Upwind Finite Difference Scheme Research Articles

Efficient numerical methods for semilinear one dimensional parabolic singularly perturbed convection-diffusion systems

In this work we deal with the numerical solution of one dimensional semilinear parabolic singularly perturbed systems of convection-diffusion type. We assume that the coupling in the convection terms is weak and also that the coupling reaction terms are nonlinear. In the case of considering different small diffusion parameters at each equation with different orders of magnitude, the exact solution usually shows overlapping boundary layers on the outflow of the spatial domain. To approximate it, we construct a numerical scheme which combines the upwind finite difference scheme, defined on a piecewise uniform Shishkin mesh, to discretize in space and a linearized version of the fractional implicit Euler method together with an appropriate splitting by components to discretize in time. Then, the fully discrete method is uniformly convergent with respect to both diffusion parameters, having first order in time and almost first order in space. The choice of this time integrator provokes that only tridiagonal linear systems must be solved at each time step; in this way, the computational cost of the algorithm is considerably lower than this one associated to classical implicit methods. The numerical results obtained for different test problems corroborate in practice the good performance of the numerical algorithm.

An upwind finite difference scheme for singularly perturbed parabolic convection-diffusion problems with discontinuous initial conditions on a piecewise-uniform mesh

An upwind finite difference scheme for singularly perturbed parabolic convection-diffusion problems with discontinuous initial conditions on a piecewise-uniform mesh

Upwind reproducing kernel collocation method for convection-dominated problems

Because of the best approximation property, traditional Bubnov–Galerkin numerical methods have proven immensely successful in modeling self-adjoint problems, such as heat conduction, elasticity, and so on. However, a numerical instability arises in these (and central finite difference) methods for problems with strong convection. In this class of problems, the convective transport term can lead to large spurious oscillations but can be handled by the class of Petrov–Galerkin methods. In particular, the upwind-type schemes and their variational and subgrid descendants have been substantially developed over the years for an effective weak-form Galerkin solution that precludes these instabilities. Nevertheless, the scale of development of upwind methods for strong-form collocation is substantially smaller, where numerical oscillations are also observed when they are straightforwardly applied to convection-dominated problems without special treatment. To this end, this paper presents a new upwind collocation method. First, the connection between the upwind finite difference scheme and the gradient smoothing technique in meshfree methods is established. It is then shown that selecting the collocation points as meshfree nodal points is not optimal; selecting the collocation points according to the flow direction and Péclet number is then studied. The upwind effect is achieved without introducing artificial parameters and is trivial to generalize for multi-dimensional cases. Cross-wind diffusion is also not observed in the solution. An error analysis is presented, and the effectiveness of the proposed methodology is well demonstrated by the steady and unsteady numerical examples.

Richardson Extrapolation Technique for Singularly Perturbed Convection-diffusion Problem with Non-local Boundary Conditions

This study presents the Richardson extrapolation techniques for solving singularly perturbed convection -diffusion problems (SPCDP) with non-local boundary conditions. A numerical approach is presented using an upwind finite difference scheme a piecewise-uniform (Shishkin) mesh. To handle the non-local boundary conditions, the trapezoidal rule is applied. The study establishes an error bound for numerical solutions and determines the numerical approximation for scaled derivatives. To enhance convergence and accuracy, we utilize Richardson extrapolation. This elevates accuracy from firs-order to second order convergence

Numerical Analysis of Singularly Perturbed System of parabolic convection-diffusion IBVP system with boundary layers

In this paper, we get the numerical solution of a singularly perturbed system of boundary layer-exhibiting parabolic convection-diffusion problems. The backward-Euler method and an upwind finite difference scheme make up the suggested numerical approach for the time and spatial derivatives, respectively. For the spatial discretization, we analyse the scheme on a piecewise uniform Shishkin mesh in order to establish uniform convergence with regard to the perturbation parameters. The stability analysis for the suggested method is provided, and a parameter-uniform error estimate is generated. We have conducted some numerical tests to verify the theoretical findings.

