Upwind Finite Difference Scheme Research Articles

Rate of convergence of finite difference approximations for degenerate ordinary differential equations

In this paper we study finite difference approximations for the following linear stationary convection-diffusion equations: \[ 1 2 σ 2 ( x ) u ( x ) + b ( x ) u ′ ( x ) − u ( x ) = − f ( x ) , x ∈ R , {1\over 2}\sigma ^2(x)u(x) + b(x)u’(x) - u(x) =-f(x),\quad x\in \mathbb {R}, \] where σ \sigma is allowed to be degenerate. We first propose a new weighted finite difference scheme, motivated by approximating the diffusion process associated with the equation in the strong sense. We show that, under certain conditions, this scheme converges with the first order rate and that such a rate is sharp. To the best of our knowledge, this is the first sharp result in the literature. Moreover, by using the connection between our scheme and the standard upwind finite difference scheme, we get the rate of convergence of the latter, which is also new.

Effect of viscous dissipation on mixed convection flow of water near its density maximum in a rectangular enclosure with isothermal wall

PurposeTo investigate the effect of viscous dissipation on unsteady, combined convective heat transfer to water near its density maximum in a rectangular cavity.Design/methodology/approachThe upwind finite difference scheme along with successive over relaxation iteration technique is used to solve the governing equations for mixed convection flow of water with density maximum inversion in a rectangular cavity.FindingsThe effect of viscous dissipation was to increase the fluid temperature and resulted in the formation of vortex motion near the lower part of the cavity in an opposite direction to the central vortex. An increase in the Eckert number and Reynolds number of the flow resulted in augmented surface heat transfer rates from the top heated surface.Research limitations/implicationThe analysis is valid for unsteady, two dimensional laminar flow. Isothermal conditions are assumed for the top and bottom walls. An extension to unsteady three dimensional flow case is left for future work.Practical implicationsThe method is very useful to analyze nuclear reactor thermal/hydraulic loss of coolant transients, energy conservation, ventilation of rooms, solar energy collection, cooling of electronic equipment, dispersion of waste heat in estuaries and crystal growth in liquids.Originality/valueThe results of this study may be of interest to engineers interested in heat transfer augmentation of mixed convection in window cavities.

DIFFUSIVITY OF TWO-COMPONENT ISOTHERMAL FINITE DIFFERENCE LATTICE BOLTZMANN MODELS

Diffusion equations are derived for an isothermal lattice Boltzmann model with two components. The first-order upwind finite difference scheme is used to solve the evolution equations for the distribution functions. When using this scheme, the numerical diffusivity, which is a spurious diffusivity in addition to the physical diffusivity, is proportional to the lattice spacing and significantly exceeds the physical value of the diffusivity if the number of lattice nodes per unit length is too small. Flux limiter schemes are introduced to overcome this problem. Empirical analysis of the results of flux limiter schemes shows that the numerical diffusivity is very small and depends quadratically on the lattice spacing.

Numerical and Physical Diffusion: Can Wave Prediction Models Resolve Directional Spread?

Abstract A new semi-Lagrangian advection scheme called multistep ray advection is proposed for solving the spectral energy balance equation of ocean surface gravity waves. Existing so-called piecewise ray methods advect wave energy over a single time step using “pieces” of ray trajectories, after which the spectrum is updated with source terms representing various physical processes. The generalized scheme presented here allows for an arbitrary number N of advection time steps along the same rays, thus reducing numerical diffusion, and still including source-term variations every time step. Tests are performed for alongshore uniform bottom topography, and the effects of two types of discretizations of the wave spectrum are investigated, a finite-bandwidth representation and a single frequency and direction per spectral band. In the limit of large N, both the accuracy and computation cost of the method increase, approaching a nondiffusive fully Lagrangian scheme. Even for N = 1 the semi-Lagrangian scheme test results show less numerical diffusion than predictions of the commonly used first-order upwind finite-difference scheme. Application to the refraction and shoaling of narrow swell spectra across a continental shelf illustrates the importance of controlling numerical diffusion. Numerical errors in a single-step (Δt = 600 s) scheme implemented on the North Carolina continental shelf (typical swell propagation time across the shelf is about 3 h) are shown to be comparable to the angular diffusion predicted by the wave–bottom Bragg scattering theory, in particular for narrow directional spectra, suggesting that the true directional spread of swell may not always be resolved in existing wave prediction models, because of excessive numerical diffusion. This diffusion is effectively suppressed in cases presented here with a four-step semi-Lagrangian scheme, using the same value of Δt.

