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 new alternating group explicit-implicit （nAGEI） scheme for dispersive equations with a periodic boundary condition is derived. This new unconditionally stable scheme has a fourth-order truncation er...In this paper, a new alternating group explicit-implicit （nAGEI） scheme for dispersive equations with a periodic boundary condition is derived. This new unconditionally stable scheme has a fourth-order truncation error in space and a convergence ratio faster than some known alternating methods such as ASEI and AGE. Comparison in accuracy with the AGEI and AGE methods is presented in the numerical experiment.展开更多
The computation of the radiative transfer equation is expensive mainly due to two stiff terms:the transport term and the collision operator.The stiffness in the former comes from the fact that particles(such as photon...The computation of the radiative transfer equation is expensive mainly due to two stiff terms:the transport term and the collision operator.The stiffness in the former comes from the fact that particles(such as photons)travel at the speed of light,while that in the latter is due to the strong scattering in the optically thick region.We study the fully implicit scheme for this equation to account for the stiffness.The main challenge in the implicit treatment is the coupling between the spacial and angular coordinates that requires the large size of the to-be-inverted matrix,which is also ill-conditioned and not necessarily symmetric.Our main idea is to utilize the spectral structure of the ill-conditioned matrix to construct a pre-conditioner,which,along with an exquisite split of the spatial and angular dependence,significantly improve the condition number and allows a matrix-free treatment.We also design a fast solver to compute this pre-conditioner explicitly in advance.Our method is shown to be efficient in both diffusive and free streaming limit,and the computational cost is comparable to the state-of-the-art method.Various examples including anisotropic scattering and two-dimensional problems are provided to validate the effectiveness of our method.展开更多
The alternating direction implicit(ADI)method is a highly efficient technique for solving multi-dimensional time dependent initial-boundary value problems on rectangles.When the ADI technique is coupled with orthogona...The alternating direction implicit(ADI)method is a highly efficient technique for solving multi-dimensional time dependent initial-boundary value problems on rectangles.When the ADI technique is coupled with orthogonal spline collocation(OSC)for discretization in space we not only obtain the global solution efficiently but the discretization error with respect to space variables can be of an arbitrarily high order.In[2],we used a Crank Nicolson ADI OSC method for solving general nonlinear parabolic problems with Robin’s boundary conditions on rectangular polygons and demonstrated numerically the accuracy in various norms.A natural question that arises is:Does this method have an extension to non-rectangular regions?In this paper,we present a simple idea of how the ADI OSC technique can be extended to some such regions.Our approach depends on the transfer of Dirichlet boundary conditions in the solution of a two-point boundary value problem(TPBVP).We illustrate our idea for the solution of the heat equation on the unit disc using piecewise Hermite cubics.展开更多
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National Natural Science Foundation of China (No.10671113)
文摘In this paper, a new alternating group explicit-implicit （nAGEI） scheme for dispersive equations with a periodic boundary condition is derived. This new unconditionally stable scheme has a fourth-order truncation error in space and a convergence ratio faster than some known alternating methods such as ASEI and AGE. Comparison in accuracy with the AGEI and AGE methods is presented in the numerical experiment.
基金The work of Q.Li is supported in part by a start-up fund from UW-Madison and National Science Foundation under the grant DMS-1619778The work of L.Wang is supported in part by the National Science Foundation under the grant DMS-1620135Both authors would like to express gratitude to the support from the NSF research network grant RNMS11-07444(KI-Net).We also thank Professors Shi Jin,Jim Morel and Cory Hauck for fruitful discussions.
文摘The computation of the radiative transfer equation is expensive mainly due to two stiff terms:the transport term and the collision operator.The stiffness in the former comes from the fact that particles(such as photons)travel at the speed of light,while that in the latter is due to the strong scattering in the optically thick region.We study the fully implicit scheme for this equation to account for the stiffness.The main challenge in the implicit treatment is the coupling between the spacial and angular coordinates that requires the large size of the to-be-inverted matrix,which is also ill-conditioned and not necessarily symmetric.Our main idea is to utilize the spectral structure of the ill-conditioned matrix to construct a pre-conditioner,which,along with an exquisite split of the spatial and angular dependence,significantly improve the condition number and allows a matrix-free treatment.We also design a fast solver to compute this pre-conditioner explicitly in advance.Our method is shown to be efficient in both diffusive and free streaming limit,and the computational cost is comparable to the state-of-the-art method.Various examples including anisotropic scattering and two-dimensional problems are provided to validate the effectiveness of our method.
基金This work was supported by grant no.13328 from the Petroleum Institute,Abu Dhabi,UAE.
文摘The alternating direction implicit(ADI)method is a highly efficient technique for solving multi-dimensional time dependent initial-boundary value problems on rectangles.When the ADI technique is coupled with orthogonal spline collocation(OSC)for discretization in space we not only obtain the global solution efficiently but the discretization error with respect to space variables can be of an arbitrarily high order.In[2],we used a Crank Nicolson ADI OSC method for solving general nonlinear parabolic problems with Robin’s boundary conditions on rectangular polygons and demonstrated numerically the accuracy in various norms.A natural question that arises is:Does this method have an extension to non-rectangular regions?In this paper,we present a simple idea of how the ADI OSC technique can be extended to some such regions.Our approach depends on the transfer of Dirichlet boundary conditions in the solution of a two-point boundary value problem(TPBVP).We illustrate our idea for the solution of the heat equation on the unit disc using piecewise Hermite cubics.