Efficient Upwind Finite-Difference Schemes for Wave Equations on Overset Grids

Efficient Upwind Finite-Difference Schemes for Wave Equations on Overset Grids

Numerical Study of Natural Convection Flow in Rectangular Cavity with Viscous Dissipation and Internal Heat Generation for Different Aspect Ratios

Numerical simulations have been performed to investigate the influence of constant volumetric heat generation and viscous dissipation on the unsteady natural convection flow of an incompressible Newtonian fluid contained in a rectangular cavity. The left vertical wall of the cavity is cooled, while the right vertical wall is heated, and the bottom and top walls are adiabatic. A numerical technique based on the implicit finite difference method (IFDM), along with an upwind finite difference scheme and an iterative successive over relaxation (SOR) technique, is employed to solve the governing equations numerically. The effect of physical parameters, namely the modified Rayleigh number (103≤Ra≤107), aspect ratio (1≤A≤4), Prandtl number (Pr=0.7, 1.0, 6.2, 15), volumetric internal heat generation parameter (Qλ=0, 1), and Eckert number (0≤Ec≤10−6), on the streamlines and isotherms are discussed graphically. Variations of maximum stream function, as well as average and local Nusselt number, are also discussed. The results show that the increase in Eckert number from 0 to 10−4 causes the average heat transfer to decrease, while Pr=0.71, Ra=104, and Qλ=0. Additionally, the average heat transfer decreases as the cavity width increases from 1 to 4, while Pr=0.71, Ra = 5×104, Ec=10−6 and Qλ=1. The results of the numerical model used here are in excellent accord with earlier findings.

Improved fast iterative method for higher calculation accuracy of traveltimes

Fast Iterative Method (FIM), a grid-based ray-tracing algorithm, has a computational efficiency far exceeding that of conventional fast marching method (FMM) in serial computing, because it adopts an iterative algorithm and can be calculated in parallel. However, the finite difference scheme of the FIM only employs the first-order Godunov upwind finite difference scheme, resulting in low calculation accuracy of arrival traveltimes. This paper proposes a new algorithm of FIM, which utilizes second-order and mixed-order difference schemes under the parallel computing structure, effectively improves the calculation accuracy of traveltimes, and basically maintains the original computational efficiency. Especially, improved finite difference calculation in the diagonal direction of square grids and the double-grid technique near the source are applied to further reduce overall calculation error. Numerical experiments show that for the given model, the calculation error can be reduced from 2 ms of the original FIM calculation to 0.05 ms, and the calculation accuracy is increased by 40 times, while the time consumption increases only 27%.

Computational instability of sedimentation-consolidation model based on an interaction coefficient

Serious computational instability problems are usually encountered in solving the sedimentation-consolidation model based on an interaction coefficient (SCMIC). In this study, the computational instability behavior of SCMIC is investigated with given upwind finite difference scheme. Results show that significant numerical errors occur when low grid density is used. By analyzing the variations of governing equation coefficients with time, it is found that the computational instability is due to the highly nonlinear variation of d σ ′ d e with e ( σ ′ is effective stress and e is void ratio), which causes significant errors in calculating the average value of d σ ′ d e . By eliminating the effect of the high nonlinearity of d σ ′ d e , the computational instability of SCMIC can be completely resolved. Such a resolution method for the computational instability overcomes the restriction of grid density, which can facilitate the simulation of simultaneous sedimentation and consolidation in the geotechnical engineering field.

An adaptive grid method for a singularly perturbed convection-diffusion equation with a discontinuous convection coefficient

<abstract><p>In this paper, an adaptive grid method is put forward to solve a singularly perturbed convection-diffusion problem with a discontinuous convection coefficient. First, this problem is discretized by using an upwind finite difference scheme on an arbitrary nonuniform grid except the fixed jump point. Then, a first-order maximum norm a posterior error estimate is derived. Further, based on this a posteriori error estimation and the mesh equidistribution principle, an adaptive grid generation algorithm is constructed. Finally, some numerical experiments are presented that support our theoretical estimate.</p></abstract>

