In this paper,we propose a finite volume Hermite weighted essentially non-oscillatory(HWENO)method based on the dimension by dimension framework to solve hyperbolic conservation laws.It can maintain the high accuracy ...In this paper,we propose a finite volume Hermite weighted essentially non-oscillatory(HWENO)method based on the dimension by dimension framework to solve hyperbolic conservation laws.It can maintain the high accuracy in the smooth region and obtain the high resolution solution when the discontinuity appears,and it is compact which will be good for giving the numerical boundary conditions.Furthermore,it avoids complicated least square procedure when we implement the genuine two dimensional(2D)finite volume HWENO reconstruction,and it can be regarded as a generalization of the one dimensional(1D)HWENO method.Extensive numerical tests are performed to verify the high resolution and high accuracy of the scheme.展开更多
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...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.展开更多
In this paper,a high-order moment-based multi-resolution Hermite weighted essentially non-oscillatory(HWENO)scheme is designed for hyperbolic conservation laws.The main idea of this scheme is derived from our previous...In this paper,a high-order moment-based multi-resolution Hermite weighted essentially non-oscillatory(HWENO)scheme is designed for hyperbolic conservation laws.The main idea of this scheme is derived from our previous work[J.Comput.Phys.,446(2021)110653],in which the integral averages of the function and its first order derivative are used to reconstruct both the function and its first order derivative values at the boundaries.However,in this paper,only the function values at the Gauss-Lobatto points in the one or two dimensional case need to be reconstructed by using the information of the zeroth and first order moments.In addition,an extra modification procedure is used to modify those first order moments in the troubledcells,which leads to an improvement of stability and an enhancement of resolution near discontinuities.To obtain the same order of accuracy,the size of the stencil required by this moment-based multi-resolution HWENO scheme is still the same as the general HWENO scheme and is more compact than the generalWENO scheme.Moreover,the linear weights are not unique and are independent of the node position,and the CFL number can still be 0.6whether for the one or two dimensional case,which has to be 0.2 in the two dimensional case for other HWENO schemes.Extensive numerical examples are given to demonstrate the stability and resolution of such moment-based multi-resolution HWENO scheme.展开更多
In this paper,we developed a class of the fourth order accurate finite volume Hermite weighted essentially non-oscillatory(HWENO)schemes based on the work(Computers&Fluids,34:642-663(2005))by Qiu and Shu,with Tota...In this paper,we developed a class of the fourth order accurate finite volume Hermite weighted essentially non-oscillatory(HWENO)schemes based on the work(Computers&Fluids,34:642-663(2005))by Qiu and Shu,with Total Variation Diminishing Runge-Kutta time discretization method for the two-dimensional hyperbolic conservation laws.The key idea of HWENO is to evolve both with the solution and its derivative,which allows for using Hermite interpolation in the reconstruction phase,resulting in a more compact stencil at the expense of the additional work.The main difference between this work and the formal one is the procedure to reconstruct the derivative terms.Comparing with the original HWENO schemes of Qiu and Shu,one major advantage of new HWENOschemes is its robust in computation of problem with strong shocks.Extensive numerical experiments are performed to illustrate the capability of the method.展开更多
This paper further considers weighted essentially non-oscillatory(WENO)and Hermite weighted essentially non-oscillatory(HWENO)finite volume methods as limiters for Runge-Kutta discontinuous Galerkin(RKDG)methods to so...This paper further considers weighted essentially non-oscillatory(WENO)and Hermite weighted essentially non-oscillatory(HWENO)finite volume methods as limiters for Runge-Kutta discontinuous Galerkin(RKDG)methods to solve problems involving nonlinear hyperbolic conservation laws.The application discussed here is the solution of 3-D problems on unstructured meshes.Our numerical tests again demonstrate this is a robust and high order limiting procedure,which simultaneously achieves high order accuracy and sharp non-oscillatory shock transitions.展开更多
In this paper,we present a new type of Hermite weighted essentially nonoscillatory(HWENO)schemes for solving the Hamilton-Jacobi equations on the finite volume framework.The cell averages of the function and its first...In this paper,we present a new type of Hermite weighted essentially nonoscillatory(HWENO)schemes for solving the Hamilton-Jacobi equations on the finite volume framework.The cell averages of the function and its first one(in one dimension)or two(in two dimensions)derivative values are together evolved via time approaching and used in the reconstructions.And the major advantages of the new HWENO schemes are their compactness in the spacial field,purely on the finite volume framework and only one set of small stencils is used for different type of the polynomial reconstructions.Extensive numerical tests are performed to illustrate the capability of the methodologies.展开更多
基金supported by the NSFC grant 12101128supported by the NSFC grant 12071392.
