Standard Upwind Scheme Research Articles

PARAMETER-UNIFORM HYBRID NUMERICAL SCHEME FOR SINGULARLY PERTURBED PARABOLIC CONVECTION-DIFFUSION PROBLEMS WITH DISCONTINUOUS INITIAL CONDITIONS

This article presents a parameter-uniform hybrid numerical technique for singularly perturbed parabolic convection-diffusion problems (SPPCDP) with discontinuous initial conditions (DIC). It utilizes the classical backward-Euler technique for time discretization and a hybrid finite difference scheme ( which is a proper combination of the midpoint upwind scheme in the outer regions and the classical central difference scheme in the interior layer regions(generated by the DIC)) for spatial discretization. The scheme produces parameter-uniform numerical approximations on a piecewise- uniform Shishkin mesh. When the perturbation parameter ε(0 < ε ≪ 1) is small, it becomes difficult to solve these problems using the classical numerical methods (standard central difference or a standard upwind scheme) on uniform meshes because discontinuous initial conditions frequently appear in the solutions of this class of problems. The method is shown to converge uniformly in the discrete supremum with nearly second-order spatial accuracy. The suggested method is subjected to a stability study, and parameter-uniform error estimates are generated. In order to support the theoretical findings, numerical results are presented.

An almost second order uniformly convergent method for a two-parameter singularly perturbed problem with a discontinuous convection coefficient and source term

<p>In this paper, we discuss a higher-order convergent numerical method for a two-parameter singularly perturbed differential equation with a discontinuous convection coefficient and a discontinuous source term. The presence of perturbation parameters generates boundary layers, and the discontinuous terms produce interior layers on both sides of the discontinuity. In order to obtain a higher-order convergent solution, a hybrid monotone finite difference scheme is constructed on a piecewise uniform Shishkin mesh, which is adapted inside the boundary and interior layers. On this mesh (including the point of discontinuity), the present method is almost second-order parameter-uniform convergent. The current scheme is compared with the standard upwind scheme, which is used at the point of discontinuity. The numerical experiments based on the proposed scheme show higher-order (almost second-order) accuracy compared to the standard upwind scheme, which provides almost first-order parameter-uniform convergence.</p>

Consistent MPFA Discretization for Flow in the Presence of Gravity

AbstractA standard practice used in the industry to discretizing the gravity term in the two‐phase Darcy flow equations is to apply an upwind strategy. In this paper, we show that this can give a persistent unphysical flux field and an incorrect pressure distribution. As a solution to this problem, we present a new consistent discretization of flow, termed Gravitationally Consistent Multipoint Flux Approximation (GCMPFA), which is valid for both single‐ and two‐phase flows. The discretization is based on the idea that the gravitational term in the flow equations is treated as part of the discrete flux operator and not as a right‐hand side. Here, the traditional formulation representing pressure as a potential is extended to the case including gravity by introducing an additional set of right‐hand side to the local linear system solved in the MPFA construction, thus obtaining an expression of the fluxes in terms of jumps in cell‐center gravities. Numerical examples showing the convergence of the method are provided for both single‐ and two‐phase flows. For two‐phase flow, we show how our new method is capable of eliminating the unphysical fluxes arising when using a standard upwind scheme, thus converging to the correct pressure distribution.

PARAMETER-UNIFORM IMPROVED HYBRID NUMERICAL SCHEME FOR SINGULARLY PERTURBED PROBLEMS WITH INTERIOR LAYERS

In this paper, we consider a class of singularly perturbed convection-diffusion boundary-value problems with discontinuous convection coefficient which often occur as mathematical models for analyzing shock wave phenomena in gas dynamics. In general, interior layers appear in the solutions of this class of problems and this gives rise to difficulty while solving such problems using the classical numerical methods (standard central difference or standard upwind scheme) on uniform meshes when the perturbation parameter ε is small. To achieve better numerical approximation in solving this class of problems, we propose a new hybrid scheme utilizing a layer-resolving piecewise-uniform Shishkin mesh and the method is shown to be ε-uniformly stable. In addition to this, it is proved that the proposed numerical scheme is almost second-order uniformly convergent in the discrete supremum norm with respect to the parameter ε. Finally, extensive numerical experiments are conducted to support the theoretical results. Further, the numerical results obtained by the newly proposed scheme are also compared with the hybrid scheme developed in the paper [Z.Cen, Appl. Math. Comput., 169(1): 689-699, 2005]. It shows that the current hybrid scheme exhibits a significant improvement over the hybrid scheme developed by Cen, in terms of the parameter-uniform order of convergence.

Multiquadric RBF-FD method for the convection-dominated diffusion problems base on Shishkin nodes

In this paper, a new hybrid scheme based multiquadric radial basis function-generated finite difference (RBF-FD) method with Shishkin nodes is proposed to solve stationary convection-dominated diffusion problems, i.e., the boundary layer problems, which combines the midpoint upwind scheme on the coarse grid with a standard central scheme on the fine grid. Numerical results demonstrate the hybrid scheme with Shishkin nodes has an absolute advantage in accurate over standard upwind scheme with equidistant nodes. Moreover, by varying the shape parameter in multiquadric RBF-FD method, the convergence rates of numerical solutions have been further improved compared with the corresponding FD method.

Investigation of all-speed schemes for turbulent simulations with low-speed features

Turbulence plays a key role in the aerospace design process. It is common that incompressible and compressible flows coexist in turbulent flows around aerospace vehicles. However, most upwind schemes in compressible solvers were designed to capture shock waves and have been proved to have difficulties in predicting low-speed flow regions. In order to overcome this defect, many all-speed schemes have been proposed. This paper investigates the properties of the all-speed schemes when applying to Reynolds averaged Navier–Stokes simulations with important low-speed features. First, the correctness of our code is validated. Then four test cases are adopted to evaluate the scheme performance, including a Mach 2.85 compression ramp, the NACA 4412 airfoil, a Mach 2.92 ramped cavity and a three-dimensional surface-mounted cube. Grid-converged results from the all-speed schemes show good agreement with the experimental data and remarkable improvement when compared to standard upwind schemes. Moreover, different from the traditional preconditioning methods, the all-speed schemes are simple to realize and free from the cut-off strategy or any problem-dependent parameter. Therefore, they are expected to be widely implemented into compressible solvers and applied to all-speed turbulent flow simulations.

Preconditioning and Uniform Convergence for Convection-Diffusion Problems Discretized on Shishkin-Type Meshes

A one-dimensional linear convection-diffusion problem with a perturbation parameter ɛ multiplying the highest derivative is considered. The problem is solved numerically by using the standard upwind scheme on special layer-adapted meshes. It is proved that the numerical solution is ɛ-uniform accurate in the maximum norm. This is done by a new proof technique in which the discrete system is preconditioned in order to enable the use of the principle where “ɛ-uniform stability plus ɛ-uniform consistency implies ɛ-uniform convergence.” Without preconditioning, this principle cannot be applied to convection-diffusion problems because the consistency error is not uniform in ɛ. At the same time, the condition number of the discrete system becomes independent of ɛ due to the same preconditioner; otherwise, the condition number of the discrete system before preconditioning increases when ɛ tends to 0. We obtained such results in an earlier paper, but only for the standard Shishkin mesh. In a nontrivial generalization, we show here that the same proof techniques can be applied to the whole class of Shishkin-type meshes.

A stable FSI algorithm for light rigid bodies in compressible flow

In this article we describe a stable partitioned algorithm that overcomes the added mass instability arising in fluid–structure interactions of light rigid bodies and inviscid compressible flow. The new algorithm is stable even for bodies with zero mass and zero moments of inertia. The approach is based on a local characteristic projection of the force on the rigid body and is a natural extension of the recently developed algorithm for coupling compressible flow and deformable bodies [1–3]. The new algorithm advances the solution in the fluid domain with a standard upwind scheme and explicit time-stepping. The Newton–Euler system of ordinary differential equations governing the motion of the rigid body is augmented by added mass correction terms. This system, which is very stiff for light bodies, is solved with an A-stable diagonally implicit Runge–Kutta scheme. The implicit system (there is one independent system for each body) consists of only 3d+d2 scalar unknowns in d=2 or d=3 space dimensions and is fast to solve. The overall cost of the scheme is thus dominated by the cost of the explicit fluid solver. Normal mode analysis is used to prove the stability of the approximation for a one-dimensional model problem and numerical computations confirm these results. In multiple space dimensions the approach naturally reveals the form of the added mass tensors in the equations governing the motion of the rigid body. These tensors, which depend on certain surface integrals of the fluid impedance, couple the translational and angular velocities of the body. Numerical results in two space dimensions, based on the use of moving overlapping grids and adaptive mesh refinement, demonstrate the behavior and efficacy of the new scheme. These results include the simulation of the difficult problems of shock impingement on an ellipse and a more complex body with appendages, both with zero mass.

A low-Mach number fix for Roe’s approximate Riemann solver

We present a low-Mach number fix for Roe’s approximate Riemann solver (LMRoe). As the Mach number Ma tends to zero, solutions to the Euler equations converge to solutions of the incompressible equations. Yet, standard upwind schemes do not reproduce this convergence: the artificial viscosity grows like 1/Ma, leading to a loss of accuracy as Ma → 0. With a discrete asymptotic analysis of the Roe scheme we identify the responsible term: the jump in the normal velocity component Δ U of the Riemann problem. The remedy consists of reducing this term by one order of magnitude in terms of the Mach number. This is achieved by simply multiplying Δ U with the local Mach number. With an asymptotic analysis it is shown that all discrepancies between continuous and discrete asymptotics disappear, while, at the same time, checkerboard modes are suppressed. Low Mach number test cases show, first, that the accuracy of LMRoe is independent of the Mach number, second, that the solution converges to the incompressible limit for Ma → 0 on a fixed mesh and, finally, that the new scheme does not produce pressure checkerboard modes. High speed test cases demonstrate the fall back of the new scheme to the classical Roe scheme at moderate and high Mach numbers.

Continuous elliptic and multi-dimensional hyperbolic Darcy-flux finite-volume methods

Elliptic and hyperbolic Darcy-flux approximations are presented. Families of flux-continuous finite-volume methods are investigated for the elliptic full-tensor pressure equation with general discontinuous coefficients. Full pressure continuity across control-volume interfaces is built into the methods leading to an important distinction from the earlier pointwise continuous methods. The families of quasi-positive methods significantly reduce spurious oscillations (induced by earlier schemes) in discrete pressure solutions for strongly anisotropic full-tensor fields. Anisotropy favoring triangulation and non-linear flux splitting are also shown to be effective for computing solutions free of spurious oscillations. Multi-dimensional upwind schemes that reduce cross-wind numerical diffusion induced by the standard upwind scheme are also presented for hyperbolic Darcy-flux approximation.

On the dissipation mechanism of upwind-schemes in the low Mach number regime: A comparison between Roe and HLL

It is well known that standard upwind schemes for the Euler equations face a number of problems in the low Mach number regime: stiffness, cancellation and accuracy problems. A new aspect of the accuracy problem, presented in this paper, is the dependence on the type of flux solver: while the accuracy of the HLL scheme massively decreases for Ma → 0 on a given triangular mesh, the Roe scheme remains accurate, i.e. flows of arbitrarily small Mach numbers can – at least in principle – be simulated on a fixed triangular mesh. We give an asymptotic analysis of this phenomenon and present a number of numerical results.

The influence of cell geometry on the accuracy of upwind schemes in the low mach number regime

It is well known, that standard upwind schemes for the Euler equations face a number of problems in the low Mach number regime: stiffness, cancellation and accuracy problems. A new aspect, presented in this paper, is the dependence on the cell geometry: applied on a triangular grid, the accuracy problem disappears, i.e. flows of arbitrarily small Mach numbers can be simulated on a fixed mesh. We give an asymptotic analysis of this, up to date unknown, phenomenon for the first-order Roe scheme and present a number of numerical results.

False diffusion in numerical simulation of combustion processes in tangential-fired furnace

Numerical simulation serves as one of the most important tools for analyzing coal combustion in Tangentially Fired Furnaces (TFF) with NUMERICAL FALSE DIFFUSION as one key problem that degrades the simulation accuracy, especially for complex flow patterns. False diffusion often completely compromises the accuracy, leading to erroneous predictions. This paper reviews various methods to reduce the numerical diffusion. In computational fluid dynamics (CFD), false diffusion originates from a truncation error of the Taylor series approximation of the derivative and multidimensional discretization effects. Higher-order upwind convective schemes were designed to reduce truncation errors, while grid line adjusting methods were developed to reduce crossflow diffusion. This paper compares numerical and experimental results for isothermal flows to evaluate these methods. Results with the standard upwind scheme in a rectangular Cartesian mesh are compared with results in body-fitted meshes for comprehensive combustion processes in a TFF. Analysis of the false diffusion effect in the x, y, z directions and the artificial viscosity distribution in a rectangular mesh shows where the false diffusion overtakes the real physical diffusion and where the mesh must be refined or grid line must be adjusted to improve TFF combustion simulations.

On a convection–diffusion problem with a weak layer

In this article we consider a model linear convection–diffusion problem with a weak layer. We analyze the singular-perturbation nature of the problem and show that no special precautions are required to cope with the weak layer: a standard upwind scheme on a (quasi-)uniform mesh is sufficient. We give a simple analysis for the method. Thus highlighting that not all problems with a small parameter multiplying the highest-order derivative are suitable for studying boundary-layer phenomena.

Preconditioned iterative methods and finite difference schemes for convection–diffusion

We conduct experimental study on numerical solution of the two dimensional convection–diffusion equation discretized by three finite difference schemes: the traditional central difference scheme, the standard upwind scheme and the fourth-order compact scheme. We study the computed accuracy achievable by each scheme, the algebraic properties of the coefficient matrices arising from different schemes and the performance of the Gauss–Seidel iterative method, the preconditioned GMRES iterative method, and the multigrid method, for solving linear systems arising from these schemes.

A naturally upwinded conservative procedure for the incompressible Navier-Stokes equations on non-staggered grids

Upwind methods have been previously adopted in incompressible flow calculations in an attempt to eliminate the need for artificial viscosity while maintaining high accuracy. Standard upwind methods found in the literature are primarily concerned with accurate modeling of the convection terms and generally neglect the viscous terms in the analysis. A new procedure is developed here which seeks to include the viscous terms so that regions of significant gradients will not be over-dissipated. The approach taken here is an extension of the work of Refs [1–5]in which interpolating functions are derived from direct integration of linearized forms of the governing equations. The resulting functions are then used as interpolants for differencing the full equations. During the process, no attempt is made to deliberately upwind bias the differencing. However, because of the manner in which coefficients of the differencing are functions of the cell Reynolds number, the scheme introduces its own upwind bias. This property was first observed by Roscoe [1] in which the observation that this class of schemes preserves the hyperbolic/elliptic nature of the original system of equations was made. The present work yields, for the first time, a fully conservative Roscoe-type method. Although the method approaches the development of the finite difference equations in a different manner than traditional upwind procedures, it achieves higher accuracy in a manner analogous to the MUSCL approach [6] used in standard upwind schemes. The procedure also uses a unique discretization of the continuity equation which involves pressure terms as in Ref. [5]. This equation is fully conservative and satisfies mass conservation to machine zero. The new scheme is first demonstrated on the 1-D, viscous, Burger's equation and results are compared with a first and second-order Roe scheme. It is then applied to the 2-D, incompressible Navier-Stokes equations for non-staggered grids. Results for the driven cavity problem are compared with proven methods and are found to be in excellent agreement.

The midpoint upwind scheme

A modified upwind scheme is considered for a singularly perturbed two-point boundary value problem whose solution has a single boundary layer. The scheme is analyzed on an arbitrary mesh. It is then analyzed on a Shishkin mesh and precise convergence bounds are obtained, which show that the scheme is superior to the standard upwind scheme. A variant of the scheme on the same Shishkin mesh is proved to achieve even better convergence behaviour.

The Midpoint Upwind Scheme

Abstract A modified upwind scheme is considered for a singularly perturbed two-point boundary value problem whose solution has a single boundary layer. The scheme is analyzed on an arbitrary mesh. It is then analyzed on a Shishkin mesh and precise convergence bounds are obtained, which show that the scheme is superior to the standard upwind scheme. A variant of the scheme on the same Shishkin mesh is proved to achieve even better convergence behaviour.

Simulation of a simple reacting gas flow with an upwind scheme

Good results have been obtained using the Random Choice Method (RCM) in the computation of reacting gas flow problems. The RCM is an unfamiliar method and difficult to program. The question arises as to whether a simpler difference approximation can obtain as effective results with less computational difficulty. Among all difference schemes upwind methods have been proven to have excellent properties. Thus, such methods serve as models for the effectiveness of all difference schemes. A standard upwind scheme modified to include a fractional heat conduction step is used to compute solutions of one dimensional compressible fluid flow equations with a finite heat conduction coefficient. The gas is assumed to be chemically reacting and thus to deposit energy in the field. Comparison is made to the known qualitative behavior of the solutions for different ratios of the reaction rate and the heat conduction coefficient. This difference scheme is seen to compare unfavorably with the RCM.