Convergence to Steady-State Solutions of the New Type of High-Order Multi-resolution WENO Schemes: a Numerical Study

A new type of high-order multi-resolution weighted essentially non-oscillatory(WENO)schemes(Zhu and Shu in J Comput Phys,375:659-683,2018)is applied to solve for steady-state problems on structured meshes.Since the classical WENO schemes(Jiang and Shu in J Comput Phys,126:202-228,1996)might suffer from slight post-shock oscillations(which are responsible for the residue to hang at a truncation error level),this new type of high-order finite-difference and finite-volume multi-resolution WENO schemes is applied to control the slight post-shock oscillations and push the residue to settle down to machine zero in steady-state simulations.This new type of multi-resolution WENO schemes uses the same large stencils as that of the same order classical WENO schemes,could obtain fifth-order,seventh-order,and ninth-order in smooth regions,and could gradually degrade to first-order so as to suppress spurious oscillations near strong discontinuities.The linear weights of such new multi-resolution WENO schemes can be any positive numbers on the condition that their sum is one.This is the first time that a series of unequal-sized hierarchical central spatial stencils are used in designing high-order finitedifference and finite-volume WENO schemes for solving steady-state problems.In comparison with the classical fifth-order finite-difference and finite-volume WENO schemes,the residue of these new high-order multi-resolution WENO schemes can converge to a tiny number close to machine zero for some benchmark steady-state problems.

Properties of High-Order Finite Difference Schemes and Idealized Numerical Testing

Construction of high-order difference schemes based on Taylor series expansion has long been a hot topic in computational mathematics, while its application in comprehensive weather models is still very rare. Here, the properties of high-order finite difference schemes are studied based on idealized numerical testing, for the purpose of their application in the Global/Regional Assimilation and Prediction System(GRAPES) model. It is found that the pros and cons due to grid staggering choices diminish with higher-order schemes based on linearized analysis of the one-dimensional gravity wave equation. The improvement of higher-order difference schemes is still obvious for the mesh with smooth varied grid distance. The results of discontinuous square wave testing also exhibits the superiority of high-order schemes. For a model grid with severe non-uniformity and non-orthogonality, the advantage of high-order difference schemes is inapparent, as shown by the results of two-dimensional idealized advection tests under a terrain-following coordinate. In addition, the increase in computational expense caused by high-order schemes can be avoided by the precondition technique used in the GRAPES model. In general, a high-order finite difference scheme is a preferable choice for the tropical regional GRAPES model with a quasi-uniform and quasi-orthogonal grid mesh.

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.

High-order targeted essentially non-oscillatory scheme for two-fluid plasma model

The weakly ionized plasma flows in aerospace are commonly simulated by the single-fluid model,which cannot describe certain nonequilibrium phenomena by finite collisions of particles,decreasing the fidelity of the solution.Based on an alternative formulation of the targeted essentially non-oscillatory(TENO)scheme,a novel high-order numerical scheme is proposed to simulate the two-fluid plasmas problems.The numerical flux is constructed by the TENO interpolation of the solution and its derivatives,instead of being reconstructed from the physical flux.The present scheme is used to solve the two sets of Euler equations coupled with Maxwell's equations.The numerical methods are verified by several classical plasma problems.The results show that compared with the original TENO scheme,the present scheme can suppress the non-physical oscillations and reduce the numerical dissipation.

High-Order Semi-Lagrangian WENO Schemes Based on Non-polynomial Space for the Vlasov Equation

In this paper,we present a semi-Lagrangian(SL)method based on a non-polynomial function space for solving the Vlasov equation.We fnd that a non-polynomial function based scheme is suitable to the specifcs of the target problems.To address issues that arise in phase space models of plasma problems,we develop a weighted essentially non-oscillatory(WENO)scheme using trigonometric polynomials.In particular,the non-polynomial WENO method is able to achieve improved accuracy near sharp gradients or discontinuities.Moreover,to obtain a high-order of accuracy in not only space but also time,it is proposed to apply a high-order splitting scheme in time.We aim to introduce the entire SL algorithm with high-order splitting in time and high-order WENO reconstruction in space to solve the Vlasov-Poisson system.Some numerical experiments are presented to demonstrate robustness of the proposed method in having a high-order of convergence and in capturing non-smooth solutions.A key observation is that the method can capture phase structure that require twice the resolution with a polynomial based method.In 6D,this would represent a signifcant savings.

