Dgm-fd: a finite difference scheme based on the discontinuous galerkin method

DGM-FD: A finite difference scheme based on the discontinuous Galerkin method applied to wave propagation

T HE field of direct simulation of acoustic waves propagation in the presence of a fluid in motion holds high standards of requirements. This is especially the case when it comes to industrial computational aeroacoustics (CAA), where methods must be able to copewith realistic, complex geometries while involving a high order of accuracy and yet result in affordable solvers in terms of computational resources. In that context, different computational methods have emerged over the years. Resulting from various theoretical backgrounds and approaches, they naturally hold specific characteristics, advantages, and drawbacks. Among them, discontinuous Galerkin (DG)methods [1–4] and finite difference (FD)methods [5– 7] are widely spread. The main characteristics of DG and FD methods are roughly recalled in Table 1. DGmethods arewell adapted to take into account complex geometries as they can deal with unstructured meshes [4,8]. Furthermore, their formulation naturally allows local-order refinement and is well suited for parallel computing. On the other hand, FD methods preferably run on structured meshes and are particularly efficient on Cartesian grids. They are easier to implement and less demanding in terms of CPU resources yet show good numerical dispersion and dissipation properties. Both these methods have been extensively studied, successfully implemented, and applied to that range of problems in industrial configurations. Based on this, a new family of questions have been raised. In particular, the possibility of coupling solvers of different natures in order to locally take advantage of the qualities of each method has already been studied byUtzmann et al. [9–11] in the field of CAA. We will also focus on hybridization techniques of DG/FD schemes based on a computational domain-decomposition approach. A typical motivation behind this is to be able to approximate the solution in the close neighborhood of complex obstacles on an unstructured DG mesh and compute the rest of the field on a Cartesian FD grid in order to alleviate computational time and resources. The purpose of this study is to investigate natural questions raised by such a coupling in order to gather informations on the behavior of the resulting hybrid solver in precision and stability. In this paper, we introduce two two-dimensional (2-D) DG timedomain (DGTD)/FD time-domain (FDTD) hybridization algorithms in the context of the resolution of the linearized Euler equations (LEEs) in two different kinds of geometrical configurations. Our results mainly focus on validation aspects, driven by numerical experiments that were conducted on academic test cases, as theoretical results on the interaction of these two schemes presently seem out of reach in a general case. We also present the behavior of our hybrid solver in the context of an acoustic benchmark problem that consists in the diffraction of an acoustic source by complex-geometries obstacles. The paper is organized as follows. In Sec. II, we present our physical modeling and both the FDTD andDGTD schemes that are used in this study.We also recall their compared properties, advantages, and drawbacks. In Sec. III, after presenting the numerical and theoretical issues raised by the design of a DG/FD coupled solver, we focus on our hybridization strategy and its numerical validation. In Sec. IV, we present an application of our hybrid solver to a complex-geometry acoustic test case, which was proposed at the Fourth CAA Workshop on Benchmark Problems [12].

