Order Spatial Accuracy Research Articles

A time marching strategy for solving parabolic and elliptic equations with Neumann boundary conditions

In the community of computational fluid dynamics, pressure Poisson equation with Neumann boundary condition is usually encountered when solving the incompressible Navier–Stokes equations in a segregated approach such as SIMPLE, PISO, and projection methods. To deal with Neumann boundary conditions more naturally and to retain high order spatial accuracy as well, a sixth-order accurate combined compact difference scheme developed on staggered grids (NSCCD6) is adopted to solve the parabolic and elliptic equations subject to Neumann boundary conditions. The staggered grid system is usually used when solving the incompressible Navier–Stokes equations. By adopting the combined compact difference concept, there is no need to discretize Neumann boundary conditions with one-sided discretization scheme which is of lower accuracy order. The conventional Crank–Nicolson scheme is applied in this study for temporal discretization. For two-dimensional cases, D’yakonov alternating direction implicit scheme is adopted. A newly proposed time step changing strategy is adopted to improve convergence rate when solving the steady state solutions of the parabolic equation. High accuracy order of the currently proposed NSCCD6 scheme for one- and two-dimensional cases are shown in this article.

Parameter-uniform fractional step hybrid numerical scheme for 2D singularly perturbed parabolic convection–diffusion problems

This article focuses on developing and analyzing an efficient numerical scheme for solving two-dimensional singularly perturbed parabolic convection–diffusion initial-boundary value problems exhibiting a regular boundary layer. For approximating the time derivative, we use the Peaceman–Rachford alternating direction implicit method on uniform mesh and for the spatial discretization, a hybrid finite difference scheme is proposed on a special rectangular mesh which is tensor-product of piecewise-uniform Shishkin meshes in the spatial directions. We prove that the numerical scheme converges uniformly with respect to the perturbation parameter $$\varepsilon $$ and also attains almost second-order spatial accuracy in the discrete supremum norm. Finally, numerical results are presented to validate the theoretical results. In addition to this, numerical experiments are conducted to demonstrate the effect of the time-dependent boundary conditions in the order of convergence numerically by introducing the classical evaluation of the boundary data; and also the improvement in the spatial order of accuracy of the present method by considering the Bakhvalov–Shishkin mesh in the spatial directions.

Direct Numerical Simulation of Flow over Periodic Hills up to Re H = 10 , 595 $\text {Re}_{H}= 10{,}595$

We present fully resolved computations of flow over periodic hills at the hill-Reynolds numbers $\text {Re}_{H}= 5{,}600$ and $\text {Re}_{H}= 10{,}595$ with the highest fidelity to date. The calculations are performed using spectral incompressible discontinuous Galerkin schemes of $8^{\text {th}}$ and $7^{\text {th}}$ order spatial accuracy, $3^{\text {rd}}$ order temporal accuracy, as well as 34 and 180 million grid points, respectively. We show that the remaining discretization error is small by comparing the results to h- and p-coarsened simulations. We quantify the statistical averaging error of the reattachment length, as this quantity is widely used as an ‘error norm’ in comparing numerical schemes. The results exhibit good agreement with the experimental and numerical reference data, but the reattachment length at $\text {Re}_{H}= 10{,}595$ is predicted slightly shorter than in the most widely used LES references. In the second part of this paper, we show the broad range of capabilities of the numerical method by assessing the scheme for underresolved simulations (implicit large-eddy simulation) of the higher Reynolds number in a detailed h/p convergence study.

New polyhedral discretisation methods applied to the Richards equation: CDO schemes in Code_Saturne

This paper shows an application of a vertex-based Compatible Discrete Operator (CDO) scheme to simulate the Richards equation on meshes using polyhedral cells. Second order spatial accuracy is reached for a verification test case using two meshes made of different types of polyhedral cells. A second validation test case dealing with a variably saturated soil inside a vertical column has been simulated, with three advected passive pollutants being released. Results are in good agreement with the reference for both types of mesh. MPI scalability has also been assessed at scale up to 98,304 cores of a Cray X30 for a 1.7 B cell mesh for the simulation of the aforementioned case, and nearly linear speed-up is observed. Finally the implementation of hybrid MPI-OpenMP has been tested on 2 types of Intel Xeon Phi Knights Landing co-processors, showing that the best configuration for the code is obtained when all the physical cores are filled by MPI tasks and the 4 hyperthreads by OpenMP threads.

