for Basic Research
文摘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.
基金supported by the National Key Research and Development Plan(grant No.2016YFB0200700)the National Natural Science Foundation of China(grant Nos.11372342,11572342,and 11672321)the National Key Project GJXM92579.
文摘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.
基金This work was supported by the National Natural Science Foundation of China(Nos.91530118,91130003,11021101,11290142,11471310,11601032,11301234,11271171)the Provincial Natural Science Foundation of Jiangxi(Nos.20142BCB23009,20161ACB20006,20151BAB201012).
文摘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.
基金Project supported by the National Natural Science Foundation of China(No.50479053)
文摘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.
基金supported by National Natural Science Foundation of China(91130030).The authors are grateful to Dr.Jie Wu for his help on the manuscript.
文摘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.
基金This work is supported by AFRL VA Summer Faculty Research Program.The authors thank Drs.Poggie,Gaitonde,Visbal for their support through VA Summer Faculty Program.
文摘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.
文摘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.
