Upwind Difference Research Articles

Higher Order Uniformly Convergent Numerical Algorithm for Time-Dependent Singularly Perturbed Differential-Difference Equations

This paper presents a higher order uniformly convergent discretization for second order singularly perturbed parabolic differential equation with delay and advance terms. The retarded terms are approximated by the Taylor series which give rise to a nearby singularly perturbed parabolic differential equation. A hybrid numerical algorithm based on implicit Euler scheme is proposed to discretize the time variable and a combined finite difference scheme made out of modified upwind and central difference schemes to discretize the spatial variable on a piecewise uniform mesh of Shishkin type in space and a uniform mesh in time. The existence and uniqueness of a solution for the proposed hybrid algorithm is analyzed. It is proved that the algorithm is $$\epsilon $$ -uniformly convergent of almost second order in space, $$O(N^{-2} (\ln N)^2)$$ and first order in time, $$O(M^{-1})$$ . A few numerical experiments supporting the results are presented. The efficiency of the proposed hybrid algorithm is demonstrated by comparing upwind and modified upwind algorithm.

High Resolution Compact Finite Difference Schemes for Convection Dominated Problems

In this short article, the upwind and central compact finite difference schemes for spatial discretization of the first-order derivative are analyzed. Comparison of the schemes is provided and the best discretization scheme for convection dominated problems is suggested.

A uniformly convergent numerical scheme for singularly perturbed differential equation with integral boundary condition arising in neural network

This article deals with a singularly perturbed quasilinear boundary value problem with integral boundary condition which arises in neural network. The problem is discretised by using an upwind finite difference scheme on a non-uniform mesh obtained via equidistribution of a monitor function. We prove that the method is first order convergent in the discrete maximum norm independent of perturbation parameter. The parameter uniform convergence is confirmed by numerical computations.

Linear combination of the Upwind and Standard Leapfrog difference schemes with weight coefficients obtained by minimizing the approximation error 

Linear combination of the Upwind and Standard Leapfrog difference schemes with weight coefficients obtained by minimizing the approximation error 

Numerical analysis of stability for a nonlinear size-structured population model with elastic growth

This article is concerned with an upwind difference scheme for a nonlinear size-dependent population model, into which an elastic growth is incorporated; that is, a decrease of size is possible. We make a through analysis of convergence for the numerical method, and use it as an auxiliary tool in the investigation of stability of population steady states. Two examples are presented.

Parameter uniform numerical method for a system of two coupled singularly perturbed parabolic convection-diffusion equations

In this paper, we propose a numerical scheme for a system of two linear singularly perturbed parabolic convection-diffusion equations. The presented numerical scheme consists of a classical backward-Euler scheme on a uniform mesh for the time discretization and an upwind finite difference scheme on an arbitrary nonuniform mesh for the spatial discretization. Then, for the time semidiscretization scheme, an a priori and an a posteriori error estimations in the maximum norm are obtained. It should be pointed out that the a posteriori error bound is suitable to design an adaptive algorithm, which is used to generate an adaptive spatial grid. It is proved that the method converges uniformly in the discrete maximum norm with first-order time and spatial accuracy, respectively, for the fully discrete scheme. At last, some numerical results are given to validate the theoretical results.

Analysis of internal flow field for a three‐stage centrifugal fan under various operating conditions

Centrifugal fan in series is the core facility for pneumatic transportation. Its operation is required to be adjusted real time as the flow rate of transported materials changes. For optimisation performance, internal flow of a type 700 three-stage fan at rated rotation speed is analysed with FLUENT6.3 for various total pressure. Multiple reference frame (MRF) model is used for the impellers rotation in simulation, and Re-Normalisation group (RNG) κ-ɛ turbulence model and Roe-FDS flux type with first-order upwind difference scheme are used for calculation of compressible flow under various total pressure. Flow characteristics are obtained and differences of total pressure and velocity distribution in every impeller are analysed in different conditions, and velocity distribution on meridian plane as well as section of wind guide plates and volute are compared. Results show that there are many vortexes and backflows in fan with higher total pressure, and the lower pressure, the more smoothly flows, besides flow at inlet and outlet of impellers and in guide plates is seriously influenced by total pressure. Curves of P-Q and P-η at 4600 r/min are predicted based on simulation, which is helpful reducing the impact caused by off-design operation and improving efficiency.