A high order optimized upwind Disperion-RelationPreserving (DRP) finite difference scheme is developed and implemented for multi-dimensional acoustic problems. A sequence of numerical simulations of acoustic wave propagation problems is carried out to evaluate the robustness and accuracy of the optimized upwind DRP scheme and its potential for solving complex aeroacoustics problems. The results of the upwind DRP scheme are compared with that of the central DRP scheme and the analytic solutions whenever it is possible. It is concluded from the current investigations that the optimized upwind DRP scheme not only can be successfully implemented to multi-dimensional aeroacoustic computations, but can accurately predict acoustics wave propagation without adding an artificial damping term. Introduction In contrast to Computational Fluid Dynamics (CFD), which has advanced to a fairly mature state, Computational Aeroacoustics (CAA) has only recently emerged as a separate area of study. Although Aeroacoustics problems are governed by the same equations as those in aerodynamics, acoustic waves have their own characteristics which makes the computation challenging. Acoustic waves are inherently unsteady, and their amplitudes are several order smaller than the mean flow and their frequencies are generally very high. These require that computational schemes be high order in both space and time with least dispersion and dissipation [1,2]. Many CFD schemes such as MacCormack scheme, upwind schemes and ENO schemes etc., have been extended to high order using more stencil points and applied to the computations of acoustic problems [3-7]. Many compact and noncompact optimized schemes [8-12] including Tam and Webb's Dispersion-Relation-Preserving (DRP) scheme, which were designed for the linear acoustic waves, were recently reviewed by Zigg [13]. It has been shown that for waves with high wavenumbers (short waves) the optimized schemes require less grid points per wavelength (PPW) than traditional high order CFD methods. This property of requiring less PPW is essential in CAA, since large computational domain is usually required. Most optimized schemes, however, are restricted to central difference algorithms. This restriction inevitably leads to stability problems which mu5t be dealt with through the use of filters or explicit dissipation terms. Although the deliberate filter or dissipation terms are proved quite successful in many acoustic problems [3,9], they are problem-dependent and require the prior knowledge of the problems. Upwind schemes have been widely used in CFD and have been shown very efficient and robust. Upwind schemes ensure that waves propagate in * current address: CFD Research Corporation, Huntsville, AL Copyright© 1998, American Institute of Aeronautics and Astronautics, Inc. the correct physical direction. With the build-in dissipation upwind schemes automatically damp out high wavenumbers component of solution. Since the dissipation does not distinguish the spurious waves from real acoustic waves, upwind schemes are necessary to be optimized such that the dissipation has little effect on large range of acoustic waves. Because of the success of the optimized upwind DRP scheme in solving one-dimensional aeroacoustics problems [14], the objectives of the current investigations are aimed at (1) to develop and to implement the optimized upwind DRP scheme to multi-dimensional aeroacoustics problems, (2) to evaluate the accuracy and robustness of the scheme and its potential for solving complex aeroacoustics problems. Problem Formulation 1. The Optimized upwind DRP Scheme Optimized shcemes preserve wave propagation characteristics for relative large range of wave numbers and requires less grid points per wavelength (PPW). They are usually constructed by optimizing the finite difference approximations of the space and time derivatives in the wave and frequency space [8,13]. Consider the approximation of the first spatial derivative du/dxby the finite difference equation, which is given for a uniform grid of spacing Ax. Suppose M values of u to the right and N values of u to the left of the point x are used in the finite difference equation where x is a continuous variable, i.e. du(x) 1 jAx) 0) The Fourier transform and its inverse of a function are related by dx (2)

Purpose Elastohydrodynamic lubrication (EHL) commonly occurs in highly stressed tribological pairs of mechanical components, such as rolling gears and bearings, where it reduces the friction and wear between contact surfaces. For example, EHL theory has been applied in railway engineering to analyze the wheel–rail contact behavior of high-speed trains under water-lubricated conditions. The combination of high contact pressures and water’s low viscosity significantly influences both elastic deformation and the numerical convergence of the Reynolds equation. Therefore, a robust and accurate numerical method for EHL contact problems is essential. Design/methodology/approach In this paper, a high-order finite difference method is proposed to solve the EHL line and point contact problems, whose cavitation conditions are treated by the penalty method. The highly nonlinear equations resulting from the high-order finite difference discretization are solved by the trust-region dogleg algorithm. A high-order biased upwind finite difference scheme is also presented in order to reduce the numerical dissipation and dispersion arising from the high-order upwind finite difference scheme. Findings Numerical examples demonstrate that this method achieves more accurate solutions using fewer nodes compared to other numerical methods. Furthermore, the biased upwind finite difference scheme has better accuracy than the upwind one. Originality/value In this paper, high-order centered and (biased) upwind finite difference methods combined with a state-of-the-art nonlinear solver, i.e. the trust-region dogleg algorithm, are constructed to solve the EHL line and point contact problems.

