The main purpose of this work is to find for any non-negative measure, the relations between the Gauss-Radau and Gauss-Lobatto formula and Gauss formulae for the same measure. As applications, the author obtained the ...The main purpose of this work is to find for any non-negative measure, the relations between the Gauss-Radau and Gauss-Lobatto formula and Gauss formulae for the same measure. As applications, the author obtained the explicit Gauss-Radau and Gauss-Lobatto formulae for the Jacobi weight and the Gori-Micchelli weight.展开更多
A global spherical Fourier-Legendre spectral element method is proposed to solve Poisson equations and advective flow over a sphere. In the meridional direction, Legendre polynomials are used and the region is divided...A global spherical Fourier-Legendre spectral element method is proposed to solve Poisson equations and advective flow over a sphere. In the meridional direction, Legendre polynomials are used and the region is divided into several elements. In order to avoid coordinate singularities at the north and south poles in the meridional direction, Legendre-Gauss-Radau points are chosen at the elements involving the two poles. Fourier polynomials are applied in the zonal direction for its periodicity, with only one element. Then, the partial differential equations are solved on the longitude-latitude meshes without coordinate transformation between spherical and Cartesian coordinates. For verification of the proposed method, a few Poisson equations and advective flows are tested. Firstly, the method is found to be valid for test cases with smooth solution. The results of the Poisson equations demonstrate that the present method exhibits high accuracy and exponential convergence. High- precision solutions are also obtained with near negligible numerical diffusion during the time evolution for advective flow with smooth shape. Secondly, the results of advective flow with non-smooth shape and deformational flow are also shown to be reasonable and effective. As a result, the present method is proved to be capable of solving flow through different types of elements, and thereby a desirable method with reliability and high accuracy for solving partial differential equations over a sphere.展开更多
This paper is concerned with convergence and superconvergence properties of the local discontinuous Galerkin(LDG)method for two-dimensional semilinear second-order elliptic problems of the form−Δu=f(x,y,u)on Cartesia...This paper is concerned with convergence and superconvergence properties of the local discontinuous Galerkin(LDG)method for two-dimensional semilinear second-order elliptic problems of the form−Δu=f(x,y,u)on Cartesian grids.By introducing special GaussRadau projections and using duality arguments,we obtain,under some suitable choice of numerical fuxes,the optimal convergence order in L2-norm of O(h^(p+1))for the LDG solution and its gradient,when tensor product polynomials of degree at most p and grid size h are used.Moreover,we prove that the LDG solutions are superconvergent with an order p+2 toward particular Gauss-Radau projections of the exact solutions.Finally,we show that the error between the gradient of the LDG solution and the gradient of a special Gauss-Radau projection of the exact solution achieves(p+1)-th order superconvergence.Some numerical experiments are performed to illustrate the theoretical results.展开更多
In this paper,we present optimal error estimates of the local discontinuous Galerkin method with generalized numerical fluxes for one-dimensional nonlinear convection-diffusion systems.The upwind-biased flux with the ...In this paper,we present optimal error estimates of the local discontinuous Galerkin method with generalized numerical fluxes for one-dimensional nonlinear convection-diffusion systems.The upwind-biased flux with the adjustable numerical viscosity for the convective term is chosen based on the local characteristic decomposition,which is helpful in resolving discontinuities of degenerate parabolic equations without enforcing any limiting procedure.For the diffusive term,a pair of generalized alternating fluxes is considered.By constructing and analyzing generalized Gauss-Radau projections with respect to different convective or diffusive terms,we derive optimal error estimates for nonlinear convection-diffusion systems with the symmetrizable flux Jacobian and fully nonlinear diffusive problems.Numerical experiments including long time simulations,different boundary conditions and degenerate equations with discontinuous initial data are provided to demonstrate the sharpness of theoretical results.展开更多
文摘The main purpose of this work is to find for any non-negative measure, the relations between the Gauss-Radau and Gauss-Lobatto formula and Gauss formulae for the same measure. As applications, the author obtained the explicit Gauss-Radau and Gauss-Lobatto formulae for the Jacobi weight and the Gori-Micchelli weight.
