Gas-kinetic numerical schemes for one- and two-dimensional inner flows

Several kinds of explicit and implicit finite-difference schemes directly solving the discretized velocity distribution functions are designed with precision of different orders by analyzing the inner characteristics of the gas-kinetic numerical algorithm for Boltzmann model equation. The peculiar flow phenomena and mechanism from various flow regimes are revealed in the numerical simulations of the unsteady Sod shock-tube problems and the two-dimensional channel flows with different Knudsen numbers. The numerical remainder-effects of the difference schemes are investigated aad analyzed based on the computed results. The ways of improving the computational efficiency of the gaskinetic numerical method and the computing principles of difference discretization are discussed.

Lagrangian model of zooplankton dispersion:numerical schemes comparisons and parameter sensitivity tests

This paper presents two comparisons or tests for a Lagrangian model of zooplankton dispersion:numerical schemes and time steps.Firstly,we compared three numerical schemes using idealized circulations.Results show that the precisions of the advanced Adams-Bashfold-Moulton(ABM) method and the Runge-Kutta(RK) method were in the same order and both were much higher than that of the Euler method.Furthermore,the advanced ABM method is more efficient than the RK method in computational memory requirements and time consumption.We therefore chose the advanced ABM method as the Lagrangian particle-tracking algorithm.Secondly,we performed a sensitivity test for time steps,using outputs of the hydrodynamic model,Symphonie.Results show that the time step choices depend on the fluid response time that is related to the spatial resolution of velocity fields.The method introduced by Oliveira et al.in 2002 is suitable for choosing time steps of Lagrangian particle-tracking models,at least when only considering advection.

Nonstandard Numerical Schemes for a Linear Stochastic Oscillator with Additive Noise

In order to simulate a linear stochastic oscillator with additive noise,improved nonstandard optimal(INSOPT) schemes are derived utilizing the nonstandard finite difference(NSFD)technique and the improvement technique.These proposed schemes reproduce long time features of the oscillator solution exactly.Their abilities in preserving the symplecticity,the linear growth property of the second moment and the oscillation property of the solution of the stochastic oscillator system on long time interval are studied.It can be shown that the component { x_n}_(n≥1) of the INSOPT schemes switch signs infinitely many times as n →∞,almost surely.Further,the mean-square convergence order of 1 is obtained for these INSOPT schemes.Finally,numerical experiments illustrate intuitively the results obtained in this paper.

Numerical Simulation of Modified Kortweg-de Vries Equation by Linearized Implicit Schemes

In this paper, we are going to derive four numerical methods for solving the Modified Kortweg-de Vries (MKdV) equation using fourth Pade approximation for space direction and Crank Nicolson in the time direction. Two nonlinear schemes and two linearized schemes are presented. All resulting schemes will be analyzed for accuracy and stability. The exact solution and the conserved quantities are used to highlight the efficiency and the robustness of the proposed schemes. Interaction of two and three solitons will be also conducted. The numerical results show that the interaction behavior is elastic and the conserved quantities are conserved exactly, and this is a good indication of the reliability of the schemes which we derived. A comparison with some existing is presented as well.

A numerical study for some efficient time-integration schemes using barotropic equations

Three kinds of methods, i. e., explicit, semi-implicit, and semi-implicit and semi-Lagrangian method, are tested in the time-integration of shallow-water equations on rotating sphere. Helpful results are available from experiments, especially about the accuracy and efficiency of different semi-implicit and semi-Lagrangian schemes.