An efficient numerical method for singularly perturbed time dependent parabolic 2D convection–diffusion systems

In this paper we deal with solving efficiently 2D linear parabolic singularly perturbed systems of convection–diffusion type. We analyze only the case of a system of two equations where both of them feature the same diffusion parameter. Nevertheless, the method is easily extended to systems with an arbitrary number of equations which have the same diffusion coefficient. The fully discrete numerical method combines the upwind finite difference scheme, to discretize in space, and the fractional implicit Euler method, together with a splitting by directions and components of the reaction–convection–diffusion operator, to discretize in time. Then, if the spatial discretization is defined on an appropriate piecewise uniform Shishkin type mesh, the method is uniformly convergent and it is first order in time and almost first order in space. The use of a fractional step method in combination with the splitting technique to discretize in time, means that only tridiagonal linear systems must be solved at each time level of the discretization. Moreover, we study the order reduction phenomenon associated with the time dependent boundary conditions and we provide a simple way of avoiding it. Some numerical results, which corroborate the theoretical established properties of the method, are shown.

An efficient robust numerical method for singularly perturbed Burgers’ equation

In this article, we propose a parameter–uniformly convergent numerical method for viscous Burgers’ equation. In order to find a numerical approximation to Burgers’ equation, we linearize the equation to obtain sequence of linear PDEs. The linear PDEs are solved by a finite difference scheme, which comprises of the backward-difference scheme for the time derivative and upwind finite difference scheme for the spatial derivatives. Layer-adapted nonuniform meshes are invoked at each time level to exhibit layer nature of the solution. The nonuniform meshes are obtained by equidistribution of a positive integrable monitor function, which involves the derivative of the solution. It is shown that the methods converges uniformly with respect to the perturbation parameter. Numerical experiments are carried out to validate the ε-uniform error estimate of O(N−1+Δt).

Numerical modeling of lock-exchange gravity/turbidity currents by a high-order upwinding combined compact difference scheme

This study presents two-dimensional direct numerical simulations for sediment-laden current with higher density propagating forward through a lighter ambient water. The incompressible NavierStokes equations including the buoyancy force for the density difference between the light and heavy fluids are solved by a finite difference scheme based on a structured mesh. The concentration transport equations are used to explore such rich transport phenomena as gravity and turbidity currents. Within the framework of an Upwinding Combined Compact finite Difference (UCCD) scheme, rigorous determination of weighting coefficients underlies the modified equation analysis and the minimization of the numerical modified wavenumber. This sixth-order UCCD scheme is implemented in a four-point grid stencil to approximate advection and diffusion terms in the concentration transport equations and the first-order derivative terms in the Navier-Stokes equations, which can greatly enhance convective stability and increase dispersive accuracy at the same time. The initial discontinuous concentration field is smoothed by solving a newly proposed Heaviside function to prevent numerical instabilities and unreasonable concentration values. A two-step projection method is then applied to obtain the velocity field. The numerical algorithm shows a satisfying ability to capture the generation, development, and dissipation of the Kelvin-Helmholz instabilities and turbulent billows at the interface between the current and the ambient fluid. The simulation results also are compared with the data in published literatures and good agreements are found to prove that the present numerical model can well reproduce the propagation, particle deposition, and mixing processes of lock-exchange gravity and turbidity currents.

Topology optimization of conformal structures on manifolds using extended level set methods (X-LSM) and conformal geometry theory

In this paper, we propose a new method to systematically address the issue of structural shape and topology optimization on free-form surfaces. A free-form surface, also termed manifold, is conformally mapped onto a 2D rectangle domain where the level set function is defined. With the conformal mapping, the covariant derivatives on the manifold can be represented by the Euclidean gradient operators multiplied by a scalar. Keeping this intrinsic relation in mind, we derive the Euclidean form for the Riemannian Hamilton–Jacobi equation governing the boundary evolution on the manifold, which can be solved on a 2D plane using classical level set methods, such as the upwind finite difference or fast marching method. By reducing the dimension of the problem, the topology optimization problem on the manifold embedded in the 3D space can be recast as a 2D topology optimization problem in the Euclidean space. Compared with other approaches which need project the Euclidean differential operators to the manifold, the proposed method can not only reduce the computational cost but also preserve all the advantages of conventional level set methods. The proposed method reveals the fundamental relation between topology optimization on manifolds and Euclidean planes. It provides a unified level-set-based computational framework for the generative design of conformal structures with increasing applications in different fields of interests.

