High-order Difference Scheme Research Articles

Application of a Multi-dimensional Limiting Process to Central-Upwind Schemes for Solving Hyperbolic Systems of Conservation Laws

In this paper, we study semi-discrete central-upwind difference schemes with a modified multi-dimensional limiting process (MLP) to solve two-dimensional hyperbolic systems of conservation laws. In general, high-order central difference schemes for conservation laws involve no Riemann solvers or characteristic decompositions but have a tendency to smear linear discontinuities. To overcome this drawback of central-upwind schemes, we use a MLP that uses multi-dimensional information for slope limitation to control the oscillations across discontinuities for multi-dimensional applications. Some numerical results are provided to demonstrate the performance of the proposed scheme.

Some high order difference schemes for the space and time fractional Bloch–Torrey equations

In this paper, several difference schemes are proposed for both one-dimensional and two-dimensional space and time fractional Bloch–Torrey equations. The spatial second-order scheme and the spatial fourth-order compact scheme are established, respectively. The obtained schemes can achieve the global second-order numerical accuracy in time. The unique solvability, unconditional stability and convergence of the proposed schemes are proved by the energy method. Two ADI schemes are also discussed for the two dimensional problem. Numerical examples are given to verify the numerical accuracy and efficiency of the difference schemes.

Research on Aerodynamic Noise Reduction for High-Speed Trains

A broadband noise source model based on Lighthill’s acoustic theory was used to perform numerical simulations of the aerodynamic noise sources for a high-speed train. The near-field unsteady flow around a high-speed train was analysed based on a delayed detached-eddy simulation (DDES) using the finite volume method with high-order difference schemes. The far-field aerodynamic noise from a high-speed train was predicted using a computational fluid dynamics (CFD)/Ffowcs Williams-Hawkings (FW-H) acoustic analogy. An analysis of noise reduction methods based on the main noise sources was performed. An aerodynamic noise model for a full-scale high-speed train, including three coaches with six bogies, two inter-coach spacings, two windscreen wipers, and two pantographs, was established. Several low-noise design improvements for the high-speed train were identified, based primarily on the main noise sources; these improvements included the choice of the knuckle-downstream or knuckle-upstream pantograph orientation as well as different pantograph fairing structures, pantograph fairing installation positions, pantograph lifting configurations, inter-coach spacings, and bogie skirt boards. Based on the analysis, we designed a low-noise structure for a full-scale high-speed train with an average sound pressure level (SPL) 3.2 dB(A) lower than that of the original train. Thus, the noise reduction design goal was achieved. In addition, the accuracy of the aerodynamic noise calculation method was demonstrated via experimental wind tunnel tests.

A high-order difference scheme for the fractional sub-diffusion equation

ABSTRACTBased on the Lubich's high-order operators, a second-order temporal finite-difference method is considered for the fractional sub-diffusion equation. It has been proved that the finite-difference scheme is unconditionally stable and convergent in norm by the energy method in both one- and two-dimensional cases. The rate of convergence is order of two in temporal direction under the initial value satisfying some suitable conditions. Some numerical examples are given to confirm the theoretical results.

High order schemes for the tempered fractional diffusion equations

L\'{e}vy flight models whose jumps have infinite moments are mathematically used to describe the superdiffusion in complex systems. Exponentially tempering the Levy measure of L\'{e}vy flights leads to the tempered stable L\'{e}vy processes which combine both the $\alpha$-stable and Gaussian trends; and the very large jumps are unlikely and all their moments exist. The probability density functions of the tempered stable L\'{e}vy processes solve the tempered fractional diffusion equation. This paper focuses on designing the high order difference schemes for the tempered fractional diffusion equation on bounded domain. The high order difference approximations, called the tempered and weighted and shifted Gr\"{u}nwald difference (tempered-WSGD) operators, in space are obtained by using the properties of the tempered fractional calculus and weighting and shifting their first order Gr\"{u}nwald type difference approximations. And the Crank-Nicolson discretization is used in the time direction. The stability and convergence of the presented numerical schemes are established; and the numerical experiments are performed to confirm the theoretical results and testify the effectiveness of the schemes.

