High-order Accurate Solution Research Articles

Trigonometric WENO Schemes for Hyperbolic Conservation Laws and Highly Oscillatory Problems

In this paper, we use trigonometric polynomial reconstruction, instead of algebraic polynomial reconstruction, as building blocks for the weighted essentially non-oscillatory (WENO) finite difference schemes to solve hyperbolic conservation laws and highly oscillatory problems. The goal is to obtain robust and high order accurate solutions in smooth regions, and sharp and non-oscillatory shock transitions. Numerical results are provided to illustrate the behavior of the proposed schemes. AMS subject classifications: 65M06, 65M99, 35L65

Accuracy preserving limiter for the high-order accurate solution of the Euler equations

Higher-order finite-volume methods have been shown to be more efficient than second-order methods. However, no consensus has been reached on how to eliminate the oscillations caused by solution discontinuities. Essentially non-oscillatory (ENO) schemes provide a solution but are computationally expensive to implement and may not converge well for steady-state problems. This work studies the extension of limiters used for second-order methods to the higher-order case. Requirements for accuracy and efficient convergence are discussed. A new limiting procedure is proposed. Ringleb’s flow problem is used to demonstrate that nearly nominal orders of accuracy for schemes up to fourth-order can be achieved in smooth regions using the new limiter. Results for the fourth-order accurate solution of transonic flow demonstrates good convergence properties and significant qualitative improvement of the solution relative the second-order method. The new limiter can also be successfully applied to reduce the dissipation of second-order schemes with minimal sacrifices in convergence properties relative to existing approaches.

Prediction of thermal transients in district heating systems

A prediction of thermal transients in district heating systems is important in order to adjust in an energy efficient manner the loads of heat power plants and pump stations with dynamic consumption needs during changes of the outside air temperature, the wind intensity, the solar radiation, or system start-ups and shut-downs. A model and corresponding computer code are developed with the aim to simulate the thermal transients in district heating systems. They are based on the high-order accurate numerical solution of the transient energy equation, and the hydraulic prediction of pressure and fluid flow rates within the complex pipe network. Thermal transients caused by an increase and decrease of the heat power plant load are simulated for real operating conditions of the district heating system. Predicted temperature front propagations show a good agreement with data measured at three consumer substations located in different parts of the district heating network and at different distances from the heat plant. The developed computational tool provides reliable information about time periods of temperature fronts propagation and heat distribution from the heat source to consumers within the whole district heating network.

Krylov subspace spectral methods for the time-dependent Schrödinger equation with non-smooth potentials

This paper presents modifications of Krylov Subspace Spectral (KSS) Methods, which build on the work of Gene Golub and others pertaining to moments and Gaussian quadrature to produce high-order accurate approximate solutions to the time-dependent Schrödinger equation in the case where either the potential energy or the initial data is not a smooth function. These modifications consist of using various symmetric perturbations to compute off-diagonal elements of functions of matrices. It is demonstrated through analytical and numerical results that KSS methods, with these modifications, achieve the same high-order accuracy and possess the same stability properties as they do when applied to parabolic problems, even though the solutions to the Schrödinger equation do not possess the same smoothness.

An implicit high-order hybridizable discontinuous Galerkin method for linear convection–diffusion equations

We present a hybridizable discontinuous Galerkin method for the numerical solution of steady and time-dependent linear convection–diffusion equations. We devise the method as follows. First, we express the approximate scalar variable and corresponding flux within each element in terms of an approximate trace of the scalar variable along the element boundary. We then define a unique value for the approximate trace by enforcing the continuity of the normal component of the flux across the element boundary; a global equation system solely in terms of the approximate trace is thus obtained. The high number of globally coupled degrees of freedom in the discontinuous Galerkin approximation is therefore significantly reduced. If the problem is time-dependent, we discretize the time derivative by means of backward difference formulae. This results in efficient schemes capable of producing high-order accurate solutions in space and time. Indeed, when the time-marching method is ( p + 1 ) th order accurate and when polynomials of degree p ⩾ 0 are used to represent the scalar variable, the flux and the approximate trace, we observe that the approximations for the scalar variable, the flux and the trace of the scalar variable converge with the optimal order of p + 1 in the L 2 -norm. Finally, we introduce a simple element-by-element postprocessing scheme to obtain new approximations of the flux and the scalar variable. The new approximate flux, which has a continuous inter-element normal component, is shown to converge with order p + 1 in the L 2 -norm. The new approximate scalar variable is shown to converge with order p + 2 in the L 2 -norm. For the time-dependent case, the postprocessing does not need to be applied at each time-step but only at the times for which an enhanced solution is required. Moreover, the postprocessing procedure is less expensive than the solution procedure, since it is performed at the element level. Extensive numerical results are presented to demonstrate the convergence properties of the method.

