Second-order Accurate Solutions Research Articles

Study on a grid interface algorithm for solutions of incompressible Navier–Stokes equations

Grid interface treatment is a crucial issue in solving unsteady, three-dimensional, incompressible Navier–Stokes equations by domain decomposition methods. Recently, a mass flux based interpolation (MFBI) interface algorithm was proposed for Chimera grids [Tang HS, Jones SC, Sotiropoulos F. An overset grid method for 3D unsteady incompressible flows. J Comput Phys, 2003;191:567–600] and it has been successfully applied to a variety of flows. MFBI determines velocity and pressure at grid interfaces by mass conservation and interpolation, and it is easy to implement. Compared with the commonly used standard interpolation, which directly interpolates velocity as well as pressure, the proposed interface algorithm gives fewer solution oscillations and faster convergence rates. This paper makes a study on MFBI. Starting with discussions about grid connectivity, it is shown that MFBI is second-order accurate for mass flux across grid interface. It is also derived that the scheme provides second-order accuracy for momentum flux. In addition, another version of MFBI is presented. At last, numerical examples are presented to demonstrate that MFBI honors mass flux balance at grid interfaces and it leads to second-order accurate solutions.

A Cartesian grid embedded boundary method for the heat equation and Poisson’s equation in three dimensions

We present an algorithm for solving Poisson’s equation and the heat equation on irregular domains in three dimensions. Our work uses the Cartesian grid embedded boundary algorithm for 2D problems of Johansen and Colella [A Cartesian grid embedded boundary method for Poisson’s equation on irregular domains, J. Comput. Phys. 147(2) (1998) 60–85] and extends work of McCorquodale, Colella and Johansen [A Cartesian grid embedded boundary method for the heat equation on irregular domains, J. Comput. Phys. 173 (2001) 620–635]. Our method is based on a finite-volume discretization of the operator, on the control volumes formed by intersecting the Cartesian grid cells with the domain, combined with a second-order accurate discretization of the fluxes. The resulting method provides uniformly second-order accurate solutions and gradients and is amenable to geometric multigrid solvers.

Residual distribution for general time-dependent conservation laws

We consider the second-order accurate numerical solution of general time-dependent hyperbolic conservation laws over unstructured grids in the framework of the Residual Distribution method. In order to achieve full conservation of the linear, monotone and first-order space-time schemes of (Csik et al., 2003) and (Abgrall et al., 2000), we extend the conservative residual distribution (CRD) formulation of (Csik et al., 2002) to prismatic space-time elements. We then study the design of second-order accurate and monotone schemes via the nonlinear mapping of the local residuals of linear monotone schemes. We derive sufficient and necessary conditions for the well-posedness of the mapping. We prove that the schemes obtained with the CRD formulation satisfy these conditions by construction. Thus the nonlinear schemes proposed in this paper are always well defined. The performance of the linear and nonlinear schemes are evaluated on a series of test problems involving the solution of the Euler equations and of a two-phase flow model. We consider the resolution of strong shocks and complex interacting flow structures. The results demonstrate the robustness, accuracy and non-oscillatory character of the proposed schemes. d schemes.

Drag and pressure fields for the MHD flow around a circular cylinder at intermediate Reynolds numbers

Steady incompressible flow around a circular cylinder in an external magnetic field that is aligned with fluid flow direction is studied for Re (Reynolds number) up to 40 and the interaction parameter in the range 0 ≤ N ≤ 15 (or 0 ≤ M ≤ 30), where M is the Hartmann number related to N by the relation , using finite difference method. The pressure‐Poisson equation is solved to find pressure fields in the flow region. The multigrid method with defect correction technique is used to achieve the second‐order accurate solution of complete nonlinear Navier‐Stokes equations. It is found that the boundary layer separation at rear stagnation point for Re = 10 is suppressed completely when N < 1 and it started growing again when N ≥ 9. For Re = 20 and 40, the suppression is not complete and in addition to that the rear separation bubble started increasing when N ≥ 3. The drag coefficient decreases for low values of N(<0.1) and then increases with increase of N. The pressure drag coefficient, total drag coefficient, and pressure at rear stagnation point vary with . It is also found that the upstream and downstream pressures on the surface of the cylinder increase for low values of N(<0.1) and rear pressure inversion occurs with further increase of N. These results are in agreement with experimental findings.