Solitary wave solution of nonlinear Benjamin–Bona–Mahony–Burgers equation using a high-order difference scheme

In this work, we investigate the solitary wave solution of nonlinear Benjamin–Bona–Mahony–Burgers (BBMB) equation using a high-order linear finite difference scheme. We prove that this scheme is stable and convergent with the order of \(O(\tau ^2+h^4)\). Furthermore, we discuss the existence and uniqueness of numerical solutions. Numerical results obtained from propagation of a single solitary and interaction of two and three solitary waves confirm the efficiency and high accuracy of proposed method.

Use of Weno Schemes for Simulation of the Reflected Shock Wave–Boundary Layer Interaction

Numerical simulation of the shock wave–laminar boundary layer interaction is carried out on the basis of the model problem at different Reynolds numbers. The use of WENO schemes of high order of accuracy is demonstrated. The calculated shock-wave structure of the fl ow is compared with the data available in the literature and with the data obtained from TVD schemes. Criteria of accuracy of numerical calculations that are related to the position of shock-wave structures, and also the time of solution of the problem with various difference schemes, are discussed. Recommendations on practical use of difference schemes of high order of accuracy are given.

Some high-order difference schemes for the distributed-order differential equations

Two difference schemes are derived for both one-dimensional and two-dimensional distributed-order differential equations. It is proved that the schemes are unconditionally stable and convergent in an L1(L∞) norm with the convergence orders O(τ2+h2+Δα2) and O(τ2+h4+Δα4), respectively, where τ, h and Δα are the step sizes in time, space and distributed-order variables. Several numerical examples are given to confirm the theoretical results.

Two high-order numerical algorithms for solving the multi-term time fractional diffusion-wave equations

In this paper we apply a high order difference scheme and Galerkin spectral technique for the numerical solution of multi-term time fractional partial differential equations. The proposed methods are based on a finite difference scheme in time. The time fractional derivatives which have been described in Caputo’s sense are approximated by a scheme of order O(τ3−α),1<α<2 and the space derivative is discretized with a fourth-order compact finite difference procedure and Galerkin spectral method. We prove the unconditional stability of the compact procedure by coefficient matrix property. The L∞-convergence of the compact finite difference method has been proved by the energy method. Also we obtain an error estimate for Galerkin spectral method. Numerical results are provided to verify the accuracy and efficiency of the proposed schemes.

Analysis of the dengue risk by means of a Takagi–Sugeno-style model

Dengue is an infectious disease caused by an arbovirus (transmitted by the bite of the Aedes aegypti mosquito), which occurs mainly in tropical and subtropical areas, including Brazil. Nowadays it constitutes a major public health problems in the world. This paper proposes analysis of the evolution of dengue risk, from a model of the kind Takagi–Sugeno where the consequence of each fuzzy rule is a partial differential equation – PDE. The uncertainty parameters κi, related to the risk of spread of dengue, are directly linked to the behavior of the mosquito population and they were evaluated by making use of a system based on fuzzy rules and information provided by experts. These parameters depend on the population of humans, which provides blood to the maturation of eggs, and potential mosquito breeding containers and still depend on the rainfall. The amount of rainfall presents stochastic dependence in the sampled values, and for this reason we chose the Markov chain method to model it. Researchers at the Laboratory for Spatial Analysis of Epidemiological Data (epiGeo – UNICAMP) performed studies on the dengue risk in the southern region of the city of Campinas (São Paulo state, Brazil) between 2006 and 2007, by means of statistical methods and provided information to begin this study. The numerical solution of differential equation resulting from Takagi–Sugeno inference was obtained using the weighted essentially non-oscillatory schemes, the fifth order (WENO-5) not smooth for regions of the domain and centered finite difference schemes of high order for the smooth regions, in the space discretization. Also a lifting scheme was made to define smoothness in the regions. For the time evolution we have chosen Runge–Kutta TVD (Total Variation Diminishing), of the third order. Simulations corresponding to a period of summer (December, January and February) were performed and the results have showed that an effective action to reduce the risk of dengue is to severely combat the potential mosquito breeding sites.

