Simple Upwind Scheme Research Articles

Numerical approximation of parabolic singularly perturbed problems with large spatial delay and turning point

PurposeSingular perturbation turning point problems (SP-TPPs) involving parabolic convection–diffusion Partial Differential Equations (PDEs) with large spatial delay are studied in this paper. These type of equations are important in various fields of mathematics and sciences such as computational neuroscience and require specialized techniques for their numerical analysis.Design/methodology/approachWe design a numerical method comprising a hybrid finite difference scheme on a layer-adapted mesh for the spatial discretization and an implicit-Euler scheme on a uniform mesh in the temporal variable. A combination of the central difference scheme and the simple upwind scheme is used as the hybrid scheme.FindingsConsistency, stability and convergence are investigated for the proposed scheme. It is established that the present approach has parameter-uniform convergence of OΔτ+K−2(lnK)2, where Δτ and K denote the step size in the time direction and number of mesh-intervals in the space direction.Originality/valueParabolic SP-TPPs exhibiting twin boundary layers with large spatial delay have not been studied earlier in the literature. The presence of delay portrays an interior layer in the considered problem’s solution in addition to twin boundary layers. Numerical illustrations are provided to demonstrate the theoretical estimates.

A three-dimensional coupled thermo-hydro model for geothermal development in discrete fracture networks of hot dry rock reservoirs

Three-dimensional (3D) fluid flow and thermal transport modeling in discrete fracture networks (DFNs) is critical for geothermal energy recovery from enhanced geothermal systems (EGSs). Considering nonlinear fluid flow models, a 3D coupled thermo-hydro (T-H) model based on the Galerkin finite element method was developed for DFNs. Forchheimer flow was considered in this fluid flow model, and a simplified thermal transport model was adopted based on the local thermal non-equilibrium theory. Finite element spatial discretization of fluid flow and thermal transport models in DFNs was formulated. A simple upwind scheme was adopted as the numerical strategy of the thermal transport model, avoiding its solution oscillation. In addition, the proposed numerical method for the fluid flow and thermal transport was verified by comparing the model results with experimental, analytical, and numerical solutions. This approach was further applied to the modeling of heat extraction in a Habanero EGS reservoir for a period of 40 years under different injection pressures. The results demonstrated that the proposed approach was effective for simulating coupled T-H processes in 3D DFNs. With increasing the injection pressure, the power generation of reservoir increased, but its life span decreased. Injection temperature had positive effect on life span for power generation.

Higher order robust numerical computation for singularly perturbed problem involving discontinuous convective and source term

In this article, we considered a singularly perturbed convection–diffusion equation having discontinuous convective and source terms. Due to these discontinuities, an expected behavior appears in the solution known as the interior layer behavior at the point of discontinuity. Layers are some small narrow regions of the domain where the steep variations in the solution occur as singular perturbation parameter ε approaches zero. Classical central finite difference schemes and finite element methods with piecewise polynomial basis functions on the uniform mesh are known to be inadequate to solve such problems unless an unacceptably large number of mesh points are used when ε is very small. Thus, the problem is solved numerically first by a first‐order accurate, simple upwind scheme on a specially designed piecewise uniform Shishkin mesh. Further, to enhance the order of convergence and to achieve better accuracy, we constructed a Richardson extrapolation scheme (RE scheme) whichis almost second‐order accurate. An extensive analysis has been carried out to establish the almost second‐order ε‐uniform convergence of the proposed RE scheme. Lastly, some numerical examples are considered to verify our theoretical results, and comparisons of numerical results are performed with the numerical results that exist in the literature.

Second-order parameter-uniform finite difference scheme for singularly perturbed parabolic problem with a boundary turning point

In this article, we consider a one-dimensional time-dependent singularly perturbed problem, whose solution possesses a parabolic boundary layer in the neighbourhood of the left lateral surface. A priori estimates are derived for the classical solution and its derivatives, which are useful for the stability and convergence analysis of the proposed difference scheme. The discussed scheme consists of an implicit Euler method on uniform mesh in time and a simple upwind scheme on piece-wise uniform mesh in space. Then, to improve the order of convergence, we apply the Richardson extrapolation scheme in both space and time direction. Theoretically, we prove that the proposed scheme is almost second-order parameter uniform convergent and verify it by doing some numerical experiments.

