Krylov implicit integration factor discontinuous Galerkin methods on sparse grids for high dimensional reaction-diffusion equations

Computational costs of numerically solving multidimensional partial differential equations (PDEs) increase significantly when the spatial dimensions of the PDEs are high, due to large number of spatial grid points. For multidimensional reaction-diffusion equations, stiffness of the system provides additional challenges for achieving efficient numerical simulations. In this paper, we propose a class of Krylov implicit integration factor (IIF) discontinuous Galerkin (DG) methods on sparse grids to solve reaction-diffusion equations on high spatial dimensions. The key ingredient of spatial DG discretization is the multiwavelet bases on nested sparse grids, which can significantly reduce the numbers of degrees of freedom. To deal with the stiffness of the DG spatial operator in discretizing reaction-diffusion equations, we apply the efficient IIF time discretization methods, which are a class of exponential integrators. Krylov subspace approximations are used to evaluate the large size matrix exponentials resulting from IIF schemes for solving PDEs on high spatial dimensions. Stability and error analysis for the semi-discrete scheme are performed. Numerical examples of both scalar equations and systems in two and three spatial dimensions are provided to demonstrate the accuracy and efficiency of the methods. The stiffness of the reaction-diffusion equations is resolved well and large time step size computations are obtained

The diversification of the lynx lineage during the Plio-Pleistocene-evidence from a new small Lynx from Longdan, Gansu Province, China

Altres ajuts: CERCA Programme/Generalitat de CatalunyaA new small-sized lynx from Longdan, Gansu Province, China, Lynx hei sp. nov., is described in this study. The new species displays the characteristic Lynx generic traits, such as distinct buccal grooves in the upper canine, presence of an anterior groove in the upper canine, absence of upper premolar 2, and a moderately developed mastoid process, but it is markedly smaller than the previously described Lynx issiodorensis specimens from the same site and is also smaller overall than most living species, comparable to Lynx rufus in size. The new species has a relatively wide and deep zygomatic arch, similar to that of living Lynx lynx, Lynx pardinus and Lynx canadensis but wider than that of Lynx rufus. Our phylogenetic analyses suggest that Lynx hei falls within the crown group Lynx, being the sister to Lynx rufus or, less probably, a sister to Lynx issiodorensis + three other living species of Lynx. The Plio-Pleistocene Lynx issiodorensis is supported as the ancestor of Lynx lynx, Lynx pardinus and Lynx canadensis. Our phylogenetic study suggests that Lynx diversification over the Plio-Pleistocene was achieved initially by body size differentiation, putatively forced by intraspecific competition with other carnivorans, followed by morphological divergence

Krylov SSP Integrating Factor Runge–Kutta WENO Methods

Weighted essentially non-oscillatory (WENO) methods are especially efficient for numerically solving nonlinear hyperbolic equations. In order to achieve strong stability and large time-steps, strong stability preserving (SSP) integrating factor (IF) methods were designed in the literature, but the methods there were only for one-dimensional (1D) problems that have a stiff linear component and a non-stiff nonlinear component. In this paper, we extend WENO methods with large time-stepping SSP integrating factor Runge–Kutta time discretization to solve general nonlinear two-dimensional (2D) problems by a splitting method. How to evaluate the matrix exponential operator efficiently is a tremendous challenge when we apply IF temporal discretization for PDEs on high spatial dimensions. In this work, the matrix exponential computation is approximated through the Krylov subspace projection method. Numerical examples are shown to demonstrate the accuracy and large time-step size of the present method

A Study on Stable Regularized Moving Least-Squares Interpolation and Coupled with SPH Method

The smoothed particle hydrodynamics (SPH) method has been popularly applied in various fields, including astrodynamics, thermodynamics, aerodynamics, and hydrodynamics. Generally, a high-precision interpolation is required to calculate the particle physical attributes and their derivatives for the boundary treatment and postproceeding in the SPH simulation. However, as a result of the truncation of kernel function support domain and irregular particle distribution, the interpolation using conventional SPH interpolation experiences low accuracy for the particles near the boundary and free surface. To overcome this drawback, stable regularized moving least-squares (SRMLS) method was introduced for interpolation in SPH. The surface fitting studies were performed with a variety of polyline bases, spatial resolutions, particle distributions, kernel functions, and support domain sizes. Numerical solutions were compared with the results using moving least-squares (MLS) and three SPH methods, including CSPH, K2SPH, and KGFSPH, and it was found that SRMLS not only has nonsingular moment matrix, but also obtains high-accuracy result. Finally, the capability of the algorithm coupled with SRMLS and SPH was illustrated and assessed through several numerical tests

Fixed-point iterative sweeping methods for static Hamilton-Jacobi equations

Fast sweeping methods utilize the Gauss-Seidel iterations and alternating sweeping strat-egy to achieve the fast convergence for computations of static Hamilton-Jacobi equations. They take advantage of the properties of hyperbolic PDEs and try to cover a family of characteristics of the corresponding Hamilton-Jacobi equation in a certain direction simul-taneously in each sweeping order. The time-marching approach to steady state calculation is much slower than the fast sweeping methods due to the CFL condition constraint. But this kind of fixed-point iterations as time-marching methods have explicit form and do not involve inverse operation of nonlinear Hamiltonian. So it can solve general Hamilton-Jacobi equations using any monotone numerical Hamiltonian and high order approximations eas-ily. In this paper, we adopt the Gauss-Seidel idea and alternating sweeping strategy to the time-marching type fixed-point iterations to solve the static Hamilton-Jacobi equations. Ex-tensive numerical examples verify at least a 2 ∼ 5 times acceleration of convergence even on relatively coarse grids. The acceleration is even more when the grid is further refined. More-over the Gauss-Seidel philosophy and alternating sweeping strategy improves the stability, i.e., a larger CFL number can be used. Also the computational cost is exactly the same as the time-marching scheme at each time step

Exploring the Role of the Spatial Characteristics of Visible and Near-Infrared Reflectance in Predicting Soil Organic Carbon Density

Soil organic carbon stock plays a key role in the global carbon cycle and the precision agriculture. Visible and near-infrared reflectance spectroscopy (VNIRS) can directly reflect the internal physical construction and chemical substances of soil. The partial least squares regression (PLSR) is a classical and highly commonly used model in constructing soil spectral models and predicting soil properties. Nevertheless, using PLSR alone may not consider soil as characterized by strong spatial heterogeneity and dependence. However, considering the spatial characteristics of soil can offer valuable spatial information to guarantee the prediction accuracy of soil spectral models. Thus, this study aims to construct a rapid and accurate soil spectral model in predicting soil organic carbon density (SOCD) with the aid of the spatial autocorrelation of soil spectral reflectance. A total of 231 topsoil samples (0–30 cm) were collected from the Jianghan Plain, Wuhan, China. The spectral reflectance (350–2500 nm) was used as auxiliary variable. A geographically-weighted regression (GWR) model was used to evaluate the potential improvement of SOCD prediction when the spatial information of the spectral features was considered. Results showed that: (1) The principal components extracted from PLSR have a strong relationship with the regression coefficients at the average sampling distance (300 m) based on the Moran’s I values. (2) The eigenvectors of the principal components exhibited strong relationships with the absorption spectral features, and the regression coefficients of GWR varied with the geographical locations. (3) GWR displayed a higher accuracy than that of PLSR in predicting the SOCD by VNIRS. This study aimed to help people realize the importance of the spatial characteristics of soil properties and their spectra. This work also introduced guidelines for the application of GWR in predicting soil properties by VNIRS