In the present work, the performance of four finite difference formulations for the steady-state, two-dimensional partial differential equations for fluid flows were examined. The finite difference formulations were the: i.) upwind finite difference scheme (UDS) ii.) hybrid central/upwind finite difference scheme (CUDS) iii.) quadratic, upstream-weighted finite difference scheme (QUDS) iv.) hybrid central/skewed-upwind finite difference scheme (CSUDS) The schemes (QUDS) and (CSUDS) proved to be superior to the two former discretization schemes when applied to the convective terms of the fluid flow equations to yield numerical predictions of backward-facing step flows. With both the (QUDS) and the (CSUDS) scheme, grid independence of the flow results was achieved with a largely reduced number of grid points. Also, numerical diffusion, was strongly reduced with both schemes and it is shown in the paper that the results obtained with both the (QUDS) and the (CSUDS) scheme are in close agreement.

New very high-order upwind multi-layer compact (MLC) schemes with spectral-like resolution for flow simulations

In this paper, we consider a second-order singularly perturbed differential-difference equations with mixed delay and advance parameters. At first, we approximate the model problem by an upwind finite difference scheme on a Shishkin mesh. We know that the upwind scheme is stable and its solution is oscillation free, but it gives lower order of accuracy. So, to increase the convergence, we propose a hybrid finite difference scheme, in which we use the cubic spline difference method in the fine mesh regions and a midpoint upwind scheme in the coarse mesh regions. We establish a theoretical parameter uniform bound in the discrete maximum norm. To check the efficiency of the proposed methods, we consider test problems with delay, advance and the mixed parameters and the results are in agreement with our theoretical findings.

On two upwind finite-difference schemes for hyperbolic equations in non-conservative form

Local defect correction techniques : analysis and application to combustion

This paper presents the development of the hybrid method of mimetic finite difference (MFD) and discontinuous Galerkin (DG) for polymer flooding and an in-depth study of its computational performance. The proposed hybrid method, simply denoted by MFD-DG method, initially discretizes pressure equations using the MFD method to calculate the pressure distribution in the computational domain, thereby obtaining the total two-phase flow velocities on each grid edge. Subsequently, the DG method is employed to calculate the water saturation distribution, with the result reconstructed by a slope limiter, and then the same approach is used to obtain the polymer concentration distribution. Three numerical examples involving one-dimensional (1D) problem, two-dimensional problems with anisotropic full-tensor permeability as well as complex geometry are conducted to test the performance of the MFD-DG method in the polymer flooding model. Results show that compared with classical methods like the upwind difference method and the finite volume method, the MFD-DG method proposed here has significantly higher computational accuracy. In the 1D case, its L2 error is an order of magnitude smaller, and it can achieve higher-resolution water-drive and polymer-displacement fronts. Moreover, it has a higher convergence order, which makes the accuracy improvement more prominent as the grid size reduces, highlighting the need to replace traditional methods with the MFD-DG method for polymer flooding simulation. The examples also verify that the MFD-DG method can be effectively applied to scenarios with heterogeneous full-tensor permeability distributions and complex grids.

In this chapter, finite difference (FD) approximations and the method of lines, which combine FD with available time integrators, are discussed. First, a convection diffusion-reaction PDE is used to introduce a few basic FD schemes and addresses the concept of stability of the numerical scheme. As a rule of thumb, centered FD schemes appear as good choices for approximating second-order (diffusion) operators, whereas upwind FD schemes are preferable to first-order (convection) operators. Once stability is ensured, accuracy can be adjusted by selecting appropriately the fineness of the spatial grid and the order of the FD approximation (i.e. the stencil of points on which the FD approximation is built). The calculation of FD approximation can be conveniently organized using differentiation matrices, which are easy to manipulate in MATLAB, SCILAB, or OCTAVE. FD can also be efficiently computed on arbitrarily spaced grids using an algorithm due to Fornberg. This chapter continues with a presentation of different methods to take the boundary conditions into account. Boundary conditions are an essential part of the IBVP definition, and several methods for translating the BCs in the code implementation are presented. Finally, attention is paid to the computation of the Jacobian matrix of the ODE system, which is used by various solvers.