High resolution and high order accurate solution of complex turbulent flow field in jet elements

In view of the flow field of jet elements had the complex shock wave and the vortex characteristic. One kind of strongly compact—weighted ENO scheme is used for solving this kind of complex flow field. This form is applied to the Reynolds averaged Navier—Stokes equations, its convection term uses the third and fourth order, turbulence model uses Realizable κ–e model, which has calculated 7 kinds of operating conditions. Numerical results indicated that using this scheme may succeed in capturing to Coanda effect, namely if the total pressure under the limiting value, departure of flow will occur when there is no control flow. The circumfluence and the separation of boundary can be seen distinctly, and the high order accurate complex shock wave and the vortex can be caught from the calculating results. Moreover, the numerical results are in good agreement with the experimental ones, which demonstrates that this scheme has feasibility and validity. This is essential to guide the design of jet elements.

Three-point explicit compact difference scheme with arbitrary order of accuracy and its application in CFD

Based on the successive iteration in the Taylor series expansion method, a three-point explicit compact difference scheme with arbitrary order of accuracy is derived in this paper. Numerical characteristics of the scheme are studied by the Fourier analysis. Unlike the conventional compact difference schemes which need to solve the equation to obtain the unknown derivatives in each node, the proposed scheme is explicit and can achieve arbitrary order of accuracy in space. Application examples for the convection-diffusion problem with a sharp front gradient and the typical lid-driven cavity flow are given. It is found that the proposed compact scheme is not only simple to implement and economical to use, but also is effective to simulate the convection-dominated problem and obtain high-order accurate solution in coarse grid systems.

On the Use of High-Order Accurate Solutions of PNS Schemes as Basic Flows for Stability Analysis of Hypersonic Axisymmetric Flows

Abstract High-order accurate solutions of parabolized Navier–Stokes (PNS) schemes are used as basic flow models for stability analysis of hypersonic axisymmetric flows over blunt and sharp cones at Mach 8. Both the PNS and the globally iterated PNS (IPNS) schemes are utilized. The IPNS scheme can provide the basic flow field and stability results comparable with those of the thin-layer Navier–Stokes (TLNS) scheme. As a result, using the fourth-order compact IPNS scheme, a high-order accurate basic flow model suitable for stability analysis and transition prediction can be efficiently provided. The numerical solution of the PNS equations is based on an implicit algorithm with a shock fitting procedure in which the basic flow variables and their first and second derivatives required for the stability calculations are automatically obtained with the fourth-order accuracy. In addition, consistent with the solution of the basic flow, a fourth-order compact finite-difference scheme, which does not need higher derivatives of the basic flow, is efficiently implemented to solve the parallel-flow linear stability equations in intrinsic orthogonal coordinates. A sensitivity analysis is also conducted to evaluate the effects of numerical dissipation and grid size and also accuracy of computing the basic flow derivatives on the stability results. The present results demonstrate the efficiency and accuracy of using high-order compact solutions of the PNS schemes as basic flow models for stability and transition prediction in hypersonic flows. Moreover, indications are that high-order compact methods used for basic-flow computations are sensitive to the grid size and especially the numerical dissipation terms, and therefore, more careful attention must be kept to obtain an accurate solution of the stability and transition results.

Application of a three-point explicit compact difference scheme to the incompressible navier-stokes equations

A three-point explicit compact difference scheme with high order of accuracy for solving the unsteady incompressible Navier-Stokes equations was presented. Numerical solutions are obtained for the model problem of lid-driven cavity flow and are compared with benchmark solutions found in the literature. It is discovered that the proposed three point explicit compact scheme is not only simple to implement and economical to use, but also is effective to obtain high-order accurate solution in coarse grid systems.

Large-Eddy Simulation of a Wing Tip Vortex on Overset Grids

A multiblock large-eddy simulation (LES) code with overset grid capability is reported on that has been developed primarily to study the tip vortex noise problem arising on rotating wind-turbine blades. The LES code can also be used to simulate the tip vortices around nonrotating wing geometries typically found in aeronautical applications. It employs state-of-the-art compact finite differencing, implicit spatial filtering, and nonreflecting boundary conditions on free boundaries, as well as characteristic-type boundary conditions on solid walls. High-order-accurate interpolation is used for the transfer of information among the individual component grids that make up the overset grid topology. The multiblock nature and the overset grid capability of the code allow high-order accurate numerical solutions in complex domains. Results for the tip vortex around a nonrotating wing with a rounded tip are presented to demonstrate the capability of the code. Comparisons of the numerical results with experimental data are also carried out.