Numerical Simulation of Bed Load and Suspended Load Sediment Transport Using Well-Balanced Numerical Schemes

Sediment transport can be modelled using hydrodynamic models based on shallow water equations coupled with the sediment concentration conservation equation and the bed con-servation equation.The complete system of equations is made up of the energy balance law and the Exner equations.The numerical solution for this complete system is done in a seg-regated manner.First,the hyperbolic part of the system of balance laws is solved using a finite volume scheme.Three ways to compute the numerical flux have been considered,the Q-scheme of van Leer,the HLLCS approximate Riemann solver,and the last one takes into account the presence of non-conservative products in the model.The discretisation of the source terms is carried out according to the numerical flux chosen.In the second stage,the bed conservation equation is solved by using the approximation computed for the system of balance laws.The numerical schemes have been validated making comparisons between the obtained numerical results and the experimental data for some physical experiments.The numerical results show a good agreement with the experimental data.

Numerical Simulation of Diffusion Type Traffic Flow Model Using Second-Order Lax-Wendroff Scheme Based on Exponential Velocity Density Function

In order to control traffic congestion, many mathematical models have been used for several decades. In this paper, we study diffusion-type traffic flow model based on exponential velocity density relation, which provides a non-linear second-order parabolic partial differential equation. The analytical solution of the diffusion-type traffic flow model is very complicated to approximate the initial density of the Cauchy problem as a function of x from given data and it may cause a huge error. For the complexity of the analytical solution, the numerical solution is performed by implementing an explicit upwind, explicitly centered, and second-order Lax-Wendroff scheme for the numerical solution. From the comparison of relative error among these three schemes, it is observed that Lax-Wendroff scheme gives less error than the explicit upwind and explicit centered difference scheme. The numerical, analytical analysis and comparative result discussion bring out the fact that the Lax-Wendroff scheme with exponential velocity-density relation of diffusion type traffic flow model is suitable for the congested area and shows a better fit in traffic-congested regions.

Numerical modeling of wave equation by a truncated high-order finite-difference method

Finite-difference methods with high-order accuracy have been utilized to improve the precision of numerical solution for partial differential equations. However, the computation cost generally increases linearly with increased order of accuracy. Upon examination of the finite-difference formulas for the first-order and second-order derivatives, and the staggered finite-difference formulas for the first-order derivative, we examine the variation of finite-difference coefficients with accuracy order and note that there exist some very small coefficients. With the order increasing, the number of these small coefficients increases, however, the values decrease sharply. An error analysis demonstrates that omitting these small coefficients not only maintain approximately the same level of accuracy of finite difference but also reduce computational cost significantly. Moreover, it is easier to truncate for the high-order finite-difference formulas than for the pseudospectral for- mulas. Thus this study proposes a truncated high-order finite-difference method, and then demonstrates the efficiency and applicability of the method with some numerical examples.

A High-Order Conservative Numerical Method for Gross-Pitaevskii Equation with Time-Varying Coefficients in Modeling BEC

We propose a high-order conservative method for the nonlinear Sehodinger/Gross-Pitaevskii equation with time- varying coefficients in modeling Bose Einstein condensation （BEC）. This scheme combined with the sixth-order compact finite difference method and the fourth-order average vector field method, finely describes the condensate wave function and physical characteristics in some small potential wells. Numerical experiments are presented to demonstrate that our numerical scheme is efficient by the comparison with the Fourier pseudo-spectral method. Moreover, it preserves several conservation laws well and even exactly under some specific conditions.

A High-Order Scheme for Fractional Ordinary Differential Equations with the Caputo-Fabrizio Derivative

In this paper, we consider numerical solutions of fractional ordinary diferential equations with the Caputo-Fabrizio derivative, and construct and analyze a high-order time-stepping scheme for this equation. The proposed method makes use of quadratic interpolation function in sub-intervals, which allows to produce fourth-order convergence. A rigorous stability and convergence analysis of the proposed scheme is given. A series of numerical examples are presented to validate the theoretical claims. Traditionally a scheme having fourth-order convergence could only be obtained by using block-by-block technique. The advantage of our scheme is that the solution can be obtained step by step, which is cheaper than a block-by-block-based approach.

A high-order splitting scheme for the advection diffusion equation

A high-order splitting scheme for the advection-diffusion equation of pollutants is proposed in this paper. The multidimensional advection-diffusion equation is splitted into several one-dimensional equations that are solved by the scheme. Only three spatial grid points are needed in each direction and the scheme has fourth-order spatial accuracy. Several typically pure advection and advection-diffusion problems are simulated. Numerical results show that the accuracy of the scheme is much higher than that of the classical schemes and the scheme can he efficiently solved with little programming effort.

An Application of Finite Volume WENO Schemeto Numerical Modeling of Tidal Current

A depth-averaged 2-D numerical model for unsteady tidal flow in estuaries is established by use of the finite volume WENO scheme which maintains both uniform high order accuracy and an essentially non-oscillatory shock transition on unstructured triangular grid. The third order TVD Range-Kutta method is used for time discretization. The model has been firstly tested against four cases： 1） tidal forcing, 2） seiche oscillation, 3） wind setup in a closed bay, and 4） onedimensional dam-break water flow. The results obtained in the present study compare well with those obtained from the corresponding analytic solutions idealized for the above four cases. The model is then applied to the simulation of tidal circulation in the Yangpu Bay, and detailed model calibration and verification have been conducted with measured tidal current in the spring tide, middle tide, and neap tide. The overall performance of the model is in qualitative agreement with the data observed in 2005, and it can be used to calculate the flow in estuaries and coastal waters.

