Explicit multi-symplectic method for the Zakharov-Kuznetsov equation

We propose an explicit multi-symplectic method to solve the two-dimensional Zakharov-Kuznetsov equation. The method combines the multi-symplectic Fourier pseudospectral method for spatial discretization and the Euler method for temporal discretization. It is verified that the proposed method has corresponding discrete multi-symplectic conservation laws. Numerical simulations indicate that the proposed scheme is characterized by excellent conservation.

Multi-symplectic method for generalized Boussinesq equation

The generalized Boussinesq equation that represents a group of important nonlinear equations possesses many interesting properties. Multi-symplectic formulations of the generalized Boussinesq equation in the Hamilton space are introduced in this paper. And then an implicit multi-symplectic scheme equivalent to the multi-symplectic Box scheme is constructed to solve the partial differential equations （PDEs） derived from the generalized Boussinesq equation. Finally, the numerical experiments on the soliton solutions of the generalized Boussinesq equation are reported. The results show that the multi-symplectic method is an efficient algorithm with excellent long-time numerical behaviors for nonlinear partial differential equations.

Multi-symplectic methods for membrane free vibration equation

In this paper, the multi-symplectic formulations of the membrane free vibration equation with periodic boundary conditions in Hamilton space are considered. The complex method is introduced and a semi-implicit twenty-seven-points scheme with certain discrete conservation laws-a multi-symplectic conservation law （CLS）, a local energy conservation law （ECL） as well as a local momentum conservation law （MCL） --is constructed to discrete the PDEs that are derived from the membrane free vibration equation. The results of the numerical experiments show that the multi-symplectic scheme has excellent long-time numerical behavior,

Multi-symplectic method for generalized fifth-order KdV equation

This paper considers the multi-symplectic formulations of the generalized fifth-order KdV equation in Hamiltonian space. Recurring to the midpoint rule, it presents an implicit multi-symplectic scheme with discrete multi-symplectic conservation law to solve the partial differential equations which are derived from the generalized fifth-order KdV equation numerically. The results of the numerical experiments show that this multi-symplectic algorithm is good in accuracy and its long-time numerical behaviour is also perfect.

Second order conformal multi-symplectic method for the damped Korteweg–de Vries equation

A conformal multi-symplectic method has been proposed for the damped Korteweg–de Vries(DKdV) equation, which is based on the conformal multi-symplectic structure. By using the Strang-splitting method and the Preissmann box scheme,we obtain a conformal multi-symplectic scheme for multi-symplectic partial differential equations(PDEs) with added dissipation. Applying it to the DKdV equation, we construct a conformal multi-symplectic algorithm for it, which is of second order accuracy in time. Numerical experiments demonstrate that the proposed method not only preserves the dissipation rate of mass exactly with periodic boundary conditions, but also has excellent long-time numerical behavior.

Multi-symplectic method for the coupled Schrdinger–KdV equations

In this paper, we present a multi-symplectic Hamiltonian formulation of the coupled Schrtidinger-KdV equations （CS＇KE） with periodic boundary conditions. Then we develop a novel multi-symplectic Fourier pseudospectral （MSFP） scheme for the CSKE. In numerical experiments, we compare the MSFP method with the Crank-Nicholson （CN） method. Our results show high accuracy, effectiveness, and good ability of conserving the invariants of the MSFP method.

A Review on Stochastic Multi-symplectic Methods for Stochastic Maxwell Equations

Stochastic multi-symplectic methods are a class of numerical methods preserving the discrete stochastic multi-symplectic conservation law. These methods have the remarkable superiority to conventional numerical methods when applied to stochastic Hamiltonian partial differential equations (PDEs), such as long-time behavior, geometric structure preserving, and physical properties preserving. Stochastic Maxwell equations driven by either additive noise or multiplicative noise are a system of stochastic Hamiltonian PDEs intrinsically, which play an important role in fields such as stochastic electromagnetism and statistical radiophysics. Thereby, the construction and the analysis of various numerical methods for stochastic Maxwell equations which inherit the stochastic multi-symplecticity, the evolution laws of energy and divergence of the original system are an important and promising subject. The first stochastic multi-symplectic method is designed and analyzed to stochastic Maxwell equations by Hong et al.(A stochastic multi-symplectic scheme for stochastic Maxwell equations with additive noise. J. Comput. Phys. 268:255-268, 2014). Subsequently, there have been developed various stochastic multi-symplectic methods to solve stochastic Maxwell equations. In this paper, we make a review on these stochastic multi-symplectic methods for solving stochastic Maxwell equations driven by a stochastic process. Meanwhile, the theoretical results of well-posedness and conservation laws of the stochastic Maxwell equations are included.