A splitting uniformly convergent method for one-dimensional parabolic singularly perturbed convection-diffusion systems

In this paper we deal with solving robustly and efficiently one-dimensional linear parabolic singularly perturbed systems of convection-diffusion type, where the diffusion parameters can be different at each equation and even they can have different orders of magnitude. The numerical algorithm combines the classical upwind finite difference scheme to discretize in space and the fractional implicit Euler method together with an appropriate splitting by components to discretize in time. We prove that if the spatial discretization is defined on an adequate piecewise uniform Shishkin mesh, the fully discrete scheme is uniformly convergent of first order in time and of almost first order in space. The technique used to discretize in time produces only tridiagonal linear systems to be solved at each time level; thus, from the computational cost point of view, the method we propose is more efficient than other numerical algorithms which have been used for these problems. Numerical results for several test problems are shown, which corroborate in practice both the uniform convergence and the efficiency of the algorithm.

An exponentially fitted spline method for singularly perturbed parabolic convection-diffusion problems with large time delay

This paper deals with the numerical solutions of singularly perturbed parabolic convection-diffusion problems with a large delay in time. An exponentially fitted scheme is formulated using cubic splines method in equally-spaced grids. As the delay parameter is larger than the perturbation parameter, a special type of mesh is used for the temporal variable so that the shift lies on the nodal points and an exponentially fitted cubic spline method based on Crank-Nicolson's discretization scheme is developed. The obtained scheme is conditionally stable. Parameter-uniform error estimates are derived and it is shown that the method is ε-uniformly convergent of second-order accurate in both temporal direction and spatial direction. The proposed method provides more accurate solutions than an upwind finite difference scheme in a piecewise uniform mesh.

A Fitted Numerical Method for Interior-Layer Parabolic Convection–Diffusion Problems

The aim of this paper is to construct and analyze a fitted mesh finite difference method (FMFDM) for a class of time-dependent singularly perturbed convection–diffusion–reaction problems with a turning point. While such problems have been widely studied in the case of boundary layers, little has been achieved for interior layer problems. In addition, most of works have considered the coefficient functions as the functions of space only. In this work, we study problems where the coefficient functions are dependent on both space and time variables, and their solutions exhibit an interior layer. We derive the solution bounds and their derivatives. Then we use the classical implicit Euler method to discretize the time variable with a constant step-size. This method consists of two upwind finite difference schemes defined on an appropriate non-uniform mesh of Shishkin type. This mesh is fine near the turning point and coarse elsewhere. We prove that the method is almost first-order uniformly convergent with respect to the perturbation parameter. To improve the accuracy of the proposed method, we apply Richardson extrapolation technique. Numerical simulations are provided to confirm the theoretical findings.

A study of the role of weights in the spectral properties of the least square finite difference scheme and the least square upwind finite difference scheme

AbstractIn this article, we analyze the performance of a family of nonlinear, shock‐capturing mesh‐free finite‐difference schemes in wavenumber space. These finite difference methods are based on the use of a weighted least‐square approximation procedure together with a Taylor series expansion of the unknown function. The influence of weights on the spectral properties of these schemes are studied using the approximate dispersion relation. The numerical study evidences that an appropriate choice of the weighting functions reduces dissipation and dispersion inherent to the schemes. The meshless schemes studied in this work, are then used to simulate 2D internal inviscid subsonic and transonic flows respectively in a channel with a bump. The results in 2D corroborate with the 1D analyses and reveal that when the supporting nodes closer to the reference node are assigned a greater weightage value in the computation of the 1st order spatial derivatives, better spectral properties are achieved at low wavenumber. However, the present numerical study shows that weighting function does not shift the frequency of the errors and thus in order to accelerate the convergence of these schemes, methods such as the classic multigrid can be used.