A Hybrid SIR-Fuzzy Model for Epidemic Dynamics:A Numerical Study

This study focuses on the urgent requirement for improved accuracy in diseasemodeling by introducing a newcomputational framework called the Hybrid SIR-Fuzzy Model.By integrating the traditional Susceptible-Infectious-Recovered(SIR)modelwith fuzzy logic,ourmethod effectively addresses the complex nature of epidemic dynamics by accurately accounting for uncertainties and imprecisions in both data and model parameters.The main aim of this research is to provide a model for disease transmission using fuzzy theory,which can successfully address uncertainty in mathematical modeling.Our main emphasis is on the imprecise transmission rate parameter,utilizing a three-part description of its membership level.This enhances the representation of disease processes with greater complexity and tackles the difficulties related to quantifying uncertainty in mathematical models.We investigate equilibrium points for three separate scenarios and perform a comprehensive sensitivity analysis,providing insight into the complex correlation betweenmodel parameters and epidemic results.In order to facilitate a quantitative analysis of the fuzzy model,we propose the implementation of a resilient numerical scheme.The convergence study of the scheme demonstrates its trustworthiness,providing a conditionally positive solution,which represents a significant improvement compared to current forward Euler schemes.The numerical findings demonstrate themodel’s effectiveness in accurately representing the dynamics of disease transmission.Significantly,when the mortality coefficient rises,both the susceptible and infected populations decrease,highlighting the model’s sensitivity to important epidemiological factors.Moreover,there is a direct relationship between higher Holling type rate values and a decrease in the number of individuals who are infected,as well as an increase in the number of susceptible individuals.This correlation offers a significant understanding of how many elements affect the consequences of an epidemic.Our objective is to enhance decision-making in public health by providing a thorough quantitative analysis of the Hybrid SIR-Fuzzy Model.Our approach not only tackles the existing constraints in disease modeling,but also paves the way for additional investigation,providing a vital instrument for researchers and policymakers alike.

SEVERAL NEW TYPES OF FINITE-DIFFERENCE SCHEMES FOR SHALLOW-WATER EQUATION WITH INITIAL-BOUNDARY VALUE AND THEIR NUMERICAL EXPERIMENT

This paper proposed several new types of finite-difference methods for the shallow water equation in absolute coordinate system and put forward an effective two-step predictor-corrector method, a compact and iterative algorithm for five diagonal matrix. Then the iterative method was used for a multi-grid procedure for shallow water equation. A t last, an initial-boundary value problem was considered, and the numerical results show that the linear sinusoidal wave would successively evolve into conoidal wave.

Von Neumann Stability Analysis of DG-Like and PNPM-Like Schemes for PDEs with Globally Curl-Preserving Evolution of Vector Fields

This paper examines a class of involution-constrained PDEs where some part of the PDE system evolves a vector field whose curl remains zero or grows in proportion to specified source terms.Such PDEs are referred to as curl-free or curl-preserving,respectively.They arise very frequently in equations for hyperelasticity and compressible multiphase flow,in certain formulations of general relativity and in the numerical solution of Schrödinger’s equation.Experience has shown that if nothing special is done to account for the curl-preserving vector field,it can blow up in a finite amount of simulation time.In this paper,we catalogue a class of DG-like schemes for such PDEs.To retain the globally curl-free or curl-preserving constraints,the components of the vector field,as well as their higher moments,must be collocated at the edges of the mesh.They are updated using potentials collocated at the vertices of the mesh.The resulting schemes:(i)do not blow up even after very long integration times,(ii)do not need any special cleaning treatment,(iii)can oper-ate with large explicit timesteps,(iv)do not require the solution of an elliptic system and(v)can be extended to higher orders using DG-like methods.The methods rely on a spe-cial curl-preserving reconstruction and they also rely on multidimensional upwinding.The Galerkin projection,highly crucial to the design of a DG method,is now conducted at the edges of the mesh and yields a weak form update that uses potentials obtained at the verti-ces of the mesh with the help of a multidimensional Riemann solver.A von Neumann sta-bility analysis of the curl-preserving methods is conducted and the limiting CFL numbers of this entire family of methods are catalogued in this work.The stability analysis confirms that with the increasing order of accuracy,our novel curl-free methods have superlative phase accuracy while substantially reducing dissipation.We also show that PNPM-like methods,which only evolve the lower moments while reconstructing the higher moments,retain much of the excellent wave propagation characteristics of the DG-like methods while offering a much larger CFL number and lower computational complexity.The quadratic energy preservation of these methods is also shown to be excellent,especially at higher orders.The methods are also shown to be curl-preserving over long integration times.

Higher-Order Numerical Solution of Two-Dimensional Coupled Burgers’ Equations

We proposed a higher-order accurate explicit finite-difference scheme for solving the two-dimensional heat equation. It has a fourth-order approximation in the space variables, and a second-order approximation in the time variable. As an application, we developed the proposed numerical scheme for solving a numerical solution of the two-dimensional coupled Burgers’ equations. The main advantages of our scheme are higher accurate accuracy and facility to implement. The good accuracy of the proposed numerical scheme is tested by comparing the approximate numerical and the exact solutions for several two-dimensional coupled Burgers’ equations.

