Search

A time-continuous (tc-)embedding method is first proposed for solving nonlinear scalar hyperbolic conservation laws with discontinuous solutions (shocks and rarefaction waves) on codimension 1, connected, smooth, and closed manifolds (surface PDEs or SPDEs in $${\mathbb {R}}^2$$ R 2 and $${\mathbb {R}}^3$$ R 3 ). The new embedding method improves upon the classical closest point (cp-)embedding method, which requires re-establishments of the constant-along-normal (CAN-)property of the extension function at every time step, in terms of accuracy and efficiency, by incorporating the CAN-property analytically and explicitly in the embedding equation. The tc-embedding SPDEs are solved by the second-order nonlinear central finite volume scheme with a nonlinear minmod slope limiter in space, and the third-order total variation diminished Runge-Kutta scheme in time. An adaptive nonlinear essentially non-oscillatory polynomial interpolation is used to obtain the solution values at the ghost cells. Numerical results in solving the linear wave equation and the Burgers’ equation show that the proposed tc-embedding method has better accuracy, improved resolution, and reduced CPU times than the classical cp-embedding method. The Burgers’ equation, the traffic flow problem, and the Buckley-Leverett equation are solved to demonstrate the robust performance of the tc-embedding method in resolving fine-scale structures efficiently even in the presence of a shock and the essentially non-oscillatory capturing of shocks and rarefaction waves on simple and complex shaped one-dimensional manifolds. Burgers’ equation is also solved on the two-dimensional torus-shaped and spherical-shaped manifolds.

A modified fifth order Z-type (nonlinear) weights, which consist of a linear term and a nonlinear term, in the weighted essentially non-oscillatory (WENO) polynomial reconstruction procedure for the WENO-Z finite difference scheme in solving hyperbolic conservation laws is proposed. The nonlinear term is modified by a modifier function that is based on the linear combination of the local smoothness indicators. The WENO scheme with the modified Z-type weights (WENO-D) scheme and its improved version (WENO-A) scheme are proposed. They are analyzed for the maximum error and the order of accuracy for approximating the derivative of a smooth function with high order critical points, where the first few consecutive derivatives vanish. The analysis and numerical experiments show that, they achieve the optimal (fifth) order of accuracy regardless of the order of critical point with an arbitrary small sensitivity parameter, aka, satisfy the Cp-property. Furthermore, with an optimal variable sensitivity parameter, they have a quicker convergence and a significant error reduction over the WENO-Z scheme. They also achieve an improved balance between the linear term, which resolves a smooth function with the fifth order upwind central scheme, and the modified nonlinear term, which detects potential high gradients and discontinuities in a non-smooth function. The performance of the WENO schemes, in terms of resolution, essentially non-oscillatory shock capturing and efficiency, are compared by solving several one- and two-dimensional benchmark shocked flows. The results show that they perform overall as well as, if not slightly better than, the WENO-Z scheme.

Euler equations with a gravitational source term (PDEs) admit a hydrostatic equilibrium state where the source term exactly balances the flux gradient. The property of exact preservation of the equilibria is highly desirable when the PDEs are numerically solved. Li and Xing (J Comput Phys 316:145–163, 2016) proposed a high-order well-balanced characteristic-wise finite volume weighted essentially non-oscillatory (FV-WENO) scheme for the cases of isothermal equilibrium and polytropic equilibrium. On the contrary to what was claimed, the scheme is not well-balanced. The root of the problem is the precarious effects of a non-zero sensitivity parameter in the nonlinear weights of the WENO polynomial reconstruction procedure (WENO operator). The effects are identified in the theoretical proof for the well-balanced scheme and verified numerically on a coarse mesh resolution and a long time simulation of the PDEs. In this study, two simple yet effective numerical techniques derived from the multiplicative-invariance (MI) property of a WENO operator are invoked to rectify the sensitivity parameter’s dependency yielding a correct proof for the sensitivity parameter-independent (characteristic-wise) well-balanced FV-WENO scheme. The (non-)well-balanced nature of the schemes is demonstrated with several one- and two-dimensional benchmark steady state problems and a small perturbation over the steady state problems. Moreover, the one-dimensional Sod problem under the gravitational field is also simulated for showing the performance of the well-balanced FV-WENO scheme in capturing shock, contact discontinuity, and rarefaction wave in an essentially non-oscillatory nature. It also indicates that the numerical scheme with the third-order Runge–Kutta time-stepping scheme should take the CFL number less than 0.5 to mitigate the Gibbs oscillations at the shock without increasing the numerical dissipation artificially in the Lax–Friedrichs numerical flux.

