SEMI-IMPLICIT SPECTRAL DEFERRED CORRECTION METHODS BASED ON SECOND-ORDER TIME INTEGRATION SCHEMES FOR NONLINEAR PDES

In[20],a semi-implicit spectral deferred correction(SDC)method was proposed,which is efficient for highly nonlinear partial differential equations(PDEs).The semi-implicit SDC method in[20]is based on first-order time integration methods,which are corrected iteratively,with the order of accuracy increased by one for each additional iteration.In this paper,we will develop a class of semi-implicit SDC methods,which are based on second-order time integration methods and the order of accuracy are increased by two for each additional iteration.For spatial discretization,we employ the local discontinuous Galerkin(LDG)method to arrive at fully-discrete schemes,which are high-order accurate in both space and time.Numerical experiments are presented to demonstrate the accuracy,efficiency and robustness of the proposed semi-implicit SDC methods for solving complex nonlinear PDEs.

Arbitrary High-Order Fully-Decoupled Numerical Schemes for Phase-Field Models of Two-Phase Incompressible Flows

Due to the coupling between the hydrodynamic equation and the phase-field equation in two-phase incompressible flows,it is desirable to develop efficient and high-order accurate numerical schemes that can decouple these two equations.One popular and efficient strategy is to add an explicit stabilizing term to the convective velocity in the phase-field equation to decouple them.The resulting schemes are only first-order accurate in time,and it seems extremely difficult to generalize the idea of stabilization to the second-order or higher version.In this paper,we employ the spectral deferred correction method to improve the temporal accuracy,based on the first-order decoupled and energy-stable scheme constructed by the stabilization idea.The novelty lies in how the decoupling and linear implicit properties are maintained to improve the efficiency.Within the framework of the spatially discretized local discontinuous Galerkin method,the resulting numerical schemes are fully decoupled,efficient,and high-order accurate in both time and space.Numerical experiments are performed to validate the high-order accuracy and efficiency of the methods for solving phase-field models of two-phase incompressible flows.

High-Order Decoupled and Bound Preserving Local Discontinuous Galerkin Methods for a Class of Chemotaxis Models

In this paper,we explore bound preserving and high-order accurate local discontinuous Galerkin(LDG)schemes to solve a class of chemotaxis models,including the classical Keller-Segel(KS)model and two other density-dependent problems.We use the convex splitting method,the variant energy quadratization method,and the scalar auxiliary variable method coupled with the LDG method to construct first-order temporal accurate schemes based on the gradient flow structure of the models.These semi-implicit schemes are decoupled,energy stable,and can be extended to high accuracy schemes using the semi-implicit spectral deferred correction method.Many bound preserving DG discretizations are only worked on explicit time integration methods and are difficult to get high-order accuracy.To overcome these difficulties,we use the Lagrange multipliers to enforce the implicit or semi-implicit LDG schemes to satisfy the bound constraints at each time step.This bound preserving limiter results in the Karush-Kuhn-Tucker condition,which can be solved by an efficient active set semi-smooth Newton method.Various numerical experiments illustrate the high-order accuracy and the effect of bound preserving.

Determination of micro yttrium in an ytterbium matrix by inductively coupled plasma atomic emission spectrometry and wavelet transform

In the determination of trace yttrium (Y) in an ytterbium (Yb) matrix byinductively coupled plasma atomic emission spectrometry (ICP-AES), the most prominent line ofyttrium, Y 371.030 nm line, suffers from strong interference due to an emission line of ytterbium.In mis work, a method based on wavelet transform was proposed for the spectral interferencecorrection. Haar wavelet was selected as the mother wavelet. The discrete detail after the thirddecomposition, D3, was chosen for quantitative analysis based on the consideration of bothseparation degree and peak height. The linear correlation coefficient between the height of the leftpositive peak in D3 and the concentration of Y was calculated to be 0.9926. Six synthetic sampleswere analyzed, and the recovery for yttrium varied from 96.3 percent to 110.0 percent. The amountsof yttrium in three ytterbium metal samples were determined by the proposed approach with an averagerelative standard deviation (RSD) of 2.5 percent, and the detection limit for yttrium was 0.016percent. This novel correction technique is fast and convenient, since neither complicated modelassumption nor time-consuming iteration is required. Furthermore, it is not affected by thewavelength drift inherent in monochromators that will severely reduce the accuracy of resultsobtained by some chemometric methods.

