High-Order Bound-Preserving Finite Difference Methods for Multispecies and Multireaction Detonations

In this paper,we apply high-order finite difference(FD)schemes for multispecies and multireaction detonations(MMD).In MMD,the density and pressure are positive and the mass fraction of the ith species in the chemical reaction,say zi,is between 0 and 1,withΣz_(i)=1.Due to the lack of maximum-principle,most of the previous bound-preserving technique cannot be applied directly.To preserve those bounds,we will use the positivity-preserving technique to all the zi'is and enforceΣz_(i)=1 by constructing conservative schemes,thanks to conservative time integrations and consistent numerical fluxes in the system.Moreover,detonation is an extreme singular mode of flame propagation in premixed gas,and the model contains a significant stiff source.It is well known that for hyperbolic equations with stiff source,the transition points in the numerical approximations near the shocks may trigger spurious shock speed,leading to wrong shock position.Intuitively,the high-order weighted essentially non-oscillatory(WENO)scheme,which can suppress oscillations near the discontinuities,would be a good choice for spatial discretization.However,with the nonlinear weights,the numerical fluxes are no longer“consistent”,leading to nonconservative numerical schemes and the bound-preserving technique does not work.Numerical experiments demonstrate that,without further numerical techniques such as subcell resolutions,the conservative FD method with linear weights can yield better numerical approximations than the nonconservative WENO scheme.

Numerical Study by Imposing the Finite Difference Method for Unsteady Casson Fluid Flow with Heat Flux

This article presents an investigation into the flow and heat transfer characteristics of an impermeable stretching sheet subjected to Magnetohydrodynamic Casson fluid. The study considers the influence of slip velocity, thermal radiation conditions, and heat flux. The investigation is conducted employing a robust numerical method that accounts for the impact of thermal radiation. This category of fluid is apt for characterizing the movement of blood within an industrial artery, where the flow can be regulated by a material designed to manage it. The resolution of the ensuing system of ordinary differential equations (ODEs), representing the described problem, is accomplished through the application of the finite difference method. The examination of flow and heat transfer characteristics, including aspects such as unsteadiness, radiation parameter, slip velocity, Casson parameter, and Prandtl number, is explored and visually presented through tables and graphs to illustrate their impact. On the stretching sheet, calculations, and descriptions of the local skin-friction coefficient and the local Nusselt number are conducted. In conclusion, the findings indicate that the proposed method serves as a straightforward and efficient tool for exploring the solutions of fluid models of this kind.

Implicit finite difference method for fractional percolation equation with Dirichlet and fractional boundary conditions

An implicit finite difference method is developed for a one-dimensional frac- tional percolation equation （FPE） with the Dirichlet and fractional boundary conditions. The stability and convergence are discussed for two special cases, i.e., a continued seep- age flow with a monotone percolation coefficient and a seepage flow with the fractional Neumann boundary condition. The accuracy and efficiency of the method are checked with two numerical examples.

Numerical simulation of standing wave with 3D predictor-corrector finite difference method for potential flow equations

A three-dimensional （3D） predictor-corrector finite difference method for standing wave is developed. It is applied to solve the 3D nonlinear potential flow equa- tions with a free surface. The 3D irregular tank is mapped onto a fixed cubic tank through the proper coordinate transform schemes. The cubic tank is distributed by the staggered meshgrid, and the staggered meshgrid is used to denote the variables of the flow field. The predictor-corrector finite difference method is given to develop the difference equa- tions of the dynamic boundary equation and kinematic boundary equation. Experimental results show that, using the finite difference method of the predictor-corrector scheme, the numerical solutions agree well with the published results. The wave profiles of the standing wave with different amplitudes and wave lengths are studied. The numerical solutions are also analyzed and presented graphically.

A Comparative Study of the Difference Method and the Enzyme-Hydrolyzed Casein Method for Determining True Amino Acid Digestibilities and Endogenous Amino Acid Losses in Duck Feed

In this study, we examined the varia- tions between the difference method and the enzyme- hydrolyzed casein method for determining endogenous amino acid loss and the true amino acid digestibility in ducks fed normal protein-containing diets. These methods were compared to the nitrogen-free （N-free） diet method. The difference method was based on soy- bean meal as the only protein source, with the experi- mental diets containing crude protein levels at 15% and 20%. The enzyme-hydrolyzed casein method was based on enzyme-hydrolyzed casein meal as the pro- tein source, with the experimental diet containing a crude protein level of 17.5%. The N-free diet was prepared with starches and paper fibers. In each meth- od,64 Tianfu meat drakes （7-weeks-old） with an av- erage body weight of 2.77±0.16 kg were used and divided into four groups, and fed four different diets. Each group contained four replicates of four drakes and they were force fed trial diets according to the Sirbald method for detecting their apparent amino aciddigestibility, endogenous amino acid loss and true a- mino acid digestibility. The results demonstrated that using the difference, enzyme-hydrolyzed casein and N-free diet methods, endogenous amino acid losses were 0. 9946,1. 2243 and 0. 9297 mg/g dry matter in- take （ DMI）, respectively. The true amino acid digest- ibility measured by the difference method was 88.93 %±4.43 %. Using the enzyme-hydrolyzed ca- sein method with two dietary crude protein levels of 15% and 20%, the digestibility was 91.15%±4.33% and 91.97%±4. 16%, respectively, and by the N-free diet methods with two dietary crude protein levels of 15% and 20% ,it was 88.55%±4.29% and 88.82 %±4.61%, respectively. The results suggested that when the dietary protein level was 15% to 20 %, the true amino acid digestibility and endogenous ami- no acid loss as determined by the difference method was more accurate than the values determined by the enzyme-hydrolyzed casein method.