The Reynolds-averaged Navier-Stokes equations with the equations of the k-.epsilon. turbulence model are solved numerically in a general curvilinear system for a three-dimensional turbulent flow around an Ahmed body. The simulation is especially aimed at the evaluation of three finite differencing schemes for the convection term, which include the upwind differencing scheme(UDS), the second order upwind differencing scheme(SOU scheme) and the QUICK scheme. The drag coefficient, the velocity and pressure fields are found to be changed considerably with the adopted finite differencing schemes. It is clearly demonstrated that the large difference between computation and experiment in the drag coefficient is due to relatively high predicted values of pressure drag from both front part and vertical rear end base. The results also show that the simulation with the QUICK or SOU scheme predicts fairly well the flow field and gives more accurate drag coefficient than other finite differencing scheme.

A nodal immersed finite element-finite difference method

High order WENO and DG methods for time-dependent convection-dominated PDEs: A brief survey of several recent developments

Modified finite volume methods of Osher and Chakravarthy (MOC) and Sanders and Li (MSL) and a finite element method of Lin and Chin are investigated on several aeroacoustic problems. Higher order accuracy is obtained with the monotonic upwind-centered scheme for conservation laws approach. The tested problems include oblique shock reflection, linear wave convection, monopole radiation, vortex preservation, and blade-vortex interaction. Based on the order of accuracy, stability, grid nonuniformity, and dissipation property of each scheme, it is concluded that the MOC scheme is the most suitable scheme among the schemes tested for aeroacoustic computations. We also conclude that the MSL scheme needs to be improved on problems of convergence and small wiggles before it is used in computational aeroacoustics. In the blade-vortex interaction problem, two sound waves, transonic and compressibility waves, found in recent experiments are simulated. N recent years, considerable progress has been made in the numerical analysis of fluid dynamics. Usually, numerical methods which solve the Euler/Navier-Stokes equations for aerodynamic flows fall into three major classes: finite difference, finite volume, and finite element methods. Recently, successful methods have employed higher order upwind interpolations and limiter functions to obtain algorithms possessing higher resolution and higher stability bounds. The inherently dissipative nature of upwind schemes and limiter functions is beneficial at or near shock waves. Specifically, the implementation of limiter functions make schemes more stable for computing solutions with strong shock waves. In general, an upwind scheme with a limiter function is formally second- or thirdorder accurate but is locally first-order accurate in regions of high localized gradients (such as shocks) and in many cases at local extrema also. The loss of formal accuracy near shocks is not usually serious, but for aeroacoustic calculations the dissipation at extrema will be harmful. Upwind schemes may be classified into two classes: monotonic upwind-centered scheme for conservation laws (MUSCL) and nonMUSCL.1'4 The present work is focused on the MUSCL-based schemes. We have studied the performance of the following three basic schemes: 1) the finite element method of Lin and Chin (LC)5 and Lin et al.,6 2) the finite volume method of Osher and Chakravarthy (OC)7, and 3) the finite volume method of Sanders and Li (SL). 8'9 The popularity of the OC scheme for aeroacoustic computations is likely to increase since it is formally of third order and is stable in the computation of strong shock waves. The SL scheme is formally of fourth order and is a suitable method for the aeroacoustic computation of low-speed flows. An overall assessment of this scheme is available in Refs. 8 and 9. The LC scheme is formally of second order for both uniform and nonuniform meshes. This scheme has been extensively tested on both inviscid and viscous flows.5'6 It has been shown that the scheme is capable of computation of steady and unsteady flows. In this paper, we have modified the limiter functions in the OC and SL schemes to obtain two new schemes, the modified OC (MOC) and the modified SL (MSL) schemes. In this effort a detailed investigation is performed to evaluate the capability of those schemes. Several test problems are studied, including oblique shock reflection, linear wave convection, monopole radiation, vortex preservation, and blade-vortex interaction. Details