Optimal Simple and Implementable Monetary and Fiscal Rules

The goal of this paper is to compute optimal monetary and fiscal policy rules in a real business cycle model augmented with sticky prices, a demand for money, taxation, and stochastic government consumption. We consider simple policy rules whereby the nominal interest rate is set as a function of output and inflation, and taxes are set as a function of total government liabilities. We require policy to be implementable in the sense that it guarantees uniqueness of equilibrium. We do away with a number of empirically unrealistic assumptions typically maintained in the related literature that are used to justify the computation of welfare using linear methods. Instead, we implement a second-order accurate solution to the model. Our main findings are: First, the size of the inflation coefficient in the interest-rate rule plays a minor role for welfare. It matters only insofar as it affects the determinacy of equilibrium. Second, optimal monetary policy features a muted response to output. More importantly, interest rate rules that feature a positive response of the nominal interest rate to output can lead to significant welfare losses. Third, the optimal fiscal policy is passive. However, the welfare losses associated with the adoption of an active fiscal stance are negligible.

Discretization of Integral Equations Describing Flow in Nonprismatic Channels with Uneven Beds

Application of the finite-volume method in one dimension for open channel flow predictions mandates the direct discretization of integral equations for mass conservation and momentum balance. The integral equations include source terms that account for the forces due to changes in bed elevation and channel width, and an exact expression for these source term integrals is presented for the case of a trapezoidal channel cross section whereby the bed elevation, bottom width, and inverse side slope are defined at cell faces and assumed to vary linearly and uniformly within each cell, consistent with a second-order accurate solution. The expressions may be used in the context of any second-order accurate finite-volume scheme with channel properties defined at cell faces, and it is used here in the context of the Monotone Upwind Scheme for Conservation Laws (MUSCL)-Hancock scheme which has been adopted by many researchers. Using these source term expressions, the MUSCL-Hancock scheme is shown to preserve stationarity, accurately converge to the steady state in a frictionless flow test problem, and perform well in field applications without the need for upwinding procedures previously reported in the literature. For most applications, an approximate, point-wise treatment of the bed slope and nonprismatic source terms can be used instead of the exact expression and, in contrast to reports on other finite-volume-based schemes, will not cause unphysical oscillations in the solution.

Use of a residual distribution Euler solver to study the occurrence of transonic flow in Wells turbine rotor blades

The aim of the present study is to investigate the occurrence of transonic flow in several cascade geometries and blade sections that have been considered in the design of Wells turbine rotor blades. The calculations were performed using an implicit Euler solver for two-dimensional flow. The numerical method uses a multi-dimensional upwind matrix residual distribution scheme formulated on a new symmetrized form of the Euler equations, both in time and in space, that decouples the entropy and the enthalpy equations. Second-order accurate steady-state solutions where obtained using a compact three-point stencil. The results show that unwanted transonic flow may occur in the turbine rotor at relatively low mean-flow Mach numbers.

Finite-Volume Model for Shallow-Water Flooding of Arbitrary Topography

A model based on the finite-volume method is developed for unsteady, two-dimensional, shallow-water flow over arbitrary topography with moving lateral boundaries caused by flooding or recession. The model uses Roe's approximate Riemann solver to compute fluxes, while the monotone upstream scheme for conservation laws and predictor-corrector time stepping are used to provide a second-order accurate solution that is free from spurious oscillations. A robust, novel procedure is presented to efficiently and accurately simulate the movement of a wet/dry boundary without diffusing it. In addition, a new technique is introduced to prevent numerical truncation errors due to the pressure and bed slope terms from artificially accelerating quiescent water over an arbitrary bed. Model predictions compare favorably with analytical solutions, experimental data, and other numerical solutions for one- and two-dimensional problems.