文摘In this paper,we propose a finite volume Hermite weighted essentially non-oscillatory(HWENO)method based on the dimension by dimension framework to solve hyperbolic conservation laws.It can maintain the high accuracy in the smooth region and obtain the high resolution solution when the discontinuity appears,and it is compact which will be good for giving the numerical boundary conditions.Furthermore,it avoids complicated least square procedure when we implement the genuine two dimensional(2D)finite volume HWENO reconstruction,and it can be regarded as a generalization of the one dimensional(1D)HWENO method.Extensive numerical tests are performed to verify the high resolution and high accuracy of the scheme.
基金supported by the National Basic Research Program of China(2009CB724104)
文摘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.
基金partly supported by AFOSR grant FA9550-20-1-0055 and NSF grant DMS-2010107partly supported by NSFC grant 12071392.
文摘In this paper,a high-order moment-based multi-resolution Hermite weighted essentially non-oscillatory(HWENO)scheme is designed for hyperbolic conservation laws.The main idea of this scheme is derived from our previous work[J.Comput.Phys.,446(2021)110653],in which the integral averages of the function and its first order derivative are used to reconstruct both the function and its first order derivative values at the boundaries.However,in this paper,only the function values at the Gauss-Lobatto points in the one or two dimensional case need to be reconstructed by using the information of the zeroth and first order moments.In addition,an extra modification procedure is used to modify those first order moments in the troubledcells,which leads to an improvement of stability and an enhancement of resolution near discontinuities.To obtain the same order of accuracy,the size of the stencil required by this moment-based multi-resolution HWENO scheme is still the same as the general HWENO scheme and is more compact than the generalWENO scheme.Moreover,the linear weights are not unique and are independent of the node position,and the CFL number can still be 0.6whether for the one or two dimensional case,which has to be 0.2 in the two dimensional case for other HWENO schemes.Extensive numerical examples are given to demonstrate the stability and resolution of such moment-based multi-resolution HWENO scheme.
基金the National Natural Science Foundation of China(Grant No.10671097)the European project ADIGMA on the development of innovative solution algorithms for aerodynamic simu-lations+1 种基金Scientific Research Foundation for the Returned Overseas Chinese ScholarsState Education Ministry and the Natural Science Foundation of Jiangsu Province(Grant No.BK2006511)
文摘In this paper,we developed a class of the fourth order accurate finite volume Hermite weighted essentially non-oscillatory(HWENO)schemes based on the work(Computers&Fluids,34:642-663(2005))by Qiu and Shu,with Total Variation Diminishing Runge-Kutta time discretization method for the two-dimensional hyperbolic conservation laws.The key idea of HWENO is to evolve both with the solution and its derivative,which allows for using Hermite interpolation in the reconstruction phase,resulting in a more compact stencil at the expense of the additional work.The main difference between this work and the formal one is the procedure to reconstruct the derivative terms.Comparing with the original HWENO schemes of Qiu and Shu,one major advantage of new HWENOschemes is its robust in computation of problem with strong shocks.Extensive numerical experiments are performed to illustrate the capability of the method.
基金The research was partially supported by NSFC grant 10931004,10871093,11002071 and the European project ADIGMA on the development of innovative solution algorithms for aerodynamic simulations.
文摘This paper further considers weighted essentially non-oscillatory(WENO)and Hermite weighted essentially non-oscillatory(HWENO)finite volume methods as limiters for Runge-Kutta discontinuous Galerkin(RKDG)methods to solve problems involving nonlinear hyperbolic conservation laws.The application discussed here is the solution of 3-D problems on unstructured meshes.Our numerical tests again demonstrate this is a robust and high order limiting procedure,which simultaneously achieves high order accuracy and sharp non-oscillatory shock transitions.
基金supported by NSFC grants 10931004,11372005,11002071 and 91230110.
文摘In this paper,we present a new type of Hermite weighted essentially nonoscillatory(HWENO)schemes for solving the Hamilton-Jacobi equations on the finite volume framework.The cell averages of the function and its first one(in one dimension)or two(in two dimensions)derivative values are together evolved via time approaching and used in the reconstructions.And the major advantages of the new HWENO schemes are their compactness in the spacial field,purely on the finite volume framework and only one set of small stencils is used for different type of the polynomial reconstructions.Extensive numerical tests are performed to illustrate the capability of the methodologies.
