In this paper,a class of discrete Gronwall inequalities is proposed.It is efciently applied to analyzing the constructed L1/local discontinuous Galerkin(LDG)fnite element meth-ods which are used for numerically solvin...In this paper,a class of discrete Gronwall inequalities is proposed.It is efciently applied to analyzing the constructed L1/local discontinuous Galerkin(LDG)fnite element meth-ods which are used for numerically solving the Caputo-Hadamard time fractional difusion equation.The derived numerical methods are shown to beα-robust using the newly estab-lished Gronwall inequalities,that is,it remains valid whenα→1^(−).Numerical experiments are given to demonstrate the theoretical statements.展开更多
In this paper,an efficient method is proposed to solve the Caputo diffusion equation with a variable coefficient.Since the solution of such an equation in general has a typical weak singularity near the initial time t...In this paper,an efficient method is proposed to solve the Caputo diffusion equation with a variable coefficient.Since the solution of such an equation in general has a typical weak singularity near the initial time t=0,the time-fractional derivative with order in(0,1)is discretized by L2-1_(σ)formula on nonuniform meshes.For the spatial derivative,the local discontinuous Galerkin(LDG)method is employed.A complete theoretical analysis of the numerical stability and convergence of the derived scheme is given using a discrete fractional Gronwall inequality.Numerical experiments demonstrate the validity of the established scheme and the accuracy of the theoretical analysis results.展开更多
The latitudinal diversity gradient(LDG)is one of the most notable biodiversity patterns in biogeography.The metabolic theory of ecology(MTE)explains ecological patterns,including the LDG.However,little is known about ...The latitudinal diversity gradient(LDG)is one of the most notable biodiversity patterns in biogeography.The metabolic theory of ecology(MTE)explains ecological patterns,including the LDG.However,little is known about whether the LDG remains stable over time as climate warming progresses and whether MTE remains applicable to clarify this pattern.In this study,forest data spanning temperate,subtropical,and tropical zones across China were used to analyze long-term changes in the LDG of tree species over 2005-2020.Based on the MTE framework,spatial scales were considered to assess temperature dependence of typical forest trees species.Our results show that species richness decreased with increasing latitude,and that temperature was the primary driver of this change.Although temperature in China has significantly increased over the past two decades,the LDG of tree species has remained stable.However,there was a decrease in species richness in tropical regions over time.With predictions of the MTE,the logarithm of typical forest tree species richness exhibited negative linear relationships with the inverse of ambient temperature,indicating temperature dependence of species richness.However,the relationship remained stable and was strongly influenced by spatial scale,intensifying as spatial scale increased.The findings emphasize the important role of temperature in shaping the LDG.The effects of spatial scale,in particular,should be considered when biodiversity management plans are developed for future climate change.展开更多
In this paper,the second in a series,we improve the discretization of the higher spatial derivative terms in a spectral volume(SV)context.The motivation for the above comes from[J.Sci.Comput.,46(2),314–328],wherein t...In this paper,the second in a series,we improve the discretization of the higher spatial derivative terms in a spectral volume(SV)context.The motivation for the above comes from[J.Sci.Comput.,46(2),314–328],wherein the authors developed a variant of the LDG(Local Discontinuous Galerkin)flux discretization method.This variant(aptly named LDG2),not only displayed higher accuracy than the LDG approach,but also vastly reduced its unsymmetrical nature.In this paper,we adapt the LDG2 formulation for discretizing third derivative terms.A linear Fourier analysis was performed to compare the dispersion and the dissipation properties of the LDG2 and the LDG formulations.The results of the analysis showed that the LDG2 scheme(i)is stable for 2nd and 3rd orders and(ii)generates smaller dissipation and dispersion errors than the LDG formulation for all the orders.The 4th order LDG2 scheme is howevermildly unstable:as the real component of the principal eigen value briefly becomes positive.In order to circumvent the above,a weighted average of the LDG and the LDG2 fluxes was used as the final numerical flux.Even a weight of 1.5%for the LDG(i.e.,98.5%for the LDG2)was sufficient tomake the scheme stable.Thisweighted scheme is still predominantly LDG2 and hence generated smaller dissipation and dispersion errors than the LDG formulation.Numerical experiments are performed to validate the analysis.In general,the numerical results are very promising and indicate that the approach has a great potential for higher dimension Korteweg-de Vries(KdV)type problems.展开更多
基金supported by the National Natural Science Foundation of China under Grant No.12101266.