A New Second Order Numerical Scheme for Solving Forward Backward Stochastic Differential Equations with Jumps

In this paper, we propose a new second order numerical scheme for solving backward stochastic differential equations with jumps with the generator linearly depending on . And we theoretically prove that the convergence rates of them are of second order for solving and of first order for solving and in norm.

Comparative Thermal Performance in SiO_(2)–H_(2)O and (MoS_(2)–SiO_(2))–H_(2)O Over a Curved Stretching Semi-Infinite Region: A Numerical Investigation

The investigation of Thermal performance in nanofluids and hybrid nanofluids over a curved stretching infinite region strengthens its roots in engineering and industry.Therefore,the comparative thermal analysis in SiO_(2)–H_(2)O and(MoS_(2)–SiO_(2))–H_(2)O is conducted over curved stretching surface.The model is reduced in the dimensional version via similarity transformation and then treated numerically.The velocity and thermal behavior for both the fluids is decorated against the preeminent parameters.From the analysis,it is examined that the motion of under consideration fluids declines against Fr and
.The thermal performance enhances for higher volumetric fraction and
.Further,it is noticed that thermal performance prevailed in(MoS_(2)–SiO_(2))–H_(2)O throughout the analysis.Therefore,(MoS_(2)–SiO_(2))–H_(2)O is better for industrial and engineering uses where high heat transfer is required to accomplished different processes of production.

Incompressible Flow and Heat Transfer over a Plate: A Hybrid Integral Domain-Discretized Numerical Procedure

This work deals with incompressible two-dimensional viscous flow over a semi-infinite plate ac-cording to the approximations resulting from Prandtl boundary layer theory. The governing non-linear coupled partial differential equations describing laminar flow are converted to a self-simi- lar type third order ordinary differential equation known as the Falkner-Skan equation. For the purposes of a numerical solution, the Falkner-Skan equation is converted to a system of first order ordinary differential equations. These are numerically addressed by the conventional shooting and bisection methods coupled with the Runge-Kutta technique. However the accompanying energy equation lends itself to a hybrid numerical finite element-boundary integral application. An appropriate complementary differential equation as well as the Green second identity paves the way for the integral representation of the energy equation. This is followed by a finite element-type discretization of the problem domain. Based on the quality of the results obtained herein, a strong case is made for a hybrid numerical scheme as a useful approach for the numerical resolution of boundary layer flows and species transport. Thanks to the sparsity of the resulting coefficient matrix, the solution profiles not only agree with those of similar problems in literature but also are in consonance with the physics they represent.

NUMERICAL EXPERIMENTS ON THE IMPACTS OF SEA SPRAY ON TROPICAL CYCLONES

The latest version of sea spray flux parameterization scheme developed by Andreas is coupled with the PSU/NCAR model MM5 in this paper. A western Pacific tropical cyclone named Nabi in 2005 is simulated using this coupled air-sea spray modeling system to study the impacts of sea spray evaporation on the evolution of tropical cyclones. The results demonstrate that sea spray can lead to a significant increase of heat fluxes in the air-sea interface, especially the latent heat flux, the maximum of which can increase by up to about 35% - 80% The latent heat flux seems to be more important than the sensible heat flux for the evolution of tropical cyclones. Regardless of whether sea spray fluxes have been considered, the model can always simulate the track of Nabi well, which seems to indicate that sea spray has little impact on the movement of tropical cyclones. However, with sea spray fluxes taken into account in the model, the intensity of a simulated tropical cyclone can have significant increase. Due to the enhancement of water vapor and heat from the sea surface to the air caused by sea spray, the warm core structure is better-defined, the minimum sea level pressure decreases and the vertical speed is stronger around the eye in the experiments, which is propitious to the development and evolution of tropical cyclones.

Development of a Numerical Scheme

In this paper, we developed a new numerical scheme which aimed to solve some initial value problems of ordinary differential equations. The full breakdown of this new numerical scheme derivation is presented. While in our subsequent research, we shall fully examine the characteristics of the scheme such as consistency, convergence and stability. Also, the implementation of this new numerical scheme shall be worked-on and comparison shall also be made with some existing methods.