Lattice Boltzmann Simulation of Transient Natural Convection in a Staggered Cavity with Four Vertically Heated Walls

The work in this manuscript deals with the numerical simulation of natural convection in a staggered cavity with the help of a recently developed two-dimensional double Multiple-Relaxation-Time (MRT) thermal Lattice Boltzmann method (LBM). In the last decade, there has been a rapid rise in the development of Lattice Boltzmann methods. However, its application in the simulation of natural convection from a staggered cavity has been carried out for the first time in this study. A careful undermining into the existing literature of heat and mass transfer reveal that study of natural convections in cross-sectional cavities is notably absent. Therefore, in this manuscript, we attempt to review the recently developed method and tried to analyze its implementation on natural convection in a staggered cavity with four differentially heated vertical walls. The problem geometry has eight boundaries. It is a staggered cavity with adiabatic horizontal walls and differentially heated vertical walls. The flow inside the thermally driven staggered cavity has been carefully studied for Rayleigh numbers 103, 104 and 105. The velocity and pressure boundary conditions are determined by a non-equilibrium extrapolation rule. As no benchmark results are available in the literature for this relatively new problem, we carry out its simulation with the help of a yet another well established scheme. This scheme is a higher-order compact (HOC) scheme with fourth order spatial accuracy and second order temporal accuracy. Our results show that there is a very good agreement between both these methods which exemplifies the accuracy and credibility of our results.

Analyses and implications of higher order finite volume methods on first-order macroscopic traffic flow models

Selection of suitable numerical scheme for solving macroscopic traffic flow models should be based on the numerical errors in various traffic parameters. Although such errors associated with state variables such as density are known to some extent, their implications to the derived parameters such as travel time and queue-length are not understood well. This paper proposes a general basis for selecting the spatial order of accuracy of a finite volume scheme for solving first-order traffic flow models. Four Measures of Accuracy that are especially relevant for traffic flow have been introduced. We illustrate that, although higher order schemes outperform the lower order ones in the quality of simulation in terms of resolution as well as computational cost, their superior performance is underestimated with the use of conventional error measures of state predictions. Localized parameters such as size and duration of the queue are more sensitive towards the order of the scheme, than domain aggregated parameters such as duration of congestion and travel time.

Alterations of Cowl Lip for the Improvement of Supersonic-Intake Performance

This paper discusses the performance enhancement of supersonic air intake model through the implementation of blunted leading edge to the cowl lip section of the model. A supersonic air intake model with sharp cowl leading edge is initially considered to numerically investigate its performance. Mach 3, supersonic intake flow through the base model has been simulated using commercial CFD package Ansys Fluent-15. Comparison of numerical predictions and experimental measurements is presented to demonstrate the correctness and accuracy of numerical frame work followed in the present study. Higher order spatial accuracy of the solver along with suitably refined mesh helped in accurate capturing of the flow field. Modification to the cowl lip is proposed as an effective method to improve the performance of the supersonic air intake. Two different blunted cowl leading edge geometries were investigated to identify the possible enhancement in performance parameters. Improvement in mass capture and combustion stability attained through the use of forward shifted blunt cowl leading edge is presented. It is also revealed through the present study that the blunt cowl leading edge can reduce the intensity of shock wave boundary layer interaction occurring at the isolator entry section. Deviation in total pressure recovery and flow distortion observed with different supersonic air intake models are also discussed with reasons for the same. This study demonstrates the scope of overall improvement in scramjet engine performance through the use of suitably positioned blunt cowl leading edge.

Flux-Conservative Hermite Methods for Simulation of Nonlinear Conservation Laws

A new class of Hermite methods for solving nonlinear conservation laws is presented. While preserving the high order spatial accuracy for smooth solutions in the existing Hermite methods, the new methods come with better stability properties. Artificial viscosity in the form of the entropy viscosity method is added to capture shocks.

