This paper investigates superconvergence properties of the direct discontinuous Galerkin(DDG)method with interface corrections and the symmetric DDG method for diffusion equations.We apply the Fourier analysis techniq...This paper investigates superconvergence properties of the direct discontinuous Galerkin(DDG)method with interface corrections and the symmetric DDG method for diffusion equations.We apply the Fourier analysis technique to symbolically compute eigenvalues and eigenvectors of the amplification matrices for both DDG methods with different coefficient settings in the numerical fluxes.Based on the eigen-structure analysis,we carry out error estimates of the DDG solutions,which can be decomposed into three parts:(i)dissipation errors of the physically relevant eigenvalue,which grow linearly with the time and are of order 2k for P^(k)(k=2,3)approximations;(ii)projection error from a special projection of the exact solution,which is decreasing over the time and is related to the eigenvector corresponding to the physically relevant eigenvalue;(iii)dissipative errors of non-physically relevant eigenvalues,which decay exponentially with respect to the spatial mesh sizeΔx.We observe that the errors are sensitive to the choice of the numerical flux coefficient for even degree P^(2)approximations,but are not for odd degree P^(3)approximations.Numerical experiments are provided to verify the theoretical results.展开更多
Parameterized dynamical systems with a simple zero eigenvalue and a couple of purely imaginary eigenvalues are considered. It is proved that this type of eigen-structure leads to torus bifurcation under certain nondeg...Parameterized dynamical systems with a simple zero eigenvalue and a couple of purely imaginary eigenvalues are considered. It is proved that this type of eigen-structure leads to torus bifurcation under certain nondegenerate conditions. We show that the discrete systems, obtained by discretizing the ODEs using symmetric, eigen-structure preserving schemes, inherit the similar torus bifurcation properties. Predholm theory in Banach spaces is applied to obtain the global torus bifurcation. Our results complement those on the study of discretization effects of global bifurcation.展开更多
The first International Symposium on Dynamics,Monitoring,and Diagnostics was held in Chongqing,China,in April 2022.The Symposium,which was attended both virtually and in person,had an audience of 2000 and was aimed at...The first International Symposium on Dynamics,Monitoring,and Diagnostics was held in Chongqing,China,in April 2022.The Symposium,which was attended both virtually and in person,had an audience of 2000 and was aimed at enhancing the intelligence of condition monitoring for engineering systems.During the Symposium,five keynote addresses were delivered by world leading experts,and this paper is comprised of summaries of these addresses to ensure that the important messages of these speakers are properly on record and readily able to be referenced.展开更多
基金supported by the National Natural Science Foundation of China(Grant Nos.11871428 and 12071214)the Natural Science Foundation for Colleges and Universities of Jiangsu Province of China(Grant No.20KJB110011)+1 种基金supported by the National Science Foundation(Grant No.DMS-1620335)and the Simons Foundation(Grant No.637716)supported by the National Natural Science Foundation of China(Grant Nos.11871428 and 12272347).
文摘This paper investigates superconvergence properties of the direct discontinuous Galerkin(DDG)method with interface corrections and the symmetric DDG method for diffusion equations.We apply the Fourier analysis technique to symbolically compute eigenvalues and eigenvectors of the amplification matrices for both DDG methods with different coefficient settings in the numerical fluxes.Based on the eigen-structure analysis,we carry out error estimates of the DDG solutions,which can be decomposed into three parts:(i)dissipation errors of the physically relevant eigenvalue,which grow linearly with the time and are of order 2k for P^(k)(k=2,3)approximations;(ii)projection error from a special projection of the exact solution,which is decreasing over the time and is related to the eigenvector corresponding to the physically relevant eigenvalue;(iii)dissipative errors of non-physically relevant eigenvalues,which decay exponentially with respect to the spatial mesh sizeΔx.We observe that the errors are sensitive to the choice of the numerical flux coefficient for even degree P^(2)approximations,but are not for odd degree P^(3)approximations.Numerical experiments are provided to verify the theoretical results.
基金The NSFC (10071030) of ChinaThe Volkswagen Foundation of Germany The Project-sponsored by SRP for ROCS, SEM (2002).
文摘Parameterized dynamical systems with a simple zero eigenvalue and a couple of purely imaginary eigenvalues are considered. It is proved that this type of eigen-structure leads to torus bifurcation under certain nondegenerate conditions. We show that the discrete systems, obtained by discretizing the ODEs using symmetric, eigen-structure preserving schemes, inherit the similar torus bifurcation properties. Predholm theory in Banach spaces is applied to obtain the global torus bifurcation. Our results complement those on the study of discretization effects of global bifurcation.
基金supported in part by the Australian Government through the Australian Research Council Discovery Project DP160103501.
文摘The first International Symposium on Dynamics,Monitoring,and Diagnostics was held in Chongqing,China,in April 2022.The Symposium,which was attended both virtually and in person,had an audience of 2000 and was aimed at enhancing the intelligence of condition monitoring for engineering systems.During the Symposium,five keynote addresses were delivered by world leading experts,and this paper is comprised of summaries of these addresses to ensure that the important messages of these speakers are properly on record and readily able to be referenced.