Efficient Finite Difference WENO Scheme for Hyperbolic Systems withNon-conservativeProducts

Higher order finite difference weighted essentially non-oscillatory(WENO)schemes have been constructed for conservation laws.For multidimensional problems,they offer a high order accuracy at a fraction of the cost of a finite volume WENO or DG scheme of the comparable accuracy.This makes them quite attractive for several science and engineering applications.But,to the best of our knowledge,such schemes have not been extended to non-linear hyperbolic systems with non-conservative products.In this paper,we perform such an extension which improves the domain of the applicability of such schemes.The extension is carried out by writing the scheme in fluctuation form.We use the HLLI Riemann solver of Dumbser and Balsara(J.Comput.Phys.304:275-319,2016)as a building block for carrying out this extension.Because of the use of an HLL building block,the resulting scheme has a proper supersonic limit.The use of anti-diffusive fluxes ensures that stationary discontinuities can be preserved by the scheme,thus expanding its domain of the applicability.Our new finite difference WENO formulation uses the same WENO reconstruction that was used in classical versions,making it very easy for users to transition over to the present formulation.For conservation laws,the new finite difference WENO is shown to perform as well as the classical version of finite difference WENO,with two major advantages:(i)It can capture jumps in stationary linearly degenerate wave families exactly.(i)It only requires the reconstruction to be applied once.Several examples from hyperbolic PDE systems with non-conservative products are shown which indicate that the scheme works and achieves its design order of the accuracy for smooth multidimensional flows.Stringent Riemann problems and several novel multidimensional problems that are drawn from compressible Baer-Nunziato multiphase flow,multiphase debris flow and twolayer shallow water equations are also shown to document the robustness of the method.For some test problems that require well-balancing we have even been able to apply the scheme without any modification and obtain good results.Many useful PDEs may have stiff relaxation source terms for which the finite difference formulation of WENO is shown to provide some genuine advantages.

Construction of a Computational Scheme for the Fuzzy HIV/AIDS Epidemic Model with a Nonlinear Saturated Incidence Rate

This work aimed to construct an epidemic model with fuzzy parameters.Since the classical epidemic model doesnot elaborate on the successful interaction of susceptible and infective people,the constructed fuzzy epidemicmodel discusses the more detailed versions of the interactions between infective and susceptible people.Thenext-generation matrix approach is employed to find the reproduction number of a deterministic model.Thesensitivity analysis and local stability analysis of the systemare also provided.For solving the fuzzy epidemic model,a numerical scheme is constructed which consists of three time levels.The numerical scheme has an advantage overthe existing forward Euler scheme for determining the conditions of getting the positive solution.The establishedscheme also has an advantage over existing non-standard finite difference methods in terms of order of accuracy.The stability of the scheme for the considered fuzzy model is also provided.From the plotted results,it can beobserved that susceptible people decay by rising interaction parameters.

Efficient WENO-Based Prolongation Strategies for Divergence-Preserving Vector Fields

Adaptive mesh refinement(AMR)is the art of solving PDEs on a mesh hierarchy with increasing mesh refinement at each level of the hierarchy.Accurate treatment on AMR hierarchies requires accurate prolongation of the solution from a coarse mesh to a newly defined finer mesh.For scalar variables,suitably high-order finite volume WENO methods can carry out such a prolongation.However,classes of PDEs,such as computational electrodynamics(CED)and magnetohydrodynamics(MHD),require that vector fields preserve a divergence constraint.The primal variables in such schemes consist of normal components of the vector field that are collocated at the faces of the mesh.As a result,the reconstruction and prolongation strategies for divergence constraint-preserving vector fields are necessarily more intricate.In this paper we present a fourth-order divergence constraint-preserving prolongation strategy that is analytically exact.Extension to higher orders using analytically exact methods is very challenging.To overcome that challenge,a novel WENO-like reconstruction strategy is invented that matches the moments of the vector field in the faces,where the vector field components are collocated.This approach is almost divergence constraint-preserving,therefore,we call it WENO-ADP.To make it exactly divergence constraint-preserving,a touch-up procedure is developed that is based on a constrained least squares(CLSQ)method for restoring the divergence constraint up to machine accuracy.With the touch-up,it is called WENO-ADPT.It is shown that refinement ratios of two and higher can be accommodated.An item of broader interest in this work is that we have also been able to invent very efficient finite volume WENO methods,where the coefficients are very easily obtained and the multidimensional smoothness indicators can be expressed as perfect squares.We demonstrate that the divergence constraint-preserving strategy works at several high orders for divergence-free vector fields as well as vector fields,where the divergence of the vector field has to match a charge density and its higher moments.We also show that our methods overcome the late time instability that has been known to plague adaptive computations in CED.