Numerical simulation with the second order compact approximation of first order derivative for the modified fractional diffusion equation

In this paper, based on the second order compact approximation formula of first order derivative, the numerical simulation method with second order temporal accuracy and fourth order spatial accuracy is developed to simulate the modified fractional diffusion equation; the convergence, stability and solvability of the numerical simulation method are analyzed by Fourier analysis and algebraic theory, respectively; the theoretical results extremely consistent with the numerical experiment.

Numerical solution of the wave equation with variable wave speed on nonconforming domains by high-order difference potentials

We solve the wave equation with variable wave speed on nonconforming domains with fourth order accuracy in both space and time. This is accomplished using an implicit finite difference (FD) scheme for the wave equation and solving an elliptic (modified Helmholtz) equation at each time step with fourth order spatial accuracy by the method of difference potentials (MDP). High-order MDP utilizes compact FD schemes on regular structured grids to efficiently solve problems on nonconforming domains while maintaining the design convergence rate of the underlying FD scheme. Asymptotically, the computational complexity of high-order MDP scales the same as that for FD.

Fluid–structure interaction solver for compressible flows with applications to blast loading on thin elastic structures

In this study, we report the development and application of a fluid–structure interaction (FSI) solver for compressible flows with large-scale flow-induced deformation of the structure. The FSI solver utilizes a partitioned approach to strongly couple a sharp interface immersed boundary method-based flow solver with an open-source finite-element structure dynamics solver. The flow solver is based on a higher-order finite-difference method using a Cartesian grid, where it employs the ghost-cell methodology to impose boundary conditions on the immersed boundary. Higher-order accuracy near the immersed boundary is achieved by combining the ghost-cell approach with a weighted least squares error method based on a higher-order approximate polynomial. We present validations for two-dimensional canonical acoustic wave scattering on a rigid cylinder at a low Mach number and for flow past a circular cylinder at a moderate Mach number. The second order spatial accuracy of the flow solver was established in a grid refinement study. The structural solver was validated according to a canonical elastostatics problem. The FSI solver was validated based on comparisons with published measurements and simulations of the large-scale deformation of a thin elastic steel panel subjected to blast loading in a shock tube. The solver correctly predicted the oscillating behavior of the tip of the panel with reasonable fidelity and the computed shock wave propagation was qualitatively consistent with the published results. In order to demonstrate the fidelity of the solver and to investigate the coupled physics of the shock–structure interaction for a thin elastic plate, we employed the solver to simulate a 6.4 kg TNT blast loading on the thin elastic plate. The initial conditions for the blast were taken from previously reported field tests. Using numerical schlieren, the shock front propagation, Mach reflection, and vortex shedding at the tip of the plate were visualized during the impact of the shock wave on the plate. We discuss the coupling between the nonlinear dynamics of the plate and blast loading. The plate oscillates under the influence of blast loading and the restoration of elastic forces. The time-varying displacement of the tip of the plate is the superimposition of two dominant frequencies, which correspond to the first and second modes of the natural frequency of a vibrating plate. The effects of the material properties and length of the plate on the flow-induced deformation are briefly discussed. The proposed FSI solver is a versatile computational tool for simulating the impact of a blast wave on thin elastic structures and the results presented in this study may facilitate the design of thin structures subjected to realistic blast loadings.

A memory-efficient staining algorithm in 3D seismic modelling and imaging

The staining algorithm has been proven to generate high signal-to-noise ratio (S/N) images in poorly illuminated areas in two-dimensional cases. In the staining algorithm, the stained wavefield relevant to the target area and the regular source wavefield forward propagate synchronously. Cross-correlating these two wavefields with the backward propagated receiver wavefield separately, we obtain two images: the local image of the target area and the conventional reverse time migration (RTM) image. This imaging process costs massive computer memory for wavefield storage, especially in large scale three-dimensional cases. To make the staining algorithm applicable to three-dimensional RTM, we develop a method to implement the staining algorithm in three-dimensional acoustic modelling in a standard staggered grid finite difference (FD) scheme. The implementation is adaptive to the order of spatial accuracy of the FD operator. The method can be applied to elastic, electromagnetic, and other wave equations. Taking the memory requirement into account, we adopt a random boundary condition (RBC) to backward extrapolate the receiver wavefield and reconstruct it by reverse propagation using the final wavefield snapshot only. Meanwhile, we forward simulate the stained wavefield and source wavefield simultaneously using the nearly perfectly matched layer (NPML) boundary condition. Experiments on a complex geologic model indicate that the RBC-NPML collaborative strategy not only minimizes the memory consumption but also guarantees high quality imaging results. We apply the staining algorithm to three-dimensional RTM via the proposed strategy. Numerical results show that our staining algorithm can produce high S/N images in the target areas with other structures effectively muted.

