Von Neumann Analysis Research Articles

An immersed boundary-regularized lattice Boltzmann method for modeling fluid–structure–acoustics interactions involving large deformation

This work presents a numerical method for modeling fluid–structure–acoustics interaction (FSAI) problems involving large deformation. The method incorporates an immersed boundary method and a regularized lattice Boltzmann method (LBM) where a multi-block technique and a nonreflecting boundary condition are implemented. The von Neumann analysis is conducted to investigate the stability of the regularized LBM. It is found that the accuracy and stability of the regularized LBM can be improved when the collision operator is computed from the Hermite polynomials up to the fourth order instead of the second order. To validate the present method, four benchmark cases are conducted: the propagation of an acoustic monopole point source, the sound generated by a stationary cylinder in a uniform flow, the sound generation of a two-dimensional insect model in hovering flight, and the sound generation of a three-dimensional flapping wing. Predictions given by the current method show a good agreement with numerical simulations and analytical solutions reported in the literature, demonstrating its capability of solving FSAI problems involving complex geometries and large deformation. Finally, the method is applied in modeling sound generation in vortex-induced vibrations of a rigid cylinder and a sphere. It is found that vortex-induced vibration can enhance the acoustic intensity by approximately four times compared to that of the stationary case for a cylinder. In contrast, both vibrating and stationary spheres exhibited relatively less intense noise, primarily within the wake. Notably, the spanwise noise propagation is only observed when the sphere is vibrating.

Investigation of the Sigma Approximation Technique for the Solution of the Time Spectral Equation System

Abstract The time spectral method (TSM) is commonly utilized to analyze periodic unsteady flows within turbomachines. However, the time spectral solutions in regions with large temporal and spatial gradients, such as wakes, shocks, and boundary layers, can be plagued with unphysical oscillations, known as the Gibbs phenomenon. This paper presents an investigation into the sigma approximation technique, which is capable of significantly attenuating the Gibbs phenomenon by dampening high-frequency nonlinear components in time spectral solutions. Central to this technique are three sigma parameters: the exponent, the cutoff number, and the number of harmonics, which collectively determine the damping effects for each frequency. However, the optimal combination of them remains to be ascertained considering the tradeoff between the accuracy of solutions and the reduction of unphysical oscillations. In this study, a two-row compressor configuration is first selected, and numerical simulations are performed to evaluate the efficacy of this technique under near-peak efficiency and near-stall operating conditions. It is found that Gibbs-type unphysical oscillations can be effectively mitigated without notably compromising solution accuracy through an optimal combination of sigma parameters. The NASA Stage 35 case study further verifies the effectiveness of the proposed optimal sigma parameter combination in preserving the accuracy of time spectral solutions based on the data from test rig. Moreover, the von Neumann analysis reveals that the sigma approximation technique can enhance the numerical stability of the time spectral equation system, thereby allowing the use of aggressive numerical parameters to accelerate convergence. The stability analysis results are verified by a two-dimensional bump under two different flow conditions and further assessed using the two-row compressor.

A high-order upwind compact difference scheme for solving the streamfunction-velocity formulation of the unsteady incompressible Navier–Stokes equations

In this paper, we propose an upwind compact difference method with fourth-order spatial accuracy and second-order temporal accuracy for solving the streamfunction-velocity formulation of the two-dimensional unsteady incompressible Navier–Stokes equations. The streamfunction and its first-order partial derivatives (velocities) are treated as unknown variables. Three types of compact difference schemes are employed to discretize the first-order partial derivatives of the streamfunction. Specifically, these schemes include the fourth-order symmetric scheme, the fifth-order upwind scheme, and the sixth-order symmetric scheme derived by combining the two parts of the fifth-order upwind scheme. As a result, the fourth-order spatial discretization schemes are established for the Laplacian term, the biharmonic term, and the nonlinear convective term, along with the Crank–Nicolson scheme for the temporal discretization. The unconditional stability characteristic of the scheme for the linear model is proved by discrete von Neumann analysis. Moreover, six numerical experiments involving three test problems with the analytic solutions, and three flow problems including doubly periodic double shear layer, lid-driven cavity flow, and dipole-wall interaction are carried out to demonstrate the accuracy, robustness, and efficiency of the present method. The results indicate that the present method not only has good numerical performance but also exhibits quite efficiency.