Semi-Implicit Spectral Deferred Correction Method Based on the Invariant Energy Quadratization Approach for Phase Field Problems

This paper presents a high order time discretization method by combining the semi-implicit spectral deferred correction method with energy stable linear schemes to simulate a series of phase field problems.We start with the linear scheme,which is based on the invariant energy quadratization approach and is proved to be linear unconditionally energy stable.The scheme also takes advantage of avoiding nonlinear iteration and the restriction of time step to guarantee the nonlinear system uniquely solvable.Moreover,the scheme leads to linear algebraic system to solve at each iteration,and we employ the multigrid solver to solve it efficiently.Numerical re-sults are given to illustrate that the combination of local discontinuous Galerkin(LDG)spatial discretization and the high order temporal scheme is a practical,accurate and efficient simulation tool when solving phase field problems.Namely,we can obtain high order accuracy in both time and space by solving some simple linear algebraic equations.

Novel Approach to Characterization of Rare Earth Complexation with 1, 5-Bis(2-Hydroxy-5-Chlorphenyl)-3-Cyanoformazan in Presence of Cetyltrimethylammonium Bromide

The ternary interaction of 1, 5 bis(2 hydroxy 5 chlorphenyl) 3 cyanoformazan (HCPCF) with cetyltrimethylammonium bromide (CTAB) and rare earths (RE: Yb, Dy, Er and Eu) was investigated at pH 9.84 by the microsurface adsorption spectral correction technique (MSASC). The aggregation of HCPCF on CTAB obeys the Langmuir isothermal adsorption and the interaction of RE with the HCPCF CTAB aggregate was first found to accord with the monolayer binding. The effects of temperature and ionic strength of solution on the aggregations were made. The binary aggregate and the ternary complex were characterized.

Physical and Chemical Matrix Effects in Soil Carbon Quantification Using Laser-Induced Breakdown Spectroscopy

Advanced field methods of carbon (C) analysis should now be capable of providing repetitive, sequential measurements for the evaluation of spatial and temporal variation at a scale that was previously unfeasible. Some spectroscopy techniques, such as laser-induced breakdown spectroscopy (LIBS), have portable features that may potentially lead to clean and rapid alternative approaches for this purpose. The goal of this study was to quantify the C content of soils with different textures and with high iron and aluminum concentrations using LIBS. LIBS emission spectra from soil pellets were captured, and the C content was estimated (emission line of C (I) at 193.03 nm) after spectral offset and aluminum spectral interference correction. This technique is highly portable and could be ideal for providing the soil C content in a heterogeneous experiment. Dry combustion was used as a reference method, and for calibration a conventional linear model was evaluated based on soil textural classes. The correlation between reference and LIBS values showed r = 0.86 for medium-textured soils and r = 0.93 for fine-textured soils. The data showed that better correlation and lower error (14%) values were found for the fine-textured LIBS model. The limit of detection (LOD) was found to be 0.32% for medium-textured soils and 0.13% for fine-textured soils. The results indicated that LIBS quantification can be affected by the texture and chemical composition of soil. Signal treatment was shown to be very important for mitigation of these interferences and to improve quantification.

HIGH ORDER LOCAL DISCONTINUOUS GALERKIN METHODS FOR THE ALLEN-CAHN EQUATION： ANALYSIS AND SIMULATION

In this paper, we present a local discontinuous Galerkin （LDG） method for the AllenCahn equation. We prove the energy stability, analyze the optimal convergence rate of k ＋ 1 in L2 norm and present the （2k＋1）-th order negative-norm estimate of the semi- discrete LDG method for the Allen-Cahn equation with smooth solution. To relax the severe time step restriction of explicit time marching methods, we construct a first order semi-implicit scheme based on the convex splitting principle of the discrete Allen-Cahn energy and prove the corresponding unconditional energy stability. To achieve high order temporal accuracy, we employ the semi-implicit spectral deferred correction （SDC） method. Combining with the unconditionally stable convex splitting scheme, the SDC method can be high order accurate and stable in our numerical tests. To enhance the efficiency of the proposed methods, the multigrid solver is adapted to solve the resulting nonlinear algebraic systems. Numerical studies are presented to confirm that we can achieve optimal accuracy of （O（hk＋1） in L2 norm and improve the LDG solution from （O（hk＋1） to （O（h2k＋1） with the accuracy enhancement post-processing technique.

