High-order Central Difference Schemes Research Articles

On the use of high order central difference schemes for differential equation based wall distance computations

A computationally efficient high-order solver is developed to compute the wall distances by solving the relevant partial differential equations, namely: Eikonal, Hamilton–Jacobi (HJ) and Poisson equations. In contrast to the upwind schemes widely used in the literature, we explore the suitability of high-order central difference schemes (explicit/compact) for the wall-distance computation. While solving the Hamilton–Jacobi equation, the high-order central difference schemes performed approximately 1.4–2.8 times faster than the upwind schemes with a marginal improvement in the solution accuracy. A new pseudo HJ formulation based on the localized artificial diffusivity (LAD) approach has been proposed. It is demonstrated to predict results with an accuracy comparable to that of the Eikonal equation and the simulations are ≈ 1.5 times faster than the baseline HJ solver using upwind schemes. A curvature correction is also incorporated in the HJ equation to correct for the near-wall errors due to concave/convex wall curvatures. We demonstrate the efficacy of the proposed methods on both the steady and unsteady test cases and exploit the unsteady wall-distance solver to estimate the instantaneous shape and burning surface area of a dendrite propellant grain in a solid propellant rocket motor.

Numerical visualisation on applying a five-stage multi-scaled square turbulence-generating grid to generating turbulence

This study presents the numerical visualisation of turbulence generated by a five-stage square-type multi-scale turbulence grid. The present study applies a turbulence grid with an increased number of stages to five, in contrast to previous studies using multi-scale turbulence grids with four stages. For the present numerical visualisation, direct numerical analysis is used in this analysis. This analysis uses the high-order central difference schemes and the third-order Runge-Kutta scheme. In addition to the multi-scale turbulence grid with five stages, the present study also covers two turbulence grids with four stages for reference. By using the external force term in the governing equations, the present multi-scale turbulence grids were formed numerically in the computational domain. The spatial homogeneity of the downstream turbulence field is approached in this work. The turbulence generated by the turbulence grids with four stages is characterised by a flow structure with a doughnut-like instantaneous velocity figure in the downstream region. By examining contour diagrams of the streamwise instantaneous velocity, the present study can see that the spatial homogeneity of the turbulence field downstream is improved by increasing the number of stages for the turbulence grid.

FFT-Based High Order Central Difference Schemes for Poisson’s Equation with Staggered Boundaries

This work concerns with the development of fast and high order algorithms for solving a single variable Poisson’s equation with rectangular domains and uniform meshes, but involving staggered boundaries. Here the staggered boundary means that the boundary is located midway between two adjacent grid nodes. Due to the popularity of staggered grids in scientific computing for solving multiple variables partial differential equations (PDEs), the planned development deserves further studies, but is rarely reported in the literature, because grand challenges exist for spectral methods, compact finite differences, and fast Fourier transform (FFT) algorithms in handling staggered boundaries. A systematic approach is introduced in this paper to attack various open problems in this regard, which is a natural generalization of a recently developed Augmented Matched Interface and Boundary (AMIB) method for non-staggered boundaries. Formulated through immersed boundary problems with zero-padding solutions, the AMIB method combines arbitrarily high order central differences with the FFT inversion. Over staggered boundaries, the proposed AMIB method can handle Dirichlet, Neumann, Robin or any combination of boundary conditions. Convergence orders in four, six and eight are numerically validated for the AMIB method in both two and three dimensions. Moreover, the proposed AMIB method performs well for some challenging problems, such as low regularity solution near boundary, PDE solution not satisfying the boundary condition, and involving both staggered and non-staggered boundaries on two ends.

FFT-based high order central difference schemes for three-dimensional Poisson's equation with various types of boundary conditions

In this paper, a unified approach is introduced to implement high order central difference schemes for solving Poisson's equation via the fast Fourier transform (FFT). Popular high order fast Poisson solvers in the literature include compact finite differences and spectral methods. However, FFT-based high order central difference schemes have never been developed for Poisson problems, because with long stencils, central differences require fictitious nodes outside the boundary, which poses a challenge to integrate boundary conditions in FFT computations. To overcome this difficulty, several layers of exterior grid lines are introduced to convert the problem to an immersed boundary problem with zero-padding solutions beyond the original cubic domain. Over the boundary of the enlarged cubic domain, the anti-symmetric property is naturally satisfied so that the FFT fast inversion is feasible, while the immersed boundary problem can be efficiently solved by the proposed augmented matched interface and boundary (AMIB) method. As the first fast Poisson solver based on high order central differences, the AMIB method can be easily implemented in any dimension, due to its tensor product nature of the discretization. As a systematical approach, the AMIB method can be made to arbitrarily high order in principle, and can handle the Dirichlet, Neumann, Robin or any combination of boundary conditions. The accuracy, efficiency, and robustness of the proposed AMIB method are numerically validated by considering various Poisson problems in two and three dimensions.