Numerical simulation with high order accuracy for the time fractional reaction–subdiffusion equation

In this paper, we propose the numerical simulation method with second order temporal accuracy and fourth order spatial accuracy for the time fractional reaction–subdiffusion equation; the stability, convergence and solvability of the numerical simulation method respectively are discussed by Fourier analysis and algebraic theory; the theoretical analysis results very consistent with the numerical experiment.

The Time Second Order Mass Conservative Characteristic FDM for Advection–Diffusion Equations in High Dimensions

We propose in this paper a time second order mass conservative algorithm for solving advection–diffusion equations. A conservative interpolation and a continuous discrete flux are coupled to the characteristic finite difference method, which enables using large time step size in computation. The advection–diffusion equations are first transformed to the characteristic form, for which the integration over the irregular tracking cells at previous time level is proposed to be computed using conservative interpolation. In order to get second order in time solution, we treat the diffusion terms by taking the average along the characteristics and use high order accurate discrete flux that are continuous at tracking cell boundaries to obtain mass conservative solution. We demonstrate the second order temporal and spatial accuracy, as well as mass conservation property by comparing results with exact solutions. Comparisons with standard characteristic finite difference methods show the excellent performance of our method that it can get much more stable and accurate solutions and avoid non-physical numerical oscillation.

A compact finite difference scheme for variable order subdiffusion equation

In this paper, we consider a variable order time subdiffusion equation. A Crank-Nicolson type compact finite difference scheme with second order temporal accuracy and fourth order spatial accuracy is presented. The stability and convergence of the scheme are strictly proved by using the discrete energy method. Finally, some numerical examples are provided, the results confirm the theoretical analysis and demonstrate the effectiveness of the compact difference method.

An improved lattice Boltzmann method for simulating advective–diffusive processes in fluids

Lattice Boltzmann methods are widely used to simulate advective–diffusive processes in fluids. Lattice Bhatnagar–Gross–Krook methods presented in the literature mostly just exhibit first order spatial accuracy and introduce errors proportional to the velocity squared. Formulations proposed to alleviate this have only been partly successful and are valid only in certain specific situations. We present and demonstrate here a formulation that produces no such second order errors. This formulation suggests that a subtle, but important, adjustment is all it takes to improve the accuracy of the method. The key to the improved accuracy of this new model is the non-standard definition of the concentration that relates to the distribution function describing the advection–diffusion in lattice Boltzmann. The main advantage of the algorithm comes to view when simulating situations where fluid density variations appear. The present formulation of the advection–diffusion algorithm will, by taking into account these fluid density variations, drastically reduce the errors produced compared to the standard formulations. We also show how a source term is included in this new formulation without it losing its second order spatial accuracy.

Exact wavefield reconstruction on finite-difference grids with minimal memory requirements

Wavefield injection in finite-difference (FD) grids can be described by the method of multiple point sources. The method teaches how synthetically generated wavefields and wavefield constituents can be reconstructed from surface recordings using a combination of monopole and dipole sources on an injection surface surrounding the model. We show how to properly record surface wavefields and inject point sources in staggered FD grids, in a way that is consistent with the order of spatial accuracy of the FD scheme. The description is general and can be used for schemes of any order. Only one or two surface wavefields are required to reconstruct the original wavefields or wavefield constituents to numerical precision, independent of the order of spatial accuracy of the FD stencil. We have applied the method for the separation of up- and downgoing wavefields and for source wavefield reconstruction for reverse time migration. Our implementation enables accurate source wavefield reconstruction with optimally minimal memory requirements.