A two-phase model for simulation of MTR type research reactor during protected and unprotected LOFA

An optimized, accurate and fast model is investigated to simulate the transient behavior of a research reactor during fast and slow LOFA (loss of flow accident). Simulation is performed in two states when the reactor scram system is initiated at critical core power and when it doesn't work properly. The latter case leads to boiling condition and hence the reactor core will be encountered with two-phase flow condition. A lumped heat conduction model for fuel and clad accompanied with a two-phase flow model for coolant are coupled with point kinetic equations to simulate reactor behavior during the accident. A combination of Runge-Kutta method and up-wind finite difference technique are adopted for the solution to the set of differential equations. The calculation performed for IAEA 10 MW benchmark problem in HEU and LEU fuel. Some important parameters such as the peak temperatures of fuel and clad are obtained for both HEU and LEU conditions during transients. Results of numerical simulations, at the first, agree well with the general physical rules and also verified through comparison with the other computer code results. According to present simulation, under both protected and unprotected LOFA transient, the peak clad surface temperatures remain in safe mode.

An 𝜀-uniform method for a class of singularly perturbed parabolic problems with Robin boundary conditions having boundary turning point

A class of singularly perturbed parabolic differential equations with Robin type boundary condition having boundary turning point is propounded on a rectangular domain in the [Formula: see text]-[Formula: see text] plane. A numerical method comprising a standard upwind finite difference scheme is formulated on a rectangular piecewise uniform fitted mesh [Formula: see text] and it is proved to be [Formula: see text]-uniform. Furthermore, it is shown that the errors are bounded in the supremum norm by [Formula: see text], where [Formula: see text] is a constant independent of [Formula: see text], [Formula: see text] and the perturbation parameter [Formula: see text]. Numerical results are given to illustrate the analytical results.

Numerical Study of the Effects of Nanoparticles on Fuel Diffusion Combustion

The work investigates diffusion combustion of nanofuel made from the mixture of the base fuel and Al2O3 nanoparticles in a two-dimensional combustor with stationary horizontal walls. The numerical study was carried out by solving the problem governing equations including the continuity, momentum, energy and species concentration equations formulated for two-dimensional unsteady-state laminar convection using the finite volume method. The fully implicit scheme was used for the discretization in time; while upwind differencing and central differencing were used to discretize the convective and diffusive terms respectively. The results show that the values of the temperature and fuel concentration throughout the domain were higher during combustion of nanofuel than that of the base fuel only. Specifically, for flow streams at Re = 100, the addition of nanoparticles of volume fraction, φ = 0.05, enhanced the outlet average temperature of the combustor by 13% over that of ordinary fuel. Also, the wall temperature at steady state for nanofuel combustion is 7.4742 compared with 6.5897 for the base fuel. The high concentration of nanofuel is an indication of incomplete combustion which implies that more heat and higher temperature can be generated by increasing the flow stream of air.

Inlet flow disturbance effects on direct numerical simulation of incompressible round jet at Reynolds number 2500

This letter reports inlet flow disturbance effects on direct numerical simulation of incompressible round jet at Reynolds number 2500. The simulation employs an accurate projection method in which a sixth order biased upwind difference scheme is used for spatial discretization of nonlinear convective terms, with a fourth order central difference scheme used in the discretization of the divergence of intermediate velocity. Carefully identifying reveals that the inlet flow disturbance has some influences on the distribution pattern of mean factor of swirling strength intermittency. With the increase of inlet disturbance magnitude jet core cone slightly shortens, observable differences occur in the centerline velocity and its fluctuations, despite the negligible impacts on the least square fitted centerline velocity decay constant (Bu) and distribution parameter (Ku) for velocity profile in self-similar region.