基金supported by the Shandong Post-Doctoral Innovation Fund(Grant No.201303064)the Qingdao Post-Doctoral Application Research Project+1 种基金the National Basic Research(973) Program of China(Grant No.2012CB417402 and 2010CB950402)the National Natural Science Foundation of China(Grant No.41176017)
文摘A global spherical Fourier-Legendre spectral element method is proposed to solve Poisson equations and advective flow over a sphere. In the meridional direction, Legendre polynomials are used and the region is divided into several elements. In order to avoid coordinate singularities at the north and south poles in the meridional direction, Legendre-Gauss-Radau points are chosen at the elements involving the two poles. Fourier polynomials are applied in the zonal direction for its periodicity, with only one element. Then, the partial differential equations are solved on the longitude-latitude meshes without coordinate transformation between spherical and Cartesian coordinates. For verification of the proposed method, a few Poisson equations and advective flows are tested. Firstly, the method is found to be valid for test cases with smooth solution. The results of the Poisson equations demonstrate that the present method exhibits high accuracy and exponential convergence. High- precision solutions are also obtained with near negligible numerical diffusion during the time evolution for advective flow with smooth shape. Secondly, the results of advective flow with non-smooth shape and deformational flow are also shown to be reasonable and effective. As a result, the present method is proved to be capable of solving flow through different types of elements, and thereby a desirable method with reliability and high accuracy for solving partial differential equations over a sphere.
基金This research was supported by the NASA Nebraska Space Grant(Federal Grant/Award Number 80NSSC20M0112).
文摘This paper is concerned with convergence and superconvergence properties of the local discontinuous Galerkin(LDG)method for two-dimensional semilinear second-order elliptic problems of the form−Δu=f(x,y,u)on Cartesian grids.By introducing special GaussRadau projections and using duality arguments,we obtain,under some suitable choice of numerical fuxes,the optimal convergence order in L2-norm of O(h^(p+1))for the LDG solution and its gradient,when tensor product polynomials of degree at most p and grid size h are used.Moreover,we prove that the LDG solutions are superconvergent with an order p+2 toward particular Gauss-Radau projections of the exact solutions.Finally,we show that the error between the gradient of the LDG solution and the gradient of a special Gauss-Radau projection of the exact solution achieves(p+1)-th order superconvergence.Some numerical experiments are performed to illustrate the theoretical results.
基金supported by Natural Science Foundation of Jiangsu Province(No.BK20170374)Nature Science Research Program for Colleges and Universities of Jiangsu Province(No.17KJB110016)Scientific Research Project for University Students of Suzhou University of Science and Technology in 2017-2018
基金supported by National Natural Science Foundation of China(Grant Nos.11971132 and 11971131)Natural Science Foundation of Heilongjiang Province(Grant No.YQ2021A002)Guangdong Basic and Applied Basic Research Foundation(Grant No.2020B1515310006)。
文摘In this paper,we present optimal error estimates of the local discontinuous Galerkin method with generalized numerical fluxes for one-dimensional nonlinear convection-diffusion systems.The upwind-biased flux with the adjustable numerical viscosity for the convective term is chosen based on the local characteristic decomposition,which is helpful in resolving discontinuities of degenerate parabolic equations without enforcing any limiting procedure.For the diffusive term,a pair of generalized alternating fluxes is considered.By constructing and analyzing generalized Gauss-Radau projections with respect to different convective or diffusive terms,we derive optimal error estimates for nonlinear convection-diffusion systems with the symmetrizable flux Jacobian and fully nonlinear diffusive problems.Numerical experiments including long time simulations,different boundary conditions and degenerate equations with discontinuous initial data are provided to demonstrate the sharpness of theoretical results.