Explicit Multi-symplectic Method for a High Order Wave Equation of KdV Type

In this paper, we consider multi-symplectic Fourier pseudospectral method for a high order integrable equation of KdV type, which describes many important physical phenomena. The multi-symplectic structure are constructed for the equation, and the conservation laws of the continuous equation are presented. The multisymplectic discretization of each formulation is exemplified by the multi-symplectic Fourier pseudospectral scheme. The numerical experiments are given, and the results verify the efficiency of the Fourier pseudospectral method.

Multi-symplectic method for the generalized(2+1)-dimensionalKdV-mKdV equation

In the present paper, a general solution involv- ing three arbitrary functions for the generalized （2＋1）- dimensional KdV-mKdV equation, which is derived from the generalized （1＋1）-dimensional KdV-mKdV equa- tion, is first introduced by means of the Wiess, Tabor, Carnevale （WTC） truncation method. And then multi- symplectic formulations with several conservation laws taken into account are presented for the generalized （2＋1）- dimensional KdV-mKdV equation based on the multi- symplectic theory of Bridges. Subsequently, in order to simulate the periodic wave solutions in terms of rational functions of the Jacobi elliptic functions derived from thegeneral solution, a semi-implicit multi-symplectic scheme is constructed that is equivalent 1：o the Preissmann scheme. From the results of the numerical experiments, we can con- clude that the multi-symplectic schemes can accurately sim- ulate the periodic wave solutions of the generalized （2＋1）- dimensional KdV-mKdV equation while preserve approxi- mately the conservation laws.

Multi-symplectic method to analyze the mixed state of Ⅱ-superconductors

The mixed state of two-band II-superconductor is analyzed by the multi-symplectic method. As to the Ginzburg-Landau equation depending on time that describes the mixed state of two-band II-superconductor, the multi-symplectic formulations with several conservation laws: a multi-symplectic conservation law, an energy con- servation law, as well as a momentum conservation law, are presented firstly; then an eighteen points scheme is constructed to simulate the multi-symplectic partial differential equations (PDEs) that are derived from the Ginzburg-Landau equation; finally, based on the simulation results, the volt-ampere characteristic curves are obtained, as well as the relationships between the temperature and resistivity of a suppositional two-band II-superconductor model under different magnetic intensi- ties. From the results of the numerical experiments, it is concluded that the notable property of the mixed state of the two-band II-superconductor is that: The trans- formation temperature decreases and the resistivity ρ increases rapidly with the increase of the magnetic intensity B. In addition, the simulation results show that the multi-symplectic method has two remarkable advantages: high accuracy and excellent long-time numerical behavior.

Multi-symplectic Runge-Kutta methods for Landau-Ginzburg-Higgs equation

Nonlinear wave equations have been extensively investigated in the last sev- eral decades. The Landau-Ginzburg-Higgs equation, a typical nonlinear wave equation, is studied in this paper based on the multi-symplectic theory in the Hamilton space. The multi-symplectic Runge-Kutta method is reviewed, and a semi-implicit scheme with certain discrete conservation laws is constructed to solve the first-order partial differential equations （PDEs） derived from the Landau-Ginzburg-Higgs equation. The numerical re- sults for the soliton solution of the Landau-Ginzburg-Higgs equation are reported, showing that the multi-symplectic Runge-Kutta method is an efficient algorithm with excellent long-time numerical behaviors.