Conservative upwind difference schemes for open channel flows—theory and applications

Conservative upwind finite-difference schemes are presented for the solution of the one-dimensional shallow water equations in open channels. Numerical results are presented and compared for different versions of the schemes when applied to a test problem comprising a channel with varying breadth and depth and friction terms. This includes consideration of the effect of treating part of the flux balance as a source, and a comparison of square-root and arithmetic averaging.

Diffuse-reflection boundary conditions for a thermal lattice Boltzmann model in two dimensions: Evidence of temperature jump and slip velocity in microchannels

We discuss the implementation of diffuse reflection boundary conditions in a thermal lattice Boltzmann model for which the upwind finite difference scheme is used to solve the set of evolution equations recovered after discretization of the velocity space. Simulation of heat transport between two parallel walls at rest shows evidence of temperature jumps at the walls that increase with Knudsen number. When the walls move in opposite directions with speeds +/- u(W), fluid velocity slip is observed at the walls, together with temperature jumps.

A MATLAB implementation of upwind finite differences and adaptive grids in the method of lines

In this paper, we report on the development of a MATLAB library for the solution of partial differential equation systems following the method of lines. In particular, we focus attention on upwind finite difference schemes and grid adaptivity, i.e., grid movement or grid refinement. Several algorithms are presented and their performance is demonstrated with illustrative examples including a fixed-bed reactor with periodic flow reversal, a model of flame propagation, and the Korteweg–de Vries equation.

A parameter-uniform implicit difference scheme for solving time-dependent Burgers’ equations

Abstract A numerical study is made for solving one dimensional time dependent Burgers’ equation with small coefficient of viscosity. Burgers’ equation is one of the fundamental model equations in the fluid dynamics to describe the shock waves and traffic flows. For high coefficient of viscosity a number of solution methodology exist in the literature [6] , [7] , [8] , [9] and [14] but for the sufficiently low coefficient of viscosity, the exist solution methodology fail and a discrepancy occurs in the literature. In this paper, we present a numerical method based on finite difference which works nicely for both the cases, i.e., low as well as high viscosity coefficient. The method comprises a standard implicit finite difference scheme to discretize in temporal direction on uniform mesh and a standard upwind finite difference scheme to discretize in spacial direction on piecewise uniform mesh. The quasilinearzation process is used to tackle the non-linearity. An extensive amount of analysis has been carried out to obtain the parameter uniform error estimates which show that the resulting method is uniformly convergent with respect to the parameter. To illustrate the method, numerical examples are solved using the presented method and compare with exact solution for high value of coefficient of viscosity.

Convergence of Upwind Finite Difference Schemes for a Scalar Conservation Law with Indefinite Discontinuities in the Flux Function