Numerical approximations to the scaled first derivatives of the solution to a two parameter singularly perturbed problem

A singularly perturbed problem involving two singular perturbation parameters is discretized using the classical upwinded finite difference scheme on an appropriate piecewise-uniform Shishkin mesh. Scaled discrete derivatives (with scaling only used within the layers) are shown to be parameter uniformly convergent to the scaled first derivatives of the continuous solution.

Prediction of transition in temporal mixing layer using ILES

In this paper, we study the capability of implicit large eddy simulation (ILES) to capture transition. In particular, we study a planar temporal mixing layer subjected to low free stream perturbations. We simulated this problem by ILES and other approaches like Reynolds averaged Navier Stokes simulation (RANS), conventional large eddy simulation (LES) and direct numerical simulation (DNS). Qualitative and quantitative assessment of their relative performance reveals the advantage of ILES over other methods. We propose that any scheme, upwinding in this case, will be successful in simulating such flows if the discrete dispersion relationship of the linearized equation is similar to the theoretical dispersion relation. To verify our conjecture, we derived discretized dispersion relationship with upwinding and central difference scheme for a simple prototype of Navier Stokes equation namely complex Ginzburg Landau equation (CGLE). We found that the two relations and thus perturbation growth are significantly different from each other for, at least, convection dominated flows.

An extended continuum traffic model with the consideration of the optimal velocity difference

In this letter, we derived an extended continuum model from a car-following model with consideration of the optimal velocity difference called the relative optimal velocity via employing the transformation relation from microscopic variables to macroscopic ones. The stability condition of this continuum traffic model is obtained by performing linear stability analysis. Results show that the optimal velocity difference helps to improve the stability of traffic flow. We make use of the upwind finite difference scheme for simulation. The effects of the optimal velocity difference (the relative optimal velocity) and the velocity difference (the optimal velocity) on local clustering effect and instability are studied and compared each other, respectively. The spatiotemporal evolution patterns of traffic flow for the different initial density, strength of the optimal velocity difference and the velocity-difference are obtained. Their space–time diagrams reveal the local clustering effect induced by the instability of traffic flow and generate the stop&go traffic jams. Numerical simulation result indicates the unstable region is shrunken under the effect of the optimal velocity difference (the relative optimal velocity) and/or the velocity difference (the optimal velocity).

Pricing options on investment project expansions under commodity price uncertainty

<p style='text-indent:20px;'>In this work we develop PDE-based mathematical models for valuing real options on investment project expansions when the underlying commodity price follows a geometric Brownian motion. The models developed are of a similar form as the Black-Scholes model for pricing conventional European call options. However, unlike the Black-Scholes' model, the payoff conditions of the current models are determined by a PDE system. An upwind finite difference scheme is used for solving the models. Numerical experiments have been performed using two examples of pricing project expansion options in the mining industry to demonstrate that our models are able to produce financially meaningful numerical results for the two non-trivial test problems.

An upwind compact difference scheme for solving the streamfunction–velocity formulation of the unsteady incompressible Navier–Stokes equation

In this paper, an upwind compact difference method with second-order accuracy both in space and time is proposed for the streamfunction–velocity formulation of the unsteady incompressible Navier–Stokes equations. The first derivatives of streamfunction (velocities) are discretized by two type compact schemes, viz. the third-order upwind compact schemes suggested with the characteristic of low dispersion error are used for the advection terms and the fourth-order symmetric compact scheme is employed for the biharmonic term. As a result, a five point constant coefficient second-order compact scheme is established, in which the computational stencils for streamfunction only require grid values at five points at both (n)th and (n+1)th time levels. The new scheme can suppress non-physical oscillations. Moreover, the unconditional stability of the scheme for the linear model is proved by means of the discrete von Neumann analysis. Four numerical experiments involving a test problem with the analytic solution, doubly periodic double shear layer flow problem, lid driven square cavity flow problem and two-sided non-facing lid driven square cavity flow problem are solved numerically to demonstrate the accuracy and efficiency of the newly proposed scheme. The present scheme not only shows the good numerical performance for the problems with sharp gradients, but also proves more effective than the existing second-order compact scheme of the streamfunction–velocity formulation in the aspect of computational cost.