A higher order scheme for singularly perturbed delay parabolic turning point problem

PurposeThe purpose of this study is to construct and analyze a parameter uniform higher-order scheme for singularly perturbed delay parabolic problem (SPDPP) of convection-diffusion type with a multiple interior turning point.Design/methodology/approachThe authors construct a higher-order numerical method comprised of a hybrid scheme on a generalized Shishkin mesh in space variable and the implicit Euler method on a uniform mesh in the time variable. The hybrid scheme is a combination of simple upwind scheme and the central difference scheme.FindingsThe proposed method has a convergence rate of order . Further, Richardson extrapolation is used to obtain convergence rate of order two in the time variable. The hybrid scheme accompanied with extrapolation is second-order convergent in time and almost second-order convergent in space up to a logarithmic factor.Originality/valueA class of SPDPPs of convection-diffusion type with a multiple interior turning point is studied in this paper. The exact solution of the considered class of problems exhibit two exponential boundary layers. The theoretical results are supported via conducting numerical experiments. The results obtained using the proposed scheme are also compared with the simple upwind scheme.

Numerical Solution of Quasilinear Singularly Perturbed Problems by the Principle of Equidistribution

In this paper, the numerical solution and its error analysis of quasilinear singular perturbation two-point boundary value problems based on the principle of equidistribution are given. On the non-uniform grid of the uniformly distributed arc-length monitor function, the solution of the simple upwind scheme is obtained. It is proved that the adaptive simple upwind scheme based on the principle of equidistribution has uniform convergence for small perturbation parameters. Numerical experiments are carried out and the error analysis are confirmed.

The uniform convergence of upwind schemes on layer-adapted meshes for a singularly perturbed Robin BVP

In this paper, we discuss the uniform convergence of the simple upwind scheme on the Shishkin mesh and the Bakhvalov-Shishkin mesh for solving a singularly perturbed Robin boundary value problem, and investigate the midpoint upwind scheme on the Shishkin mesh and the Bakhvalov-Shishkin mesh to achieve better uniform convergence. The elaborate e-uniform pointwise estimates are proved by using the comparison principle and barrier functions. The numerical experiments support the theoretical results for the schemes on the meshes.

Richardson extrapolation technique for singularly perturbed parabolic convection–diffusion problems

This paper deals with the study of a post-processing technique for one-dimensional singularly perturbed parabolic convection–diffusion problems exhibiting a regular boundary layer. For discretizing the time derivative, we use the classical backward-Euler method and for the spatial discretization the simple upwind scheme is used on a piecewise-uniform Shishkin mesh. We show that the use of Richardson extrapolation technique improves the e-uniform accuracy of simple upwinding in the discrete supremum norm from O (N −1 ln N + Δt) to O (N −2 ln2 N + Δt 2), where N is the number of mesh-intervals in the spatial direction and Δt is the step size in the temporal direction. The theoretical result is also verified computationally by applying the proposed technique on two test examples.

Numerical methods on Shishkin mesh for singularly perturbed delay differential equations with a grid adaptation strategy

A numerical study is made for a class of boundary value problems of second-order differential equations in which the highest order derivative is multiplied by a small parameter ϵ and both the differentiated(convection) and undifferentiated(reaction) terms are with negative shift δ. We analyze three difference operators L k h , k = 1, 2, 3 a simple upwind scheme, midpoint upwind scheme and a hybrid scheme, respectively, on a Shishkin mesh to approximate the solution of the problem. Theoretical error bounds are established. The accuracy by one of the grid adaptation strategies, namely grid redistribution is examined by solving the considered problem using hybrid method for some values of ϵ and N. As a result the accuracy gets improved by this grid adaptation strategy with almost the same computational cost. A few numerical results exhibiting the performance of these three schemes are presented.