We consider the scalar conservation law with flux function discontinuous in the space variable, i.e., \begin{eqnarray} \label{eq1} u_t+(H(x)f(u)+(1-H(x))g(u))_{x} &=& 0 \quad \mbox{in } \R \times \R_{+}, \nonumber \\ u(0, x) &=& u_{0}(x) \quad \mbox{in } \R, \label{0.1} \end{eqnarray} where H is the Heaviside function and f and g are smooth with the assumptions that either f is convex and g is concave or f is concave and g is convex. The existence of a weak solution of (\ref{eq1}) is proved by showing that upwind finite difference schemes of Godunov and Enquist--Osher type converge to a weak solution. Uniqueness follows from a Kruzkhov-type entropy condition. We also provide explicit solutions to the Riemann problem for (\ref{eq1}). At the level of numerics, we give easy-to-implement numerical schemes of Godunov and Enquist--Osher type. The central feature of this paper is the modification of the singular mapping technique (the main analytical tool for these types of equations) which allows us to show that the numerical schemes converge. Equations of type (\ref{eq1}) with the above hypothesis on the flux may occur when considering the following scalar conservation law with discontinuous flux: \begin{equation} \label{eq2} \begin{array}{r@{\;}l} u_t + (k (x) f (u))_x &= 0, \\ u (0, x) &= u_0 (x), \end{array} \end{equation} with f convex and k of indefinite sign.

A semilinear Black and Scholes partial differential equation for valuing American options: approximate solutions and convergence

In [7], we proved that the American (call/put) option valuation problem can be stated in terms of one single semilinear Black and Scholes partial differential equation set in a fixed domain. The semilinear Black and Scholes equation constitutes a starting point for designing and analyzing a variety of “easy to implement” numerical schemes for computing the value of an American option. To demonstrate this feature, we propose and analyze an upwind finite difference scheme of “predictor‐corrector type” for the semilinear Black and Scholes equation. We prove that the approximate solutions generated by the predictor‐corrector scheme respect the early exercise constraint and that they converge uniformly to the American option value. A numerical example is also presented. Besides the predictor‐corrector schemes, other methods for constructing approximate solution sequences are discussed and analyzed.

Modeling the salinity of an inland coastal brackish karstic spring with a conduit‐matrix model

The salinity of an inland coastal brackish karstic spring is modeled on the basis of a simple concept of fluid exchange through head differences between a continuous porous matrix and a karst conduit. The coastal aquifer is reduced to an equivalent porous medium (matrix) naturally invaded by seawater, crossed by a single karst conduit where fresh water and brackish water mix in variable proportions and flow up into the spring. A new numerical model with an upwind explicit finite difference scheme, called salt‐water intrusion in karst conduits (SWIKAC), was developed and successfully applied to the Almyros spring of Heraklio (Crete, Greece). The good fit of the model to the observed salinity in the spring validates the proposed conceptual model of salinization. It provides a quantitative description of the seawater intrusion inside the karst conduit. The results open up new perspectives for managing the fragile and precious fresh water resources in karstic coastal zones.

An exponentially-fitted method for singularly-perturbed ordinary differential equations with turning points and parabolic problems

An exponentially-fitted method for singularly perturbed, one-dimensional parabolic equations and ordinary differential equations both of the convection–diffusion–reaction type in equally-spaced grids is presented. The method is based on the implicit discretization of the time derivative, freezing of the coefficients of the resulting ordinary differential equations at each time step, and the analytical solution of the resulting convection–diffusion–reaction differential operator. This solution is of exponential type and exact for steady, constant-coefficients convection–diffusion–reaction equations. By means of three examples for parabolic equations and two examples for ordinary differential equations with turning points, it is shown that this method provides uniformly convergent solutions with respect to the small perturbation parameter, and these solutions are more accurate and exhibit a higher order of convergence than those obtained with a boundary-layer resolving upwind finite difference scheme on a piecewise-uniform mesh.

Recursive two-levelILU preconditioner for nonsymmetric M-matrices

We develop in this paper some preconditioners for sparse non-symmetric M-matrices, which combine a recursive two-level blockILU factorization with multigrid method, we compare these preconditioners on matrices arising from discretized convection-diffusion equations using upwind finite difference schemes and multigrid orderings, some comparison theorems and experiment results are demonstrated.