Asymptotic and numerical methods for solving singularly perturbed differential difference equations with mixed shifts

This article deals with an efficient approximation method named successive complementary expansion method (SCEM) for solving singularly perturbed differential difference equations with mixed shifts. It is compared with the method of matched asymptotic expansion (MMAE) and also with the parameter uniform upwind finite difference scheme for solving such a model. The comparison shows, unlike the MMAE, the SCEM method requires no matching procedure. It requires less computation when compared to the upwind finite difference scheme on the Shishkin mesh. The error analysis is carried out to prove the robustness of the method. Some numerical experiments are provided which show the effectiveness of the proposed method.

A posteriori error estimation and adaptive strategy for a nonlinear fractional differential equation

A nonlinear fractional differential equation with Caputo fractional derivative is considered. The problem is discretized by an upwind finite difference scheme for which a posteriori error analysis in the maximum norm is derived. A partly heuristic argument based on this a posteriori error estimation leads to several suitable monitor functions, and a new monitor function is constructed to design an adaptive grid algorithm. Numerical results are presented to illustrate the performance of the presented adaptive method. Compared with the other monitor functions, the presented adaptive grid method based on the new monitor function is more suitable to solve this type of nonlinearfractional differential equation.

Robust computational method for singularly perturbed system of parabolic convection‐diffusion problems with interior layers

In this article, we present the convergence analysis of an upwind finite difference scheme for singularly perturbed system of parabolic convection-diffusion initial-boundary-value problems with discontinuous convection coefficient and source term. The proposed numerical scheme is constructed by using the implicit-Euler scheme for the time derivative on the uniform mesh, and the upwind finite difference scheme for the spatial derivatives on a layer-resolving piecewise-uniform Shishkin mesh. It is shown that the numerical solution obtained by the proposed scheme converges uniformly with respect to the perturbation parameter. The proposed numerical scheme is of almost first-order (up to a logarithmic factor) in space and first-order in time. Numerical examples are carried out to verify the theoretical results.

Physics-Capturing Discretizations for Spectral Wind-Wave Models

This paper discusses the discretization methods that have been commonly employed to solve the wave action balance equation, and that have gained a renewed interest with the widespread use of unstructured grids for third-generation spectral wind-wave models. These methods are the first-order upwind finite difference and first-order vertex-centered upwind finite volume schemes for the transport of wave action in geographical space. The discussion addresses the derivation of these schemes from a different perspective. A mathematical framework for mimetic discretizations based on discrete calculus is utilized herein. A key feature of this algebraic approach is that the process of exact discretization is segregated from the process of interpolation, the latter typically involved in constitutive relations. This can help gain insight into the performance characteristics of the discretization method. On this basis, we conclude that the upwind finite difference scheme captures the wave action flux conservation exactly, which is a plus for wave shoaling. In addition, we provide a justification for the intrinsic low accuracy of the vertex-centred upwind finite volume scheme, due to the physically inaccurate but common flux constitutive relation, and we propose an improvement to overcome this drawback. Finally, by way of a comparative demonstration, a few test cases is introduced to establish the ability of the considered methods to capture the relevant physics on unstructured triangular meshes.

A feedback design for numerical solution to optimal control problems based on Hamilton-Jacobi-Bellman equation

<p style='text-indent:20px;'>In this paper, we present a feedback design for numerical solution to optimal control problems, which is based on solving the corresponding Hamilton-Jacobi-Bellman (HJB) equation. An upwind finite-difference scheme is adopted to solve the HJB equation under the framework of the dynamic programming viscosity solution (DPVS) approach. Different from the usual existing algorithms, the numerical control function is interpolated in turn to gain the approximation of optimal feedback control-trajectory pair. Five simulations are executed and both of them, without exception, output the accurate numerical results. The design can avoid solving the HJB equation repeatedly, thus efficaciously promote the computation efficiency and save memory.</p>

奇异摄动反应扩散方程的后验误差估计及自适应算法

An adaptive grid method for singularly perturbed semilinear reactiondiffusion equations was studied. The equation was discretized by means of the upwind finite difference scheme on an arbitrary nonuniform mesh. A posteriori error estimation of the presented numerical scheme was built. In turn, the posteriori error bound was derived, and the adaptive grid generation algorithm was designed. Numerical experiments prove the effectiveness of the proposed adaptive grid method.