Solution precision of concrete temperature fields with finite difference method

With the finite difference method to calculate the temperature distribution in mass concrete structures, the solution precision will increase with a smaller step size, at the cost of computational time. In view of the basic characteristics of the finite difference method, a simple yet powerful improvement is introduced. By multiplying the adiabatic temperature function with a correction factor, the precision of the solution can be assured without an increase in the computation time. In addition, the correction rules for three types of commonly used concrete hydration formulas are investigated.

A Finite Difference Method for Determining Interdiffusivity of Aluminide Coating Formed on Superalloy

A numerical method has been developed to extract the composition-dependent interdiffusivity from the concentration profiles in the aluminide coating prepared by pack cementation. The procedure is based on the classic finite difference method (FDM). In order to simplify the model, effect of some alloying elements on interdiffusivity can be negligible. Calculated results indicate the interdiffusivity in aluminide coating strongly depends on the composition and give the formulas used to calculate interdiffusivity at 850, 950 and 1050癈. The effect on interdiffusivity is briefly discussed.

Computing Bifurcation Diagrams of Steady State Kuramoto Sivashinsky Equation by Difference Method

Utilizing difference formulae, we obtained the discrete systems of steady state Kuramoto Sivashinsky (K S) equation. Applied Newton's method and continuation technology to the systems, the bifurcated solutions are derived, and the bifurcation diagrams are constructed. All the results are successful and satisfactory.

THE UPWIND FINITE DIFFERENCE METHOD FOR MOVING BOUNDARY VALUE PROBLEM OF COUPLED SYSTEM

Coupled system of multilayer dynamics of fluids in porous media is to describe the history of oil-gas transport and accumulation in basin evolution.It is of great value in rational evaluation of prospecting and exploiting oil-gas resources.The mathematical model can be described as a coupled system of nonlinear partial differential equations with moving boundary values.The upwind finite difference schemes applicable to parallel arithmetic are put forward and two-dimensional and three-dimensional schemes are used to form a complete set.Some techniques,such as change of variables,calculus of variations, multiplicative commutation rule of difference operators,decomposition of high order difference operators and prior estimates,are adopted.The estimates in l~2 norm are derived to determine the error in the approximate solution.This method was already applied to the numerical simulation of migration-accumulation of oil resources.

Techniques for improving computational speed in numerical simulation of casting thermal stress based on finite difference method

Finite difference method (FDM) was applied to simulate thermal stress recently, which normally needs a long computational time and big computer storage. This study presents two techniques for improving computational speed in numerical simulation of casting thermal stress based on FDM, one for handling of nonconstant material properties and the other for dealing with the various coefficients in discretization equations. The use of the two techniques has been discussed and an application in wave-guide casting is given. The results show that the computational speed is almost tripled and the computer storage needed is reduced nearly half compared with those of the original method without the new technologies. The stress results for the casting domain obtained by both methods that set the temperature steps to 0.1 ℃ and 10 ℃, respectively are nearly the same and in good agreement with actual casting situation. It can be concluded that both handling the material properties as an assumption of stepwise profile and eliminating the repeated calculation are reliable and effective to improve computational speed, and applicable in heat transfer and fluid flow simulation.

Sloshing simulation of standing wave with time-independent finite difference method for Euler equations

The numerical solutions of standing waves for Euler equations with the nonlinear free surface boundary condition in a two-dimensional （2D） tank are studied. The irregular tank is mapped onto a fixed square domain through proper mapping functions. A staggered mesh system is employed in a 2D tank to calculate the elevation of the transient fluid. A time-independent finite difference method, which is developed by Bang- fuh Chen, is used to solve the Euler equations for incompressible and inviscid fluids. The numerical results agree well with the analytic solutions and previously published results. The sloshing profiles of surge and heave motion with initial standing waves are presented. The results show very clear nonlinear and beating phenomena.

ELEMENT FUNCTIONS OF DISCRETE OPERATOR DIFFERENCE METHOD

The discrete scheme called discrete operator difference for differential equations was given. Several difference elements for plate bending problems and plane problems were given. By investigating these elements, the ability of the discrete forms expressing to the element functions was talked about. In discrete operator difference method, the displacements of the elements can be reproduced exactly in the discrete forms whether the displacements are conforming or not. According to this point, discrete operator difference method is a method with good performance.