Drilling-based measuring method for the c-φ parameter of rock and its field application

The technology of drilling tests makes it possible to obtain the strength parameter of rock accurately in situ. In this paper, a new rock cutting analysis model that considers the influence of the rock crushing zone(RCZ) is built. The formula for an ultimate cutting force is established based on the limit equilibrium principle. The relationship between digital drilling parameters(DDP) and the c-φ parameter(DDP-cφ formula, where c refers to the cohesion and φ refers to the internal friction angle) is derived, and the response of drilling parameters and cutting ratio to the strength parameters is analyzed. The drillingbased measuring method for the c-φ parameter of rock is constructed. The laboratory verification test is then completed, and the difference in results between the drilling test and the compression test is less than 6%. On this basis, in-situ rock drilling tests in a traffic tunnel and a coal mine roadway are carried out, and the strength parameters of the surrounding rock are effectively tested. The average difference ratio of the results is less than 11%, which verifies the effectiveness of the proposed method for obtaining the strength parameters based on digital drilling. This study provides methodological support for field testing of rock strength parameters.

Multi-symplectic wavelet splitting method for the strongly coupled Schrodinger system

We propose a multi-symplectic wavelet splitting equations. Based on its mu]ti-symplectic formulation, method to solve the strongly coupled nonlinear SchrSdinger the strongly coupled nonlinear SchrSdinger equations can be split into one linear multi-symplectic subsystem and one nonlinear infinite-dimensional Hamiltonian subsystem. For the linear subsystem, the multi-symplectic wavelet collocation method and the symplectic Euler method are employed in spatial and temporal discretization, respectively. For the nonlinear subsystem, the mid-point symplectic scheme is used. Numerical simulations show the effectiveness of the proposed method during long-time numerical calculation.

A comparison study on structure-function relationship of polysaccharides obtained from sea buckthorn berries using different methods:antioxidant and bile acid-binding capacity

In this study,the structural characters,antioxidant activities and bile acid-binding ability of sea buckthorn polysaccharides(HRPs)obtained by the commonly used hot water(HRP-W),pressurized hot water(HRP-H),ultrasonic(HRP-U),acid(HRP-C)and alkali(HRP-A)assisted extraction methods were investigated.The results demonstrated that extraction methods had significant effects on extraction yield,monosaccharide composition,molecular weight,particle size,triple-helical structure,and surface morphology of HRPs except for the major linkage bands.Thermogravimetric analysis showed that HRP-U with filamentous reticular microstructure exhibited better thermal stability.The HRP-A with the lowest molecular weight and highest arabinose content possessed the best antioxidant activities.Moreover,the rheological analysis indicated that HRPs with higher galacturonic acid content and molecular weight showed higher viscosity and stronger crosslinking network(HRP-C,HRP-W and HRP-U),which exhibited stronger bile acid binding capacity.The present findings provide scientific evidence in the preparation technology of sea buckthorn polysaccharides with good antioxidant and bile acid binding capacity which are related to the structure affected by the extraction methods.

A Multi-Symplectic Compact Method for the Two-Component Camassa-Holm Equation with Singular Solutions

The two-component Camassa–Holm equation includes many intriguing phenomena. We propose a multi-symplectic compact method to solve the two-component Camassa–Holm equation. Based on its multi-symplectic formulation, the proposed method is derived by the sixth-order compact finite difference method in spatial discretization and the symplectic implicit midpoint scheme in temporal discretization. Numerical experiments finely describe the velocity and density variables in the two-component integrable system and distinctly display the evolvement of the singular solutions. Moreover, the proposed method shows good conservative properties during long-time numerical simulation.