A High Order Multi-Dimensional Characteristic Tracing Strategy for the Vlasov–Poisson System

In this paper, we consider a finite difference grid-based semi-Lagrangian approach for solving the Vlasov---Poisson (VP) system. Many of existing methods are based on dimensional splitting, which decouples the problem into solving linear advection problems, see Cheng and Knorr (J Comput Phys 22:330---351, 1976). However, such splitting is subject to the splitting error. If we consider multi-dimensional problems without splitting, difficulty arises in tracing characteristics with high order accuracy. Specifically, the evolution of characteristics is subject to the electric field which is determined globally from the distribution of particle density via Poisson's equation. In this paper, we propose a novel strategy of tracing characteristics high order in time via a two-stage multi-derivative prediction---correction approach and by using moment equations of the VP system. With the foot of characteristics being accurately located, we propose to use weighted essentially non-oscillatory interpolation to recover function values between grid points, therefore to update the solution at the next time level. The proposed method does not have time step restriction as the Eulerian approach and enjoys high order spatial and temporal accuracy. The performance of the proposed schemes are numerically demonstrated via classical test problems such as Landau damping and two stream instabilities.

An enhanced FIVER method for multi-material flow problems with second-order convergence rate

The finite volume (FV) method with exact two-material Riemann problems (FIVER) is an Eulerian computational method for the solution of multi-material flow problems. It is robust in the presence of large density jumps at the fluid–fluid interfaces, and the presence of large structural motions, deformations, and even topological changes at the fluid–structure interfaces. To achieve simplicity in implementation, it approximates each material interface by a surrogate surface which conforms to the control volume boundaries. Unfortunately, this approximation introduces a first-order error of the geometric type in the solution process. In this paper, it is first shown that this error causes the original version of FIVER to be inconsistent in the neighborhood of material interfaces and degrades its global order of spatial accuracy. Then, an enhanced version of FIVER is presented to rectify this issue, restore consistency, and achieve for smooth problems the desired global convergence rate. To this effect, the original definition of a surrogate material interface is retained because of its attractive simplicity. However, the solution at this interface of a two-material Riemann problem is enhanced with a simple reconstruction procedure based on interpolation and extrapolation. Next, the extrapolation component of this procedure is equipped with a limiter in order to achieve nonlinear stability for non-smooth problems. In the one-dimensional inviscid setting, the resulting FIVER method is also shown to be total variation bounded. Focusing on the context of a second-order FV semi-discretization, the nonlinear stability and second-order global convergence rate of this enhanced FIVER method are illustrated for several model multi-fluid and fluid–structure interaction problems. The potential of this computational method for complex multi-material flow problems is also demonstrated with the simulation of the collapse of an air bubble submerged in water and the comparison of the computed results with corresponding experimental data.

A high order characteristic discontinuous Galerkin scheme for advection on unstructured meshes

A new characteristic discontinuous Galerkin (CDG) advection scheme is presented. In contrast to standard discontinuous Galerkin schemes, the test functions themselves follow characteristics in order to ensure conservation and the edges of each element are also traced backwards along characteristics in order to create a swept region, which is integrated in order to determine the mass flux across the edge. Both the accuracy and performance of the scheme are greatly improved by the use of large Courant–Friedrichs–Lewy numbers for a shear flow test case and the scheme is shown to scale sublinearly with the number of tracers being advected, outperforming a standard flux corrected transport scheme for 10 or more tracers with a linear basis. Moreover the CDG scheme may be run to arbitrarily high order spatial accuracy and on unstructured grids, and is shown to give the correct order of error convergence for piecewise linear and quadratic bases on regular quadrilateral and hexahedral planar grids. Using a modal Taylor series basis, the scheme may be made monotone while preserving conservation with the use of a standard slope limiter, although this reduces the formal accuracy of the scheme to first order. The second order scheme is roughly as accurate as the incremental remap scheme with nonlocal gradient reconstruction at half the horizontal resolution. The scheme is being developed for implementation within the Model for Prediction Across Scales (MPAS) Ocean model, an unstructured grid finite volume ocean model.