An efficient numerical method on modified space-time sparse grid for time-fractional diffusion equation with nonsmooth data.

In this paper, we focus on developing a high efficient algorithm for solving d-dimension time-fractional diffusion equation (TFDE). For TFDE, the initial function or source term is usually not smooth, which can lead to the low regularity of exact solution. And such low regularity has a marked impact on the convergence rate of numerical method. In order to improve the convergence rate of the algorithm, we introduce the space-time sparse grid (STSG) method to solve TFDE. In our study, we employ the sine basis and the linear element basis for spatial discretization and temporal discretization, respectively. The sine basis can be divided into several levels, and the linear element basis can lead to the hierarchical basis. Then, the STSG can be constructed through a special tensor product of the spatial multilevel basis and the temporal hierarchical basis. Under certain conditions, the function approximation on standard STSG can achieve the accuracy order O(2-JJ) with O(2JJ) degrees of freedom (DOF) for d=1 and O(2Jd) DOF for d>1, where J denotes the maximal level of sine coefficients. However, if the solution changes very rapidly at the initial moment, the standard STSG method may reduce accuracy or even fail to converge. To overcome this, we integrate the full grid into the STSG, and obtain the modified STSG. Finally, we obtain the fully discrete scheme of STSG method for solving TFDE. The great advantage of the modified STSG method can be shown in the comparative numerical experiment.

The Accumulation of Crocin and Geniposide and Transcripts of Phytoene Synthase during Maturation of Gardenia jasminoides Fruit.

Gardenia fruit (fruit of Gardenia jasminoides Ellis) is used as a natural pigment resource and a Chinese traditional medicine. The white mesocarp turning orange or red that occurs during gardenia fruit maturation arises from the production and accumulation of the apocarotenoids, especially crocin-1, which is derived from carotenoid. Meanwhile, the major medical component geniposide is accumulated in gardenia fruit. To further our understanding of the synthetic and accumulation mechanism for crocin-1 and geniposide in gardenia fruit, the contents of crocin-1 and geniposide and the transcripts of phytoene synthase (GjPSY) profiles in gardenia fruits were examined at various stages of maturation. The concentration of crocin-1 and geniposide in gardenia fruit was determined by reversed-phase high-performance liquid chromatography (HPLC). The results showed that the concentration of crocetin-1 was increased during fruit development and the concentration of geniposide does not change significantly during maturing. The expression levels of GjPSY mRNA were examined by RT-PCR. It was revealed that GjPSY was constitutively expressed during fruit development, suggesting that the primary mechanism that controls crocin accumulation in G. jasminoides fruits during development is not correlated to the differential regulation of transcript levels of GjPSY gene.