Interference of two Bose–Einstein condensates with varying initial conditions

In this work, we study the interference effect of two independent Bose–Einstein condensates by numerical technique. Symplectic algorithm with high order difference scheme is used to solve the one-dimensional (1D) Gross–Pitaevskii (GP) equation. It is shown that the property of the interference between two condensates resembles the double-slit interference, and the fringe is dependent on the relative phase and the inter-spacing of the initial clouds. Interference of condensate consisting of different number of atoms is also presented, and the fringe is shown to be unsymmetrical. The periodic evolution of the two condensates confined in a harmonic trapping potential is studied and a much stronger interference pattern is shown.

Nonuniform-time-step explicit Runge–Kutta scheme for high-order finite difference method

Explicit Runge–Kutta method has been widely used for time-accurate simulations in aeroacoustics and aerodynamics, partly because of its strong stability properties. However, when dealing with problems involving irregular geometries or multi-scale phenomena where the grid is often refined in localized regions, a single global time step size dictated by the smallest grid size based on the explicit stability condition would inevitably result in excessive computational consumption. A local time stepping strategy that adopts time step size to the local stability requirement in different grid blocks is an effective way to reduce the cost of time integration and to increase the overall computational efficiency. In the present paper, a non-uniform time step (NUTS) explicit Runge–Kutta scheme, previously introduced in the framework of discontinuous Galerkin method, which makes the local time stepping strategy applicable to the explicit Runge–Kutta family without any interpolation and extrapolation in time, is further extended to high-order finite difference methods. The stability of non-uniform time step (NUTS) scheme in combination with high order difference schemes is studied. Numerical experiments are carried out using 1D and 2D acoustic problems for validation of the proposed approach. Computational cost reduction by the proposed algorithm is also discussed.

High Order Difference Schemes for a Time Fractional Differential Equation with Neumann Boundary Conditions

AbstractA compact finite difference scheme is derived for a time fractional differential equation subject to Neumann boundary conditions. The proposed scheme is second-order accurate in time and fourth-order accurate in space. In addition, a high order alternating direction implicit (ADI) scheme is also constructed for the two-dimensional case. The stability and convergence of the schemes are analysed using their matrix forms.

Boundary value problem for a first-order linear parabolic system

A boundary value problem for a linear parabolic system is considered. Sufficient conditions for the well-posedness of the problem are found. The spline collocation method on a uniform grid is used to construct a high-order accurate implicit difference scheme, and its absolute stability is proved.

On the practical accuracy of shock-capturing schemes

A method for estimating the accuracy of transmission by the shock-capturing difference scheme of the Rankine-Hugoniot conditions through the shock wave front is proposed. This method includes calculation of the order of integral convergence of the difference solution (instead of using its absolute value as in the L1 norm) on the intervals crossing the shock. In such integration, an error arising before the shock-wave front due to its smearing can be compensated by the similar error of the opposite sign after the wave front. Adduced examples show that this approach allows to obtain the second order of integral convergence on the intervals crossing the shock for some of the classical high-order difference schemes.

High order of accuracy difference schemes for the inverse elliptic problem with Dirichlet condition

The overdetermination problem for elliptic differential equation with Dirichlet boundary condition is considered. The third and fourth orders of accuracy stable difference schemes for the solution of this inverse problem are presented. Stability, almost coercive stability, and coercive inequalities for the solutions of difference problems are established. As a result of the application of established abstract theorems, we get well-posedness of high order difference schemes of the inverse problem for a multidimensional elliptic equation. The theoretical statements are supported by a numerical example. MSC:35N25, 39A14, 39A30, 65J22.

A fourth-order compact solution of the two-dimensional modified anomalous fractional sub-diffusion equation with a nonlinear source term

This work is concerned to the study of high order difference scheme for the solution of a two-dimensional modified anomalous sub-diffusion equation with a nonlinear source term which describes processes that become less anomalous as time progresses. The space fractional derivatives are described in the Riemann–Liouville sense. In the proposed scheme we discretize the space derivatives with a fourth-order compact scheme and use the Grünwald–Letnikov discretization of the Riemann–Liouville derivatives to obtain a fully discrete implicit scheme. We prove the stability and convergence of proposed scheme using the Fourier analysis. The convergence order of the proposed method is O(τ+hx4+hy4). Comparison of numerical results with analytical solutions demonstrates the unconditional stability and high accuracy of proposed scheme.