文摘In this paper,a class of discrete Gronwall inequalities is proposed.It is efciently applied to analyzing the constructed L1/local discontinuous Galerkin(LDG)fnite element meth-ods which are used for numerically solving the Caputo-Hadamard time fractional difusion equation.The derived numerical methods are shown to beα-robust using the newly estab-lished Gronwall inequalities,that is,it remains valid whenα→1^(−).Numerical experiments are given to demonstrate the theoretical statements.
文摘In this paper,an efficient method is proposed to solve the Caputo diffusion equation with a variable coefficient.Since the solution of such an equation in general has a typical weak singularity near the initial time t=0,the time-fractional derivative with order in(0,1)is discretized by L2-1_(σ)formula on nonuniform meshes.For the spatial derivative,the local discontinuous Galerkin(LDG)method is employed.A complete theoretical analysis of the numerical stability and convergence of the derived scheme is given using a discrete fractional Gronwall inequality.Numerical experiments demonstrate the validity of the established scheme and the accuracy of the theoretical analysis results.
基金supported by the Key Program of National Science of China(Grant No.:42030509 and 42141005)。
文摘The latitudinal diversity gradient(LDG)is one of the most notable biodiversity patterns in biogeography.The metabolic theory of ecology(MTE)explains ecological patterns,including the LDG.However,little is known about whether the LDG remains stable over time as climate warming progresses and whether MTE remains applicable to clarify this pattern.In this study,forest data spanning temperate,subtropical,and tropical zones across China were used to analyze long-term changes in the LDG of tree species over 2005-2020.Based on the MTE framework,spatial scales were considered to assess temperature dependence of typical forest trees species.Our results show that species richness decreased with increasing latitude,and that temperature was the primary driver of this change.Although temperature in China has significantly increased over the past two decades,the LDG of tree species has remained stable.However,there was a decrease in species richness in tropical regions over time.With predictions of the MTE,the logarithm of typical forest tree species richness exhibited negative linear relationships with the inverse of ambient temperature,indicating temperature dependence of species richness.However,the relationship remained stable and was strongly influenced by spatial scale,intensifying as spatial scale increased.The findings emphasize the important role of temperature in shaping the LDG.The effects of spatial scale,in particular,should be considered when biodiversity management plans are developed for future climate change.
文摘In this paper,the second in a series,we improve the discretization of the higher spatial derivative terms in a spectral volume(SV)context.The motivation for the above comes from[J.Sci.Comput.,46(2),314–328],wherein the authors developed a variant of the LDG(Local Discontinuous Galerkin)flux discretization method.This variant(aptly named LDG2),not only displayed higher accuracy than the LDG approach,but also vastly reduced its unsymmetrical nature.In this paper,we adapt the LDG2 formulation for discretizing third derivative terms.A linear Fourier analysis was performed to compare the dispersion and the dissipation properties of the LDG2 and the LDG formulations.The results of the analysis showed that the LDG2 scheme(i)is stable for 2nd and 3rd orders and(ii)generates smaller dissipation and dispersion errors than the LDG formulation for all the orders.The 4th order LDG2 scheme is howevermildly unstable:as the real component of the principal eigen value briefly becomes positive.In order to circumvent the above,a weighted average of the LDG and the LDG2 fluxes was used as the final numerical flux.Even a weight of 1.5%for the LDG(i.e.,98.5%for the LDG2)was sufficient tomake the scheme stable.Thisweighted scheme is still predominantly LDG2 and hence generated smaller dissipation and dispersion errors than the LDG formulation.Numerical experiments are performed to validate the analysis.In general,the numerical results are very promising and indicate that the approach has a great potential for higher dimension Korteweg-de Vries(KdV)type problems.
