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
Sum frequency generation spectroscopy（SFG） has been widely used to study the interfacial chemistry of aqueous salt solutions of biological or environmental importance. Most of the SFG data analysis used the same bul...Sum frequency generation spectroscopy（SFG） has been widely used to study the interfacial chemistry of aqueous salt solutions of biological or environmental importance. Most of the SFG data analysis used the same bulk refractive index for different salt concentrations despite of the variations of the refractive indices. Here we systematically investigate the influence of the refractive index on the SFG intensities at various experimental conditions. It is discovered that the SFG intensities are the most sensitive to the refractive index at solid/liquid interfaces nearby the total internal reflection geometries. At air/liquid interfaces, the effect of the refractive indices is also nonegligible. Consequently some important SFG results, such as the response of water structures to the ionic strength at the SiO2/aqueous interfaces, are necessary to be reevaluated. These conclusions on the effect of the small variations of the refractive index are generally useful for the common practice of SFG data analysis.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financially supported by the National Natural Science Foundation of China(Nos.21227802,21303216 and21473217)
文摘Sum frequency generation spectroscopy（SFG） has been widely used to study the interfacial chemistry of aqueous salt solutions of biological or environmental importance. Most of the SFG data analysis used the same bulk refractive index for different salt concentrations despite of the variations of the refractive indices. Here we systematically investigate the influence of the refractive index on the SFG intensities at various experimental conditions. It is discovered that the SFG intensities are the most sensitive to the refractive index at solid/liquid interfaces nearby the total internal reflection geometries. At air/liquid interfaces, the effect of the refractive indices is also nonegligible. Consequently some important SFG results, such as the response of water structures to the ionic strength at the SiO2/aqueous interfaces, are necessary to be reevaluated. These conclusions on the effect of the small variations of the refractive index are generally useful for the common practice of SFG data analysis.