High Order Accurate Solution of Flow Past a Circular Cylinder

A high order method is applied to time-dependent incompressible flow around a circular cylinder geometry. The space discretization employs compact fourth-order difference operators. In time we discretize with a second-order semi-implicit scheme. A large linear system of equations is solved in each time step by a combination of outer and inner iterations. An approximate block factorization of the system matrix is used for preconditioning. Well posed boundary conditions are obtained by an integral formulation of boundary data including a condition on the pressure. Two-dimensional flow around a circular cylinder is studied for Reynolds numbers in the range 7 ? R ? 180. The results agree very well with the data known from numerical and experimental studies in the literature.

High order accurate solution of the incompressible Navier–Stokes equations

High order methods are of great interest in the study of turbulent flows in complex geometries by means of direct simulation. With this goal in mind, the incompressible Navier–Stokes equations are discretized in space by a compact fourth order finite difference method on a staggered grid. The equations are integrated in time by a second order semi-implicit method. Stable boundary conditions are implemented and the grid is allowed to be curvilinear in two space dimensions. The method is extended to three dimensions by a Fourier expansion. In every time step, a system of linear equations is solved for the velocity and the pressure by an outer and an inner iteration with preconditioning. The convergence properties of the iterative method are analyzed. The order of accuracy of the method is demonstrated in numerical experiments. The method is used to compute the flow in a channel, the driven cavity and a constricted channel.

High‐order accurate numerical solutions of incompressible flows with the artificial compressibility method

AbstractA high‐order accurate, finite‐difference method for the numerical solution of incompressible flows is presented. This method is based on the artificial compressibility formulation of the incompressible Navier–Stokes equations. Fourth‐ or sixth‐order accurate discretizations of the metric terms and the convective fluxes are obtained using compact, centred schemes. The viscous terms are also discretized using fourth‐order accurate, centred finite differences. Implicit time marching is performed for both steady‐state and time‐accurate numerical solutions. High‐order, spectral‐type, low‐pass, compact filters are used to regularize the numerical solution and remove spurious modes arising from unresolved scales, non‐linearities, and inaccuracies in the application of boundary conditions. The accuracy and efficiency of the proposed method is demonstrated for test problems. Copyright © 2004 John Wiley & Sons, Ltd.

A Three-Dimensional Parallel Discontinuous Galerkin Solver for Acoustic Propagation Studies

The paper presents recent developments of a computational code for the numerical investigation of acoustic propagation. The code solves the three-dimensional linear Euler equations using a Discontinuous Galerkin (DG) method for the spatial discretization and an explicit high-order low-storage Runge-Kutta method for advancing the solution in time. Thanks to DG discretization, high-order accurate numerical solutions on arbitrary unstructured hybrid grids have been easily computed. The code has been parallelized using MPI and preliminary results on a small 10-processor Linux cluster seem very promising.

A toolkit for numerical simulation of PDEs: I. Fundamentals of generic finite-volume simulation

To understand underlying physical phenomena or to design better devices or processes, experts in various application areas within science and engineering often need to solve partial differential equations numerically. These experts have tremendous knowledge about the physical phenomena they study, but often are significantly less knowledgeable about scientific computing. A scientific computing toolkit for generic solution of PDEs would be a great benefit for such workers, requiring them only to specify the physics of their problem, with numerical issues handled by the toolkit. This paper presents just such a toolkit, based on the finite-volume method, and argues that the finite-volume method is a more user-friendly choice for such a toolkit than the finite-element method. The user specifies problem physics by providing code snippets to compute interior and boundary fluxes, source terms, and initial conditions; and by specifying constraints on the solution at the boundaries. The toolkit addresses all of the strictly numerical issues: it recovers high-order accurate solution and gradient data from the control volume averaged solution; enforces boundary conditions; integrates the user-supplied fluxes and source terms; and performs time advance. Examples are given for advection–diffusion, solid mechanics, and compressible flow problems to demonstrate the flexibility of the Advanced Numerical Simulation Library framework.

Stability and accuracy of differential quadrature method in solving dynamic problems

In this paper, the differential quadrature method is used to solve dynamic problems governed by second-order ordinary differential equations in time. The Legendre, Radau, Chebyshev, Chebyshev–Gauss–Lobatto and uniformly spaced sampling grid points are considered. Besides, two approaches using the conventional and modified differential quadrature rules to impose the initial conditions are also investigated. The stability and accuracy properties are studied by evaluating the spectral radii and truncation errors of the resultant numerical amplification matrices. It is found that higher-order accurate solutions can be obtained at the end of a time step if the Gauss and Radau sampling grid points are used. However, the conventional approach to impose the initial conditions in general only gives conditionally stable time step integration algorithms. Unconditionally stable algorithms can be obtained if the modified differential quadrature rule is used. Unfortunately, the commonly used Chebyshev–Gauss–Lobatto sampling grid points would not generate unconditionally stable algorithms.