Gravitational radiation during coalescence of neutron stars

The coalescence of components of a binary star with equal masses (M1 = M2 = M⊙) and moving in circular orbits is considered. The equation of state for degenerate neutrons is used, leading to the equation of state for an ideal gas. The initial model has zero temperature, corresponding to a polytrope with n = 1.5. To reduce the required computational time, the initial close binary is constructed using the self-consistent field method. The computations use Newtonian gas dynamics, but the back reaction of the gravitational radiation is taken into account in a PN2.5 post-Newton approximation, obtained using ADM formalism. This makes it possible to apply previous experienceof constructing high-order Godunov-type difference schemes, which are suitable for end-to-end calculations of discontinuous solutions of the gas-dynamics equations on a fixed Eulerian grid. The Poisson equations were solved using an original spherical-function expansion method. The 3D computations yielded the parameters of the gravitational signal. Near the radiation maximum, the strain amplitude is rh ∼ 4 × 104 cm, the power maximum is 4 × 1054 erg/s, and the typical radiation frequency is ≳1 kHz. The energy carried away by gravitational waves is ≳1052 erg. These parameters are of interest, since they form an inherent part of a rotational mechanism for the supernova explosion. They are also of interest for the planning of gravitational-wave detection experiments.

Compact finite difference scheme and RBF meshless approach for solving 2D Rayleigh–Stokes problem for a heated generalized second grade fluid with fractional derivatives

This article is devoted to the comparison between a high order difference scheme and radial basis functions (RBFs) meshless approach for the numerical solution of two-dimensional fractional Rayleigh–Stokes problem. The fractional derivative of problem is described in the Riemann–Liouville sense. In the high order difference scheme we discretize the spaces derivatives with a fourth-order compact scheme and use the Grünwald–Letnikov discretization of the Riemann–Liouville derivatives to obtain a fully discrete implicit scheme. Also in the RBF meshless method we discretize the time fractional derivatives of the mentioned equation by integrating both sides of it then we will use the Kansa’s approach to approximate the spatial derivatives. We prove the stability and convergence for high order difference scheme using Fourier analysis and for time-discrete scheme in RBF method via energy approach. We compare the results of compact finite difference method and RBF meshless approach in terms of accuracy and CPU time. Also we compare the numerical results of compact difference scheme with other methods in the literature to show the high accuracy and efficiency of the proposed method.

Numerical methods integrated with fuzzy logic and stochastic method for solving PDEs: An application to dengue

Currently dengue epidemics are of great relevance in Brazil and other countries of tropical and subtropical climates, because it is a disease that infects a large number of people and in its most severe form can lead to death. In this work we proposed an integrated mathematical model (SIR-type: susceptible, infected, recovered) to study the evolution both in space and time of dengue disease. The model is given by partial differential equations (PDEs) whose numerical solutions are obtained by hybrid schemes, fuzzy logic and stochastic methods. We use the hybrid explicit numerical schemes WENO-5 (weighted essentially non-oscillatory schemes, fifth order) for regions not smooth of the map and centered finite difference schemes of high order for the regions smooth in space discretization. Also a lifting scheme was made to define smoothness or not in the regions. For the time evolution, we have chosen the third order Runge–Kutta TVD (Total Variation Diminishing). The uncertain parameters related to the behavior of Aedes aegypti are extremely important for development and/or disease control. In this way for incorporating this information into the model, the parameters were estimated using fuzzy rule-based systems and information provided by specialists. Such parameters depend on the people, who provide breeding sites and blood for the maturation of the female's eggs and they depend on rain events, too. This variable, rainfall, presents stochastic dependence on the sampled values and for this reason, we chose the Markov chain method (order 2). Information on the behavior of the disease and the conditions for the proliferation of vectors in the region south of the city of Campinas were researched in the Health Department, Agronomic Institute and with experts of the Medical Sciences Faculty of UNICAMP. Simulations of many situations were performed to obtain scenarios regarding the spread of the disease, considering the characteristics of the region studied.