Robustness Study and Superior Method Development and Validation for Analytical Assay Method of Atropine Sulfate in Pharmaceutical Ophthalmic Solution

Background: The robustness is a measurement of an analytical chemical method and its ability to contain unaffected by little with deliberate variation of analytical chemical method parameters. The analytical chemical method variation parameters are based on pH variability of buffer solution of mobile phase, organic ratio composition changes, stationary phase (column) manufacture, brand name and lot number variation;flow rate variation and temperature variation of chromatographic system. The analytical chemical method for assay of Atropine Sulfate conducted for robustness evaluation. The typical variation considered for mobile phase organic ratio change, change of pH, change of temperature, change of flow rate, change of column etc. Purpose: The aim of this study is to develop a cost effective, short run time and robust analytical chemical method for the assay quantification of Atropine in Pharmaceutical Ophthalmic Solution. This will help to make analytical decisions quickly for research and development scientists as well as will help with quality control product release for patient consumption. This analytical method will help to meet the market demand through quick quality control test of Atropine Ophthalmic Solution and it is very easy for maintaining (GDP) good documentation practices within the shortest period of time. Method: HPLC method has been selected for developing superior method to Compendial method. Both the compendial HPLC method and developed HPLC method was run into the same HPLC system to prove the superiority of developed method. Sensitivity, precision, reproducibility, accuracy parameters were considered for superiority of method. Mobile phase ratio change, pH of buffer solution, change of stationary phase temperature, change of flow rate and change of column were taken into consideration for robustness study of the developed method. Results: The limit of quantitation (LOQ) of developed method was much low than the compendial method. The % RSD for the six sample assay of developed method was 0.4% where the % RSD of the compendial method was 1.2%. The reproducibility between two analysts was 100.4% for developed method on the contrary the compendial method was 98.4%.

Hybrid Strategy of Partitioned and Monolithic Methods for Solving Strongly Coupled Analysis of Inverse and Direct Piezoelectric and Circuit Coupling

The inverse and direct piezoelectric and circuit coupling are widely observed in advanced electro-mechanical systems such as piezoelectric energy harvesters.Existing strongly coupled analysis methods based on direct numerical modeling for this phenomenon can be classified into partitioned or monolithic formulations.Each formulation has its advantages and disadvantages,and the choice depends on the characteristics of each coupled problem.This study proposes a new option:a coupled analysis strategy that combines the best features of the existing formulations,namely,the hybrid partitioned-monolithic method.The analysis of inverse piezoelectricity and the monolithic analysis of direct piezoelectric and circuit interaction are strongly coupled using a partitioned iterative hierarchical algorithm.In a typical benchmark problem of a piezoelectric energy harvester,this research compares the results from the proposed method to those from the conventional strongly coupled partitioned iterative method,discussing the accuracy,stability,and computational cost.The proposed hybrid concept is effective for coupled multi-physics problems,including various coupling conditions.

Explicit Multi-Symplectic Methods for Hamiltonian Wave Equations

In this paper,based on the multi-symplecticity of concatenating symplectic Runge-Kutta-Nystrom(SRKN)methods and symplectic Runge-Kutta-type methods for numerically solving Hamiltonian PDEs,explicit multi-symplectic schemes are constructed and investigated,where the nonlinear wave equation is taken as a model problem.Numerical comparisons are made to illustrate the effectiveness of our newly derived explicit multi-symplectic integrators.

Numerical Dispersion Relation of Multi-symplectic Runge-Kutta Methods for Hamiltonian PDEs

Numerical dispersion relation of the multi-symplectic Runge-Kutta （MSRK） method for linear Hamiltonian PDEs is derived in the present paper, which is shown to be a discrete counterpart to that possessed by the differential equation. This provides further understanding of MSRK methods. However, much still remains to be investigated further.

Stability Analysis and Performance Evaluation of Additive Mixed-Precision Runge-Kutta Methods

Additive Runge-Kutta methods designed for preserving highly accurate solutions in mixed-precision computation were previously proposed and analyzed.These specially designed methods use reduced precision for the implicit computations and full precision for the explicit computations.In this work,we analyze the stability properties of these methods and their sensitivity to the low-precision rounding errors,and demonstrate their performance in terms of accuracy and efficiency.We develop codes in FORTRAN and Julia to solve nonlinear systems of ODEs and PDEs using the mixed-precision additive Runge-Kutta(MP-ARK)methods.The convergence,accuracy,and runtime of these methods are explored.We show that for a given level of accuracy,suitably chosen MP-ARK methods may provide significant reductions in runtime.