On an adaptive monotonic convection—diffusion flux discretization scheme

The present paper concerns numerical investigation of a two-dimensional steady convection—diffusion scalar equation. Specific attention is given to resolving spurious oscillations in a confined region of high gradient. In smooth regions, a high-order accurate solution is desired, while in regions containing a sharp gradient, the strategy of applying a monotonic capturing scheme is preferred. In this paper, we model fluxes by means of a finite element model which has spaces spanned by Legendre polynomials. The propensity to yield an irreducible diagonal-dominant stiffness matrix is an attribute of this finite element flux discretization scheme. Fundamental analyses are extensively conducted in this paper in two areas. First, by conducting a modified equation analysis, we are led to confirm the consistency of the scheme. Both the stability and monotonicity of the solutions are also addressed. A guide for judging whether the stiffness matrix can yield a monotonic solution is rooted in the theory of the M-matrix. This monotonicity study provides us with greater numerical insight into the importance of the chosen Peclet number. Beyond its critical value, oscillatory solutions are present in the flow. This monotonic region is, however, fairly restricted. It is this shortcoming which limits the finite element practioner's choices for fairly small grid size. The scope of application using the proposed upwind scheme is, thus, rather small because monotonic solutions are prohibitively expensive to compute. Circumvention of such deficiency is accomplished by making modifications to the structured grid. Several test cases have been employed to examine the grid adaption technique for the treatment of advection terms in flow regions having sharp gradients. Considerable savings in computer storage and execution time are achieved by the employed unstructured grid setting.

Continuous finite difference approximations for solving differential equations

In recent papers continuous finite difference (FD) approximations have been developed for the solution of the initial value problem (ivp) for first order ordinary differential equations (odes). They provide dense output of accurate solutions and global error estimates for the ivp economically. In this paper we show that some continuous FD formulae can be used to provide a uniform treatment of both the ivp and the boundary value problem (bvp) without using the shooting method for the latter. Higher order accurate solutions can be obtained on the same meshes with constant spacing used by one-step method without using the iterated deferred correction technique. No additional conditions are required to ensure low order continuity and this leads to fewer necessary equations than those required by most of the popular methods for bvps. No quadratures are involved in this non-overlapping piecewise continuous polynomial technique. Some computed results are given to show the effectiveness of the proposed method and global error estimates.

Computational Considerations for the Simulation of Shock-Induced Sound

The numerical study of aeroacoustic problems places stringent demands on the choice of a computational algorithm because it requires the ability to propagate disturbances of small amplitude and short wavelength. The demands are particularly high when shock waves are involved because the chosen algorithm must also resolve discontinuities in the solution. The extent to which a high-order accurate shock-capturing method can be relied upon for aeroacoustics applications that involve the interaction of shocks with other waves has not been previously quantified. Such a study is initiated in this work. A fourth-order accurate essentially nonoscillatory (ENO) method is used to investigate the solutions of inviscid, compressible flows with shocks. The design order of accuracy is achieved in the smooth regions of a steady-state, quasi-one-dimensional test case. However, in an unsteady test case, only first-order results are obtained downstream of a sound-shock interaction. The difficulty in obtaining a globally high-order accurate solution in such a case with a shock-capturing method is demonstrated through the study of a simplified, linear model problem. Some of the difficult issues and ramifications for aeroacoustic simulations of flows with shocks that are raised by these results are discussed.

High-Order Accurate Discontinuous Finite Element Solution of the 2D Euler Equations

This paper deals with a high-order accurate discontinuous finite element method for the numerical solution of the Euler equations. The method combines two key ideas which are at the basis of the finite volume and of the finite element method, the physics of wave propagation being accounted for by means of Riemann problems and accuracy being obtained by means of high-order polynomial approximations within elements. We focus our attention on two-dimensional steady-state problems and present higher order accurate (up to fourth-order) discontinuous finite element solutions on unstructured grids of triangles. In particular we show that, in the presence of curved boundaries, a meaningful high-order accurate solution can be obtained only if a corresponding high-order approximation of the geometry is employed. We present numerical solutions of classical test cases computed with linear, quadratic, and cubic elements which illustrate the versatility of the method and the importance of the boundary condition