High-order accurate large-eddy simulations of compressible viscous flow in cylindrical coordinates

Dynamic interactions of shock waves and turbulent structures occur in a wide range of applications. The accurate simulation of turbulence in these flows requires a numerical scheme with minimal dissipation whereas the shock-capturing requires a dissipative scheme to stabilise the solution in the shock-wave discontinuous regions. These contradictory requirements make simulations of these flows extremely challenging. A compatible implementation of the solid boundary conditions at the point of incepting the acoustic waves and their internalisation into hydrodynamic instabilities is also challenging.Here we present, a high-fidelity, massively parallel and accurate solver for both direct numerical simulations and large-eddy simulations of supersonic turbulent flows in cylindrical coordinates. A hybrid WENO/ high order central difference scheme is used for the spatial discretisation with a fourth-order five-stage 2N-storage Runge–Kutta for the time integration. The least square contraction of Lilly [1] is utilised for large-eddy subgrid-scale modelling. The solid surface boundary condition is implemented using the offset wall and ghost cell technique.The numerical implementation is validated through a wide range of test cases from simple to more complex cases. The numerical results show good agreement with analytical solutions and available experimental results. In the large-eddy simulation of a supersonic under-expanded impinging jet, it is observed that the conventional Ducros sensor leads to a more dissipative hybrid scheme. In this paper, a new shock sensor based on a WENO smoothness indicator is proposed and it is demonstrated to perform better especially in the simulation of the supersonic under-expanded impinging jet by limiting the expensive WENO scheme to the discontinuous regions and reducing the computational cost by approximately 10%.

High-order central difference scheme for Caputo fractional derivative

In this paper we propose a class of central difference schemes for resolving the Caputo fractional derivative. The accuracy may reach any selected integer order. More precisely, the Caputo fractional derivative operator is decomposed into symmetric and antisymmetric components. Starting from difference schemes of lower order accuracy for each component, we enhance the accuracy by a weighted average of shifted differences. The weights are calculated by matching the symbols of the scheme and the operators. We further illustrate the application of the proposed schemes to a fractional advection–diffusion equation. Together with the Crank–Nicolson algorithm, it reaches designed accuracy order, and is unconditionally stable. Numerical tests are presented to demonstrate the nice features.

Stability Analysis of the Inverse Lax–Wendroff Boundary Treatment for High Order Central Difference Schemes for Diffusion Equations

In this paper, high order central finite difference schemes in a finite interval are analyzed for the diffusion equation. Boundary conditions of the initial-boundary value problem are treated by the simplified inverse Lax–Wendroff procedure. For the fully discrete case, a third order explicit Runge–Kutta method is used as an example for the analysis. Stability is analyzed by both the Gustafsson, Kreiss and Sundstrom theory and the eigenvalue visualization method on both semi-discrete and fully discrete schemes. The two different analysis techniques yield consistent results. Numerical tests are performed to demonstrate and validate the analysis results.

Application of a Multi-dimensional Limiting Process to Central-Upwind Schemes for Solving Hyperbolic Systems of Conservation Laws

In this paper, we study semi-discrete central-upwind difference schemes with a modified multi-dimensional limiting process (MLP) to solve two-dimensional hyperbolic systems of conservation laws. In general, high-order central difference schemes for conservation laws involve no Riemann solvers or characteristic decompositions but have a tendency to smear linear discontinuities. To overcome this drawback of central-upwind schemes, we use a MLP that uses multi-dimensional information for slope limitation to control the oscillations across discontinuities for multi-dimensional applications. Some numerical results are provided to demonstrate the performance of the proposed scheme.

Consistent numerical diffusion terms for simulating compressible multicomponent flows

An interface-capturing method using a high-order central difference scheme is presented for simulations of compressible multicomponent flows. The present method adds consistent numerical diffusion terms to robustly capture interface discontinuities, while maintaining the velocity, pressure, and temperature equilibriums at interfaces. Analysis of the numerical errors generated at the interfaces leads to a proposal of new consistent numerical diffusion terms. The method solves a fully conservative form of the total mass, momentum, total energy, and species-mass, and an additional advection form of the specific heats ratio to preserve the pressure oscillation-free property at the interfaces. Several one- and two-dimensional problems are used to verify the proposed method.

Simulation of turbulent flows containing strong shocks

Simulation of turbulent flows with strong shocks is a computationally challenging problem. The requirements for a method to produce accurate results for turbulence are orthogonal to those needed to treat shocks properly. In order to prevent an unphysical rate of decay of turbulent structures, it is necessary to use a method with very low numerical dissipation. Because of this, central difference schemes are widely used. However, computing strong shocks with a central difference scheme can produce unphysical post-shock oscillations that corrupt the entire flow unless additional dissipation is added. This dissipation can be difficult to localize to the area near the shock and can lead to inaccurate treatment of the turbulence. Modern high-resolution shock-capturing methods usually use upwind algorithms to provide the dissipation necessary to stabilize shocks. However, this upwind dissipation can also lead to an unphysical rate of decay of the turbulence. This paper discusses a hybrid method for simulating turbulent flows with strong shocks that couples a high-order central difference scheme with a high-resolution shock-capturing method. The shock-capturing method is used only in the vicinity of discontinuities in the flow, whereas the central difference scheme is used in the remainder of the computational domain. Results of this new method will be shown for a variety of test problems. Preliminary results for a realistic application involving detonation in gas-particle flows will also be presented.