In Jung et al. (Appl Numer Math 61:77–91, 2011), an iterative adaptive multi-quadric radial basis function (IAMQ-RBF) method has been developed for edges detection of the piecewise analytical functions. For a uniformly spaced mesh, the perturbed Toeplitz matrices, which are modified by those columns where the shape parameters are reset to zero due to the appearance of edges at the corresponding locations, are created. Its inverse must be recomputed at each iterative step, which incurs a heavy $$O(n^3)$$ computational cost. To overcome this issue of efficiency, we develop a fast direct solver (IAMQ-RBF-Fast) to reformulate the perturbed Toeplitz system into two Toeplitz systems and a small linear system via the Sherman–Morrison–Woodbury formula. The $$O(n^2)$$ Levinson–Durbin recursive algorithm that employed Yule–Walker algorithm is used to find the inverse of the Toeplitz matrix fast. Several classical benchmark examples show that the IAMQ-RBF-Fast based edges detection method can be at least three times faster than the original IAMQ-RBF based one. And it can capture an edge with fewer grid points than the multi-resolution analysis (Harten in J Comput Phys 49:357–393, 1983) and just as good as if not better than the L1PA method (Denker and Gelb in SIAM J Sci Comput 39(2):A559–A592, 2017). Preliminary results in the density solution of the 1D Mach 3 extended shock–density wave interaction problem solved by the hybrid compact-WENO finite difference scheme with the IAMQ-RBF-Fast based shocks detection method demonstrating an excellent performance in term of speed and accuracy, are also shown.

The fifth, seventh and ninth order characteristic-wise alternative weighted essentially non-oscillatory (AWENO) finite difference schemes are applied to the fully conservative (FC) form and the overestimated quasi-conservative (OQC) form of the compressible multicomponent flows. Several linear and nonlinear numerical operators such as the linear Lax–Friedrichs operator and linearized nonlinear WENO operator and their mathematical properties are defined in order to build a general mathematical (numerical) framework for identifying the necessary and sufficient conditions required in maintaining the equilibriums of certain physical relevant properties discretely. In the case of OQC form, the AWENO scheme with the modified flux can be rigorously proved to maintain the equilibriums of velocity, pressure and temperature. Furthermore, we also show that the FC form cannot maintain the equilibriums without an additional advection equation of auxiliary variable involving the specific heat ratio. Extensive one- and two-dimensional classical benchmark problems, such as the moving material interface problem, multifluid shock-density interaction problem and shock-R22-bubble interaction problem, verify the theoretical results and also show that the AWENO schemes demonstrate less dissipation error and higher resolution than the classical WENO-Z scheme in the splitting form (Nonomura and Fujii in J Comput Phys 340:358–388, 2017).

The recretohalophyte with specialized salt-secreting structures including salt glands and salt bladders can secrete salt from their bodies and easily adapt themselves to many kinds of salt habitats. Salt glands and salt bladders, arose from dermatogen cells, are excretory organs specially adapted for dealing with ionic homeostasis in the cells of recretohalophytes. The main function of salt glands or salt bladders is to secrete excess ions that invade the plant. The structures of salt glands or salt bladders differ among plant species. In addition to structural differences, salt glands also differ in their secretion abilities. In this review, we mainly focus on recent progress in the mechanism of salt excretion of salt glands and salt bladders, and in particular, emphasize the vesicle-mediated secretion systems from the vacuole to the plasmalemma and the possibly involved membrane-bound translocating proteins for salt secretion of plant gland secretory cell.

In this paper, we shall mainly study the p-solvable finite group in terms of p-local rank, and a group theoretic characterization will be given of finite p-solvable groups with p-local rank two. Theorem A Let G be a finite p-solvable group with p-local rank plr(G) = 2 and O p (G) = 1. If P is a Sylow p-subgroup of G, then P has a normal subgroup Q such that P/Q is cyclic or a generalized quaternion 2-group and the p-rank of Q is at most two. Theorem B Let G be a finite p-solvable group with O p (G) = 1. Then the p-length l p (G) < plr(G); if in addition plr(G) = 1 p (G) and p > 5 is odd, then plr(G) = 0 or 1.