Numerical methods for analysis of structure and ground vibration from moving loads

An overview of the main theoretical aspects of finite-element and boundary-element modelling of the response to moving loads is given. The moving loads represent sources of noise and vibration generated by moving vehicles, and the analysis describes the propagation of the disturbances generated in soil. A finite-element time-domain analysis in convected coordinates with a simple upwind scheme is presented, including a special set of boundary conditions permitting the passage of outgoing waves in the convected coordinate system. The modification of frequency-dependent damping to convected coordinates is described, and the convected formulation of boundary elements is presented and used for illustrating the effect of ‘high-speed’ motion. Finally, a procedure for the coupling of a local finite-element model with a boundary-element model of an exterior, or open, domain is described. The paper uses recent results from the Danish research programme ‘Damping Mechanisms in Dynamics of Structures and Materials’ as a basis for a general discussion and review of the recent literature on the subject.

Hybrid method for numerical solution of singularly perturbed delay differential equations

This paper is concerned with the numerical technique for a singularly perturbed second-order differential–difference equation of the convection–diffusion type with small delay parameter δ whose solution has a single boundary layer. We analyse three difference operators L k N , k = 1, 2, 3 a simple upwind scheme, midpoint upwind scheme and a hybrid scheme, respectively, on a Shishkin mesh to approximate the solution of the problem. The hybrid algorithm uses central difference in the boundary layer region and midpoint upwind scheme outside the boundary layer. We have concluded that the hybrid scheme gives better accuracy. The paper concludes with a few numerical results exhibiting the performance of these three schemes.

Comparison of a higher order method and the simple upwind and non-monotone methods for singularly perturbed boundary value problems

In this paper, we will present a new scheme for discretization of singularly perturbed boundary value problems based on finite difference methods. This method is a combination of simple upwind scheme and central difference method on a special non-uniform mesh (Shishkin mesh) for the space discretization. Numerical results show that the convergence of method is uniform with respect to singular perturbation parameter and has a higher order of convergence.

A uniformly convergent scheme on a nonuniform mesh for convection–diffusion parabolic problems

In this paper we construct a numerical method to solve one-dimensional time-dependent convection–diffusion problem with dominating convection term. We use the classical Euler implicit method for the time discretization and the simple upwind scheme on a special nonuniform mesh for the spatial discretization. We show that the resulting method is uniformly convergent with respect to the diffusion parameter. The main lines for the analysis of the uniform convergence carried out here can be used for the study of more general singular perturbation problems and also for more complicated numerical schemes. The numerical results show that, in practice, some of the theoretical compatibility conditions seem not necessary.

Finite volume element methods for non-definite problems

The error estimates for finite volume element method applied to 2 and 3-D non-definite problems are derived. A simple upwind scheme is proven to be unconditionally stable and first order accurate.

Evaluation of a bounded high-resolution scheme for combustor flow computations

The present paper is concerned with the application of a high-resolution scheme for a finite volume discretization method. The capability of the chosen scheme is evaluated for the numerical simulation of three-dimensional flowfields. Special emphasis lies in the calculation of the processes in the mixing zone of gas turbine combustors. Thus, numerical results are compared with detailed measurements of velocity and temperature in a model mixing zone. The velocity field as well as the temperature field strongly depend on the scheme used for the discretization of the momentum and energy equations. The proposed high-resolution scheme and the well-known QUICK scheme yield similar results, which are superior to those obtained with the simple UPWIND scheme. For the simulation of turbulent transport, the standard fc, € model is employed. The effect of numerical diffusion caused by the usual UPWIND discretization of the convective transport of A and e is assessed by the application of the high-resolution scheme of the discretization of the A and e equations. As a result, it is found that the velocity and temperature fields are insensitive to the scheme that is applied for the discretization of the A and e equations.