Solving 2D damped Kuramoto-Sivashinsky with multiple relaxation time lattice Boltzmann method

Lattice Boltzmann method (LBM) is a powerful fluid flow solver. Using this method to solve other PDEs might be a difficult task. The first challenge is to find a suitable local equilibrium distribution function (EDF) capable of recovering the desired PDE. The next difficulty arises from the explicit nature of LBM. The conditional stability of the LBM algorithm affects the numerical solution accuracy. Damped Kuramoto–Sivashinsky (DKS) equation is a fourth–order PDE that recently challenged many numerical methods' abilities. This equation is highly sensitive to a parameter that causes three states of solutions in a small interval of freedom. In this paper, we challenged LBM to solve the two–dimensional DKS equation by finding EDF using the Chapman–Enskog analysis up to the fourth–order. Also, the von Neumann analysis and a simple genetic algorithm are applied to find reliable values for the free parameters. Furthermore, a modification on image-based ghost node method is proposed for implementation of the boundary conditions in the complex geometries.

Length-scales for efficient CFL conditions in high-order methods with distorted meshes: Application to local-timestepping for [formula omitted]-multigrid

We propose a strategy to estimate the maximum stable time-steps for explicit time-stepping methods for hyperbolic systems in a high-order flux reconstruction framework. The strategy is derived through a von-Neumann analysis (VNA) framework for the advection–diffusion equation on skewed two- and three-dimensional meshes. It directly incorporates the spatial polynomial- and mesh-discretization in estimating the convective and diffusive length-scales. The strategy is extended to the density-based Navier–Stokes system of equations, taking into account the omnidirectionality of the speed of sound.We compare the performance of this strategy with three other popular choices of length-scales across a wide range of polynomial-orders, meshes of drastically varying cell-quality, and flow-physics. The proposed strategy shows robust behavior across all test-scenarios with limited variation of the maximum stable CFL-number (0.1 to 1) for polynomial-orders 1 through 10, unlike other strategies where the CFL-number varies sharply. Finally, we show the advantage of the proposed methodology for local-timestepping for p-multigrid through a RANS-modeled steady-state turbulent flow case, on a mesh with large disparity of mesh elements and aspect ratios.

A fourth order accurate numerical method for non-linear time fractional reaction–diffusion equation on a bounded domain

In the present work, a high-order numerical scheme is proposed and analyzed to solve the non-linear time fractional reaction–diffusion equation (RDE) of order α∈(0,1). The numerical scheme consists of a time-stepping cubic approximation for the time fractional derivative and a compact finite difference scheme to approximate the spatial derivative. After applying these approximations to time fractional RDE, we get a non-linear system of equations. An iterative algorithm is formulated to solve the obtained nonlinear discrete system. We analyze the unique solvability of the proposed compact finite difference scheme and discuss the stability using von Neumann analysis. Further, we prove that the scheme is convergent in the Euclidean norm with the convergence order 4−α in the temporal direction and 4 in the spatial direction using matrix analysis. Finally, the numerical experimentation is performed to demonstrate the authenticity of the proposed numerical scheme.

Comprehensive comparison between the lattice Boltzmann and Navier–Stokes methods for aerodynamic and aeroacoustic applications

In an effort to determine which Computational Fluid Dynamics method offers the best trade-off between accuracy and computational cost for aerodynamic and aeroacoustic applications, the lattice Boltzmann and finite-volume Navier–Stokes methods are compared. Unlike previous studies, the present framework enables a fair and unbiased comparison of the core capabilities of each numerical approach and focuses on schemes of practical relevance. With the aim of providing a comprehensive comparison of the methods, an extended von Neumann analysis is performed and the High–Performance Computing capacities of both methods are thoroughly discussed. In addition, it is also shown through computations on canonical test cases that a “time to solution” metric has to be considered in order to objectively assess the suitability of one particular numerical method over the other.Three main conclusions are drawn: (1) both the lattice Boltzmann and Navier–Stokes schemes exhibit an anisotropic dissipative behaviour, (2) a cell update using the lattice Boltzmann method is 2 