A High Order Adaptive Time-Stepping Strategy and Local Discontinuous Galerkin Method for the Modified Phase Field Crystal Equation

In this paper,we will develop a first order and a second order convex splitting,and a first order linear energy stable fully discrete local discontinuous Galerkin(LDG)methods for the modified phase field crystal(MPFC)equation.In which,the first order linear scheme is based on the invariant energy quadratization approach.The MPFC equation is a damped wave equation,and to preserve an energy stability,it is necessary to introduce a pseudo energy,which all increase the difficulty of constructing numerical methods comparing with the phase field crystal(PFC)equation.Due to the severe time step restriction of explicit timemarchingmethods,we introduce the first order and second order semi-implicit schemes,which are proved to be unconditionally energy stable.In order to improve the temporal accuracy,the semi-implicit spectral deferred correction(SDC)method combining with the first order convex splitting scheme is employed.Numerical simulations of the MPFC equation always need long time to reach steady state,and then adaptive time-stepping method is necessary and of paramount importance.The schemes at the implicit time level are linear or nonlinear and we solve them by multigrid solver.Numerical experiments of the accuracy and long time simulations are presented demonstrating the capability and efficiency of the proposed methods,and the effectiveness of the adaptive time-stepping strategy.

A local discontinuous Galerkin method for the pattern formation dynamical model in polymerizing action flocks

In this paper,we apply local discontinuous Galerkin methods to the pattern formation dynamical model in polymerizing action flocks.Optimal error estimates for the density and filament polarization in different norms are established.We use a semi-implicit spectral deferred correction time method for time discretization,which allows a relative large time step and avoids computation of a Jacobian matrix.Numerical experiments are presented to verify the theoretical analysis and to show the capability for simulations of action wave formation.

Adaptive local discontinuous Galerkin methods with semi-implicit time discretizations for the Navier-Stokes equations

In this paper,we present a mesh adaptation algorithm for the unsteady compressible Navier-Stokes equations under the framework of local discontinuous Galerkin methods coupled with implicit-explicit Runge-Kutta or spectral deferred correction time discretization methods.In both of the two high order semi-implicit time integration methods,the convective flux is treated explicitly and the viscous and heat fluxes are treated implicitly.The remarkable benefits of such semi-implicit temporal discretizations are that they can not only overcome the stringent time step restriction compared with time explicit methods,but also avoid the construction of the large Jacobian matrix as is done for fully implicit methods,thus are relatively easy to implement.To save computing time as well as capture the flow structures of interest accurately,a local mesh refinement(h-adaptive)technique,in which we present detailed criteria for selecting candidate elements and complete strategies to refine and coarsen them,is also applied for the Navier-Stokes equations.Numerical experiments are provided to illustrate the high order accuracy,efficiency and capabilities of the semi-implicit schemes in combination with adaptive local discontinuous Galerkin methods for the Navier-Stokes equations.

A comprehensive revisit of the ρ meson with improved Monte-Carlo based QCD sum rules

We improve the Monte-Carlo based QCD sum rules by introducing the rigorous Hoolder-inequalitydetermined sum rule window and a Breit-Wigner type parametrization for the phenomenological spectral function.In this improved sum rule analysis methodology, the sum rule analysis window can be determined without any assumptions on OPE convergence or the QCD continuum. Therefore, an unbiased prediction can be obtained for the phenomenological parameters（the hadronic mass and width etc.）. We test the new approach in the ρ meson channel with re-examination and inclusion of αs corrections to dimension-4 condensates in the OPE. We obtain results highly consistent with experimental values. We also discuss the possible extension of this method to some other channels.