Label-Free and Real-Time Monitor of Binding and Dissociation Processes between Protein A and Swine IgG by Oblique-Incidence Reflectivity Difference Method

Life science has a need for detection methods that are label-free and real-time. In this paper, we have selected staphylococcal protein A （SPA） and swine immunoglobulin G （IgG）, and monitor the bindings between SPA and swine IgG with different concentrations, as well as the dissociations of SPA-swine IgG complex in different pH values of phosphate buffer by oblique-incidence reflectivity difference （OIRD） in a label-free and real-time fashion. We obtain the ON and OFF reaction dynamic curves corresponding to the bindings and dissociations of SPA and swine IgG. Through our analysis of the experimental results, we have been able to obtain the damping coefficients and the dissociation time of SPA and swine IgG for different pH values of the phosphate buffer. The results prove that the OIRD technique is a competing method for monitoring the dynamic processes of biomolecule interaction and achieving the quantitative information of reaction kinetics.

Solution to the Dirac equation using the finite difference method

In this study,single-particle energy was examined using the finite difference method by taking 208Pb as an example.If the first derivative term in the spherical Dirac equation is discretized using a three-point formula,a one-to-one correspondence occurs between the physical and spurious states.Although these energies are exactly the same,the wave functions of the spurious states exhibit a much faster staggering than those of the physical states.Such spurious states can be eliminated when applying the finite difference method by introducing an extra Wilson term into the Hamiltonian.Furthermore,it was also found that the number of spurious states can be reduced if we improve the accuracy of the numerical differential formula.The Dirac equation is then solved in a momentum space in which there is no differential operator,and we found that the spurious states can be completely avoided in the momentum space,even without an extra Wilson term.

Investigation into Effects of Buildings near Broadcasting Antenna on Far Fields Using a Finite-Difference Method

A steady state finite difference method is used to calculate EM fields generated by an MW broadcasting antenna. The effects of buildings on the wave propagation is studied based on the numerical results and field measurements. Both the algorithm and results are useful in the design of broadcasting antennas, as well as in the selection of transmitting sites.

SECOND-ORDER ACCURATE DIFFERENCE METHOD FOR THE SINGULARLY PERTURBED PROBLEM OF FOURTH-ORDER ORDINARY DIFFERENTIAL EQUATIONS

In this paper, we construct a uniform second-order difference scheme for a class of boundary value problems of fourth-order ordinary differential equations. Finally, a numerical example is given.

Finite Difference Method Applied in Two-Dimensional Heat Conduction Problem in the Permanent Regime in Rectangular Coordinates

The solution of many conduction heat transfer problems is found by two-dimensional simplification using the analytical method since different points have different initial temperatures. The temperature at each point of a given element can be analyzed through the Heat Equation that, in some cases, converges to analytical solutions without precision and is far from the real. However, with the application of the Finite Difference Method (FDM), it is possible to solve it numerically in a relatively fast way, providing satisfactory results for the most varied boundary conditions and diverse geometries, characteristics of heat transfer problems by conduction. This study solved two problems inside a plate with and without heat generation involved in temperature distribution. Algorithms were built with the aid of the Matlab programming language, and applied to obtain a numerical solution using the FDM numerical method. The computational and analytical solutions were then compared. Under certain conditions of the parameters involved in the phenomenon of each problem, the numerical method was very efficient for presenting errors less than or equal to 0.003 and 0.03, respectively, for cases without and with heat generation.

Solution of a One-Dimension Heat Equation Using Higher-Order Finite Difference Methods and Their Stability

One-dimensional heat equation was solved for different higher-order finite difference schemes, namely, forward time and fourth-order centered space explicit method, backward time and fourth-order centered space implicit method, and fourth-order implicit Crank-Nicolson finite difference method. Higher-order schemes have complexity in computing values at the neighboring points to the boundaries. It is required there a specification of the values of field variables at some points exterior to the domain. The complexity was incorporated using Hicks approximation. The convergence and stability analysis was also computed for those higher-order finite difference explicit and implicit methods in case of solving a one dimensional heat equation. The obtained numerical results were compared with exact solutions. It is found that backward time and fourth-order centered space implicit scheme along with Hicks approximation performed well over the other mentioned higher-order approaches.

An GPU accelerated finite difference method for heat transfer simulation

The heat transfer mathematic models are widely used in iron and steel industry area. Many computational models that represent this physical process is based on finite difference methods. The simulation of these phenomena demands a high computa- tional cost. In this paper we employ GPU for the development of algorithm for a two-dimensional heat transfer problem with f＇mite difference methods. The performance evaluation has been made and the comparison between CPU and GPU were discussed. The experimental result shows that GPU can solve this problem more efficiently when we need to divide calculation material into a large number of meshes.

Numerical Verification of Transition’s Energies of Excitons in Quantum Well of ZnO with the Finite Difference Method

This paper shows that the experimental results of quantum well energy transitions can be found numerically. The cases of several ZnO-ZnMgO wells are considered and their excitonic transition energies were calculated using the finite difference method. In that way, the one-dimensional Schrödinger equation has been solved by using the BLAS and LAPACK libraries. The numerical results are in good agreement with the experimental ones.