The Sensitivity of Numerical Simulation of the East Asian Monsoon to Different Cumulus Parameterization Schemes

In this paper, a 5-level spectral AGCM is used to examine the sensitivity of simulated East Asian summer monsoon circulation and rainfall to cumulus parameterization schemes. From the simulated results of East Asian monsoon circulations and rainfalls during the summers of 1987 and 1995, it is shown that the Kuo′s convective parameterization scheme is more suitable for the numerical simulation of East Asian summer monsoon rainfall and circulation. This may be due to that the cumulus in the rainfall system is not strong in the East Asian monsoon region.

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.

A High-Order Spatiotemporal Precision-Matching Taylor–Li Scheme for Time-Dependent Problems

Based on the Taylor series method and Li’s spatial differential method, a high-order hybrid Taylor–Li scheme is proposed.The results of a linear advection equation indicate that, using the initial values of the square-wave type, a result with thirdorder accuracy occurs. However, using initial values associated with the Gaussian function type, a result with very high precision appears. The study demonstrates that, when the order of the time integral is more than three, the corresponding optimal spatial difference order could be higher than six. The results indicate that the reason for why there is no improvement related to an order of spatial difference above six is the use of a time integral scheme that is not high enough. The author also proposes a recursive differential method to improve the Taylor–Li scheme’s computation speed. A more rapid and highprecision program than direct computation of the high-order space differential item is employed, and the computation speed is dramatically boosted. Based on a multiple-precision library, the ultrahigh-order Taylor–Li scheme can be used to solve the advection equation and Burgers’ equation.

THREE HIGH-ORDER SPLITTING SCHEMES FOR 3D TRANSPORT EQUATION

Two high-order splitting schemes based on the idea of the operators splitting method are given. The three-dimensional advection-diffusion equation was split into several one-dimensional equations that were solved by these two schemes, only three computational grid points were needed in each direction but the accuracy reaches the spatial fourth-order. The third scheme proposed is based on the classical ADI scheme and the accuracy of the advection term of it can reach the spatial fourth-order. Finally, two typical numerical experiments show that the solutions of these three schemes compare well with that given by the analytical solution when the Peclet number is not bigger than 5.

NUMERICAL SCHEMES WITH HIGH ORDER OF ACCURACY FOR THE COMPUTATION OF SHOCK WAVES

High order accurate scheme is highly desirable for Slow computation with shocks. After analysis has been made for the reason of the generation of non-physical oscillations around the shock in numerical computations, a third-order, upwind biased, shock capturing scheme was proposed. Also, a new shock fitting method, called pseudo shock fitting method, was suggested, which in principle can be with any order of accuracy. Test cases for one dimensional flows show that the new method is very satisfactory.

High-order schemes for predicting computational aeroacoustic propagation with adaptive mesh refinement

High-order schemes based on block-structured adaptive mesh refinement method are prepared to solve computational aeroacoustic （CAA） problems with an aim at improving computational efficiency. A number of numerical issues associated with high-order schemes on an adaptively refined mesh, such as stability and accuracy are addressed. Several CAA benchmark problems are used to demonstrate the feasibility and efficiency of the approach.

A MIXED FINITE ELEMENT AND CHARACTERISTIC MIXED FINITE ELEMENT FOR INCOMPRESSIBLE MISCIBLE DARCY-FORCHHEIMER DISPLACEMENT AND NUMERICAL ANALYSIS

In this paper a mixed finite element-characteristic mixed finite element method is discussed to simulate an incompressible miscible Darcy-Forchheimer problem.The flow equation is solved by a mixed finite element and the approximation accuracy of Darch-Forchheimer velocity is improved one order.The concentration equation is solved by the method of mixed finite element,where the convection is discretized along the characteristic direction and the diffusion is discretized by the zero-order mixed finite element method.The characteristics can confirm strong stability at sharp fronts and avoids numerical dispersion and nonphysical oscillation.In actual computations the characteristics adopts a large time step without any loss of accuracy.The scalar unknowns and its adjoint vector function are obtained simultaneously and the law of mass conservation holds in every element by the zero-order mixed finite element discretization of diffusion flux.In order to derive the optimal 3/2-order error estimate in L^(2) norm,a post-processing technique is included in the approximation to the scalar unknowns.Numerical experiments are illustrated finally to validate theoretical analysis and efficiency.This method can be used to solve such an important problem.

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.