Finite Volume Model for Nonlevel Basin Irrigation

A finite volume model for unsteady, two-dimensional, shallow water flow is developed and applied to simulate the advance and infiltration of an irrigation wave in two-dimensional basins of complex topography. The fluxes are computed with Roe's approximate Riemann solver and the monotone upstream scheme for conservation laws is used in conjunction with predictor-corrector time-stepping to provide a second-order accurate solution. Flux-limiting is implemented to eliminate spurious oscillations and the model incorporates an efficient and robust scheme to capture the wetting and drying of the soil. Model predictions are compared with experimental data for one- and two-dimensional problems involving rough, impermeable, and permeable beds, including a poorly leveled basin.

A simple and accurate solution for calculating stresses in conical shells

In this paper, new variable transformation formulas are introduced to solve the basic governing differential equations for conical shells. By performing magnitude order analysis and neglecting the quantities with h/ R magnitude order, the basic governing differential equations for conical shells are transformed into a second-order differential equation with complex constant coefficients. By solving this second-order differential equation, a simple and accurate solution for conical shells is derived. The present solution is simpler than the exact solution because it does not use Bessel's functions, and also more accurate than the equivalent cylinder solution. Numerical examples are given to illustrate this conclusion. The simple and accurate solution provides a quick means for analyzing stresses in conical shells.

Hydrodynamics of Turbid Underflows. I: Formulation and Numerical Analysis

A mathematical model is developed for unsteady, two-dimensional, single-layer, depth-averaged turbid underflows driven by nonuniform, noncohesive sediment. The numerical solution is obtained by a high-resolution, total variation diminishing, finite-volume numerical model, which is known to capture sharp fronts accurately. The monotone upstream scheme for conservation laws is used in conjunction with predictor-corrector time-stepping to provide a second-order accurate solution. Flux-limiting is implemented to prevent the development of spurious oscillations near discontinuities. The model also possesses the capability to track the evolution and development of an erodible bed, due to sediment entrainment and deposition. This is accomplished by solving a bed-sediment conservation equation at each time step, independent of the hydrodynamic equations, with a predictor-corrector method. The model is verified by comparison to experimental data for currents driven by uniform and nonuniform sediment.

Finite-difference approximation for the velocity-vorticity formulation on staggered and non-staggered grids

In this paper, we study the finite-difference discretizations of the velocity-vorticity formulation for the Navier-Stokes equations on several grid layouts. The Cauchy-Riemann relations between the velocity and the vorticity (which imply that the velocity and the vorticity are discretely divergence-free) are satisfied discretely on one of the grids, conditionally satisfied on another grid, but violated on the rest of the grids. Using the asymptotic error analysis and numerical examples, we formally show that second-order accurate solutions can be obtained on all the grids if appropriate boundary conditions are imposed, provided that the exact solution is sufficiently smooth. For the staggered Mark-and-Cell (MAC) grid, we show that the ‘first-order’ reflection boundary condition is sufficient for obtaining a second-order accurate solution. We also show that the accuracy deteriorates to first order on the fully non-staggered grid if the reflection boundary condition is used.

A multi-step computation scheme: Decoupling kinetic processes from physical transport in water quality models

This paper presents a computation scheme that considers accuracy and efficiency in the numerical treatment of the kinetic processes in water quality models. The proposed scheme involves decoupling of physical transport and kinetic processes in the governing mass-balance equations. The decoupled equations are solved by employing multi-step computation, in which the kinetic processes are applied for the first half time step, followed by the physical transport for a full time step, and the kinetic processes are applied again for the remaining second half time step. The kinetic equations are solved numerically or analytically, which may require prior linearization of the equations. The numerical solution scheme of the kinetic equations alternates between explicit and implicit schemes, and is equivalent to the second-order accurate Crank-Nicolson solution. A hypothetical model, which is simple enough to have an analytical solution, is used to explain the concept, and to demonstrate the accuracy and computational efficiency of the proposed scheme. The proposed scheme and three other traditional solution schemes are compared to the analytical solution. The proposed scheme shows better accuracy and computational efficiency compared to other schemes.