On the spurious solutions in the high-order finite difference methods for eigenvalue problems

In this paper, the origins of spurious solutions occurring in the high-order finite difference methods are studied. Based on a uniform mesh, spurious modes are found in the high-order one-sided finite difference discretizations of many eigenvalue problems. Spurious modes are classified as spectral pollution and non-spectral pollution. The latter can be partially avoided by mesh refinement, while the former persists when the mesh is refined. Through numerical studies of some prototype eigenvalue problems, such as those of the Helmholtz and beam equations, we show that perfect central differentiation schemes do not produce any spurious modes. Nevertheless, high-order central difference schemes encounter difficulties in implementing complex boundary conditions. We further show that the central difference schemes will produce spurious modes as well, if asymmetric approximation is involved in boundary treatments. In general, central difference schemes are less likely to produce spurious modes than one-sided difference schemes.

An Energy-Stable High-Order Central Difference Scheme for the Two-Dimensional Shallow Water Equations

An energy-stable high-order central finite difference scheme is derived for the two-dimensional shallow water equations. The scheme is mathematically formulated using the semi-discrete energy method for initial boundary value problems described in Olsson (1995, Math. Comput. 64, 1035---1065): after symmetrizing the equations via a change to entropy variables, the flux derivatives are entropy-split enabling the formulation of a semi-discrete energy estimate. We show experimentally that the entropy-splitting improves the stability properties of the fully discretized equations. Thus, the dependence on numerical dissipation to keep the scheme stable for long term time integrations is reduced relative to the original unsplit form, thereby decreasing non-physical damping of solutions. The numerical dissipation term used with the entropy-split equations is in a form which preserves the semi-discrete energy estimate. A random one-dimensional dam break calculation is performed showing that the shock speed is computed correctly for this particular case, however it is an open question whether the correct shock speed will be computed in general

Direct Numerical Simulation of Hypersonic Flow over a Cavity

AbstractAn open rectangular cavity has served as a generic geometry in shock tunnel experiments with laminar flow being two‐dimensional upstream of the cavity. A high order central difference scheme with flux‐corrected transport has been employed to simulate the flow under the assumption that the inflow profile upstream of the cavity is laminar. The high order in space enables to reproduce well vortical structures and possible transition to turbulent flow.

On the Artificial Compression Method for Second-Order Nonoscillatory Central Difference Schemes for Systems of Conservation Laws

The recently proposed high-order central difference schemes for conservation laws have a tendency of smearing linear discontinuities. In principle, Harten's artificial compression method (ACM) could be used to improve resolution. We analyze why this approach has not yet been used successfully and derive a more powerful version of the ACM based on a rigorous estimate of the total variation. We discuss the potential danger of overcompression and point out directions of future algorithmic development.

Designing an Efficient Solution Strategy for Fluid Flows: II. Stable High-Order Central Finite Difference Schemes on Composite Adaptive Grids with Sharp Shock Resolution

A simple and efficient solution strategy is designed for fluid flows governed by the compressible Euler equations. It is constructed from a stable high-order central finite difference scheme on structured composite adaptive grids. This basic framework is suitable for solving smooth flows on complicated domains and is easily extendible with extra tools to handle specific flow problems. The stable high-order central difference scheme is mathematically formulated using a recently derived semi-discrete energy method for initial-boundary value problems. The high order of accuracy reduces the number of grid points required in smooth parts of the flow which leads to efficiency in both computational time and memory. A local grid adaptation technique is used to increase the grid density where required. Extra tools are developed for the sharp resolution of shocks. The grids are refined in the shock regions to retain accuracy. On the fine grids in these regions, an effective scalar artificial viscosity term is added to suppress spurious oscillations generated by the high-order central difference method. The location and orientation of shocks is determined by an easy-to-implement wavelet-based detection algorithm. The overhead of the composite adaptive grid method and detection algorithm is negligible compared to the computational kernel. The local grid adaptation with the high-order scheme is shown to increase computational efficiency. The resolution of shocks is sharp.

Cartesian Boundary Treatment of Curved Walls for High-Order Computational Aeroacoustics Schemes

It is known that the use of high-order central difference schemes on a Cartesian grid is preferable for the computation of acoustic wave propagation problems. Those schemes tend to be less dispersive and dissipative than most other types of schemes. They are also more capable of providing an accurate wave speed. A Cartesian boundary treatment for problems involving the scattering of acoustic waves by solid objects with curved boundary surfaces, designed to be used in conjunction with such high-order central difference schemes, is proposed. The development of this method is based on the observation that a solid wall actually exerts a pressure force on the fluid to keep it from flowing across the wall surface. In this method, ghost values of pressure are introduced at mesh points adjacent to the solid boundary inside the object. The ghost values are then chosen so that the solid wall boundary condition is satisfied. The method is also applicable to objects with sharp corners. Numerical examples are provided.