Curl Constraint-Preserving Reconstruction and the Guidance it Gives for Mimetic Scheme Design

Several important PDE systems,like magnetohydrodynamics and computational electrodynamics,are known to support involutions where the divergence of a vector field evolves in divergence-free or divergence constraint-preserving fashion.Recently,new classes of PDE systems have emerged for hyperelasticity,compressible multiphase flows,so-called firstorder reductions of the Einstein field equations,or a novel first-order hyperbolic reformulation of Schrödinger’s equation,to name a few,where the involution in the PDE supports curl-free or curl constraint-preserving evolution of a vector field.We study the problem of curl constraint-preserving reconstruction as it pertains to the design of mimetic finite volume(FV)WENO-like schemes for PDEs that support a curl-preserving involution.(Some insights into discontinuous Galerkin(DG)schemes are also drawn,though that is not the prime focus of this paper.)This is done for two-and three-dimensional structured mesh problems where we deliver closed form expressions for the reconstruction.The importance of multidimensional Riemann solvers in facilitating the design of such schemes is also documented.In two dimensions,a von Neumann analysis of structure-preserving WENOlike schemes that mimetically satisfy the curl constraints,is also presented.It shows the tremendous value of higher order WENO-like schemes in minimizing dissipation and dispersion for this class of problems.Numerical results are also presented to show that the edge-centered curl-preserving(ECCP)schemes meet their design accuracy.This paper is the first paper that invents non-linearly hybridized curl-preserving reconstruction and integrates it with higher order Godunov philosophy.By its very design,this paper is,therefore,intended to be forward-looking and to set the stage for future work on curl involution-constrained PDEs.

Numerical solutions of nonlinear fractional Wu-Zhang system for water surface versus three approximate schemes

This paper examines the effects of three distinct numerical schemes(Adomian Decomposition,quintic&septic Spline methods)to investigate semi-analytical and approximate solutions on Wu-Zhang(ZW)system.It describes the(1+1)-dimensional dispersive long wave in two horizontal directions on shallow waters.The ZW model is one of the fractional nonlinear partial differential equations.Conformable derivatives properties are employed to convert the nonlinear fractional partial differential equation into an ordinary differential equation with integer order so as to obtain the approximate solutions for this model.The solutions obtained for each technique were compared to reveal their relationship to their characteristics illustrated under the suitable choice of the parameters values.The obtained solutions showed the power,easiness,and effectiveness of these methods on nonlinear partial differential equations.

Al_(2)O_(3) andγAl_(2)O_(3) Nanomaterials Based Nanofluid Models with Surface Diffusion:Applications for Thermal Performance in Multiple Engineering Systems and Industries

Thermal transport investigation in colloidal suspensions is taking a significant research direction.The applications of these fluids are found in various industries,engineering,aerodynamics,mechanical engineering and medical sciences etc.A huge amount of thermal transport is essential in the operation of various industrial production processes.It is a fact that conventional liquids have lower thermal transport characteristics as compared to colloidal suspensions.The colloidal suspensions have high thermal performance due to the thermophysical attributes of the nanoparticles and the host liquid.Therefore,researchers focused on the analysis of the heat transport in nanofluids under diverse circumstances.As such,the colloidal analysis of H_(2)O composed byγAl_(2)O_(3)and Al_(2)O_(3)is conducted over an elastic cylinder.The governing flow models ofγAl_(2)O_(3)/H_(2)O and Al_(2)O_(3)/H_(2)O is reduced in the dimensionless form by adopting the described similarity transforms.The colloidal models are handled by implementing the suitable numerical technique and provided the results for the velocity,temperature and local thermal performance rate against the multiple flow parameters.From the presented results,it is shown that the velocity of Al_(2)O_(3)–H_(2)O increases promptly against a high Reynolds number and it decreases for high-volume fraction.The significant contribution of the volumetric fraction is examined for thermal enhancement of nanofluids.The temperature of Al_(2)O_(3)–H_(2)O andγAl_(2)O_(3)–H_(2)O significantly increases against a higherϕ.Most importantly,the analysis shows thatγAl_(2)O_(3)–H_(2)O has a high local thermal performance rate compared to Al_(2)O_(3)–H_(2)O.Therefore,it is concluded thatγAl_(2)O_(3)–H_(2)O is a better heat transfer fluid and is suitable for industrial and technological uses.