Implicit time integration of a class of constrained hybrid formulations—Part I: Spectral stability theory

Incomplete field formulations have recently been the subject of intense research because of their potential in coupled analysis of independently modeled substructures, adaptive refinement, domain decomposition and parallel processing. This paper presents a spectral stability theory for the differential/algebraic dynamic systems associated with these formulations, discusses the design and analysis of suitable time-integration algorithms, and emphasizes the treatment of the inter-subdomain linear constraint equations. These constraints are shown to introduce a destabilizing effect in the dynamic system that can be analyzed by investigating the behavior of the time-integration algorithm at infinite and zero frequencies. Three different approaches for constructing penalty-free unconditionally stable second-order accurate solution procedures for this class of hybrid formulations are presented, analyzed and illustrated with numerical examples. In particular, the advantages of the Hilber-Hughes-Taylor (HHT) method and its generalized version (Generalized a) are highlighted. The family of problems discussed in this paper can also be viewed as model problems for the more general case of hybrid formulations with non-linear constraints. For example, it is shown numerically in this paper that the theoretical results predicted by the spectral stability theory also apply to non-linear multibody dynamics formulations. Therefore, some of the algorithms outlined in this work are important alternatives to the popular technique consisting of transforming differential/algebraic equations into ordinary differential equations via the introduction of a stabilization term that depends on arbitrary constants, and that influences the computed solution.

Numerical computations of steady homenthalpic non-equilibrium flows

An upwind finite-volume approximation scheme applicable to finite-element-type unstructured grids is proposed for the solution of steady, inviscid non-equilibrium flow when it can be assumed that the total specific enthalpy is constant throughout the entire domain. This is the case, in particular, for external non-radiating flows in which the freestream conditions are constant. In the present approach, the energy equation in differential form is replaced by its first integral which plays the role of a known algebraic constraint. The wellposeness of the restriction of the van Leer and Steger-Warming flux-vector splittings to this case is proved assuming the equivalent-γ can locally be treated as a constant. The efficiency of the proposed method is assessed by numerical experiments related to first-order and MUSCL-type second-order accurate solutions to hypersonic flow problems.

Support-Operator Finite-Difference Algorithms for General Elliptic Problems

An algorithm is developed for discretizing boundary-value problems given by a general linear elliptic second order partial-differential equation with general mixed or Robin boundary conditions in general logically rectangular grids. The continuum problem can be written as an operator equation where the operator is self adjoint and positive definite. The discrete approximations have the same property. Consequently, the matrices for the discrete problem are symmetric and positive definite. Also, the scheme has a nearest neighbor stencil. Consequently, the most powerful linear solvers can be applied. In smooth grids, the algorithm produces second-order accurate solutions. It is the generality of the problem (general matrix coefficients, general boundary conditions, general logically rectangular grids) that makes finding such an algorithm difficult. The algorithm, which is a combination of the method of support operators and the mapping method, overcomes certain difficulties of the individual methods, producing a high-quality algorithm for solving general elliptic problems.