An exponentially-fitted method for singularly perturbed, one-dimensional, parabolic problems

An exponentially-fitted method for singularly perturbed, one-dimensional, linear, convection–diffusion–reaction equations in equally-spaced grids is presented. The method is based on the implicit discretization of the time derivative, freezing of the coefficients of the resulting ordinary differential equations at each time step, and the analytical solution of the resulting convection–diffusion differential operator. This solution is of exponential type and exact for steady, constant-coefficients convection–diffusion equations with constant sources. By means of three examples, it is shown that this method provides uniformly convergent solutions with respect to both the time step and the small perturbation parameter, and these solutions are more accurate and exhibit a higher order of convergence than those obtained with an upwind finite difference scheme in a piecewise uniform mesh that is boundary-layer resolving.

An upwind compact mpact approach with group velocity control for compressible flow fields

AbstractIn this paper we construct an upwind compact finite difference scheme with group velocity control for better simulation of compressible flow fields. Compared with traditional difference schemes, compact schemes have higher accuracy for the same stencil width. By means of the characteristic analysis of the operators, the group velocity of wave packets will be controlled to suppress the non‐physical oscillations in numerical solutions. In numerical simulation of the 3D compressible flow fields the third‐order accurate upwind compact operator is used to approximate the derivatives in the convection terms of the compressible N–S equations, the traditional finite difference scheme is used to approximate the viscous terms. Numerical solutions indicate that the method is satisfactory. Copyright © 2004 John Wiley & Sons, Ltd.

A Class of Singularly Perturbed Convection-Diffusion Problems with a Moving Interior Layer. An a Posteriori Adaptive Mesh Technique

AbstractWe study numerical approximations for a class of singularly perturbed convection-diffusion type problems with a moving interior layer. In a domain (segment) with a moving interface between two subdomains, we consider an initial boundary value problem for a singularly perturbed parabolic convection-diffusion equation. Convection fluxes on the subdomains are directed towards the interface. The solution of this problem has a moving transition layer in the neighbourhood of the interface. Unlike problems with a stationary layer, the solution exhibits singular behaviour also with respect to the time variable. Well-known upwind finite difference schemes for such problems do not converge ε-uniformly in the uniform norm. In the case of rectangular meshes which are (a priori or a posteriori ) locally condensed in the transition layer. However, the condition for convergence can be considerably weakened if we take the geometry of the layer into account, i.e., if we introduce a new coordinate system which captures the interface. For the problem in such a coordinate system, one can use either an a priori, or an a posteriori adaptive mesh technique. Here we construct a scheme on a posteriori adaptive meshes (based on the solution gradient), whose solution converges ‘almost ε-uniformly’.

Convergence of a finite difference method for combustion model problems

We study a finite difference scheme for a combustion model problem. A projection scheme near the combustion wave, and the standard upwind finite difference scheme away from the combustion wave are applied. Convergence to weak solutions with a combustion wave is proved under the normal Courant-Friedrichs-Lewy condition. Some conditions on the ignition temperature are given to guarantee the solution containing a strong detonation wave or a weak detonation wave. Convergence to strong detonation wave solutions for the random projection method is also proved.

Finite difference method for a combustion model

We study a projection and upwind finite difference scheme for a combustion model problem. Convergence to weak solutions is proved under the Courant-Friedrichs-Lewy condition. More assumptions are given on the ignition temperature; then convergence to strong detonation wave solutions or to weak detonation wave solutions is proved.

Well-posedness in BV t and convergence of a difference scheme for continuous sedimentation in ideal clarifier-thickener units

We consider a scalar conservation law modeling the settling of particles in an ideal clarifier-thickener unit. The conservation law has a nonconvex flux which is spatially dependent on two discontinuous parameters. We suggest to use a Kružkov-type notion of entropy solution for this conservation law and prove uniqueness (L1 stability) of the entropy solution in the BVt class (functions W(x,t) with ∂tW being a finite measure). The existence of a BVt entropy solution is established by proving convergence of a simple upwind finite difference scheme (of the Engquist-Osher type). A few numerical examples are also presented.

Finite difference method for detonation waves

We study an upwind finite difference scheme for detonation waves. Convergences to weak and strong solutions are discussed with some extra hypothesis. A numerical example shows that beyond these hypothesis, the situation could be more complicated and interesting.