A DECOMPOSITION FINITE DIFFERENCE METHOD FOR THE FOURTH ORDER ACCURATE SOLUTION OF POISSON'S EQUATION ON GENERAL REGIONS

We present a new, rapid, and easily parallelized finite difference method for computing fourth-order accurate solutions of Poisson's equation, Δu=f, on irregular regions. This method is a generalization of a method due to Qun and Tao for obtaining fourth-order accurate solutions of Poisson's equation on rectangular regions by forming a linear combination of two second-order accurate solutions obtained by calculations with different stencils.In our method we first embed the irregular region in a larger rectangular region and define a discontinuous extension. Using a method we presented previously, we could compute two second-order accurate approximations using the same two second-order accurate stencils as Qun and Tao did. However, because the functions we compute, and therefore the truncation errors, are discontinuous, taking the same linear combination of the two solutions will not provide a fourth-order accurate solution. We show how one can correct for these discontinuities in truncation error by local computations, and how one can obtain a fourth-order accurate solution. We also provide results of numerical experiments.

Entrance and Stationary Roughness Effects on the Load Carrying Capacity of a Wide Wedge Gas Bearing

The objective of this work is the determination of the effects of surface roughness amplitude and inlet conditions on ultra-thin, compressible, isothermal, infinitely-wide gas bearings. The method of study is numerical in nature and consists of a second-order accurate finite difference solution of the Reynolds equation of lubrication without molecular slip for a range of bearing numbers spanning six orders of magnitude. The motivation for this work comes from the magnetic disk drive industry where ever decreasing head flying heights are being sought to increase the recording density. Past studies by these authors of the infinitely-wide air bearing with stationary roughness have shown that, unlike in the case of a smooth bearing, the load peaks at a finite bearing number comparable to that at which current rigid disk drives operate. This suggests that it may be possible to arrange the roughness pattern in such a way as to cause the slider to separate from the hard disk more rapidly, minimizing wear to both surfaces. A wedge bearing with stationary sinusoidal roughness is studied for different roughness amplitudes and two phases. It is shown that for all configurations considered, the load exhibits a peak unlike in the case of the smooth bearing where the load monotonically reaches a peak at an infinite gas bearing number. Two rough sliders with flat tapers smoothly attached to their leading edge are also studied to answer questions regarding the role that the inlet condition plays in the resulting magnitude and behavior of the generated load. The leading taper allows the bearing to dynamically determine the entrained flow rate and maximum pressure as well as to self prescribe the inlet condition at the leading edge of the first roughness wave. The inlet conditions prescribed by the developing flow in the flat taper region still give rise to a peak in the load. The addition of the smooth taper, however, causes an overall decrease (increase) in the load when the roughness waves are entirely above (below) the plane of the taper compared to the results of the rough bearing with no taper. It is demonstrated that all the considered roughness patterns result in a peak load at a finite bearing number. Of special interest are two bearing configurations: one composed of a smooth taper followed by a transversely roughened slider and the other is a rough slider with a transverse roughness pattern whose slope at the inlet of the bearing is negative. Both are shown to achieve a maximum lifting force at low bearing numbers, providing rapid separation while alleviating the narrow rail manufacturing problem.

The Numerical Solution of Second-Order Boundary Value Problems on Nonuniform Meshes

In this paper, we examine the solution of second-order, scalar boundary value problems on nonuniform meshes. We show that certain commonly used difference schemes yield second-order accurate solutions despite the fact that their truncation error is of lower order. This result illuminates a limitation of the standard stability, consistency proof of convergence for difference schemes defined on nonuniform meshes. A technique of reducing centered-difference approximations of first-order systems to discretizations of the underlying scalar equation is developed. We treat both vertex-centered and cell-centered difference schemes and indicate how these results apply to partial differential equations on Cartesian product grids.

The numerical solution of second-order boundary value problems on nonuniform meshes

In this paper, we examine the solution of second-order, scalar boundary value problems on nonuniform meshes. We show that certain commonly used difference schemes yield second-order accurate solutions despite the fact that their truncation error is of lower order. This result illuminates a limitation of the standard stability, consistency proof of convergence for difference schemes defined on nonuniform meshes. A technique of reducing centered-difference approximations of first-order systems to discretizations of the underlying scalar equation is developed. We treat both vertex-centered and cell-centered difference schemes and indicate how these results apply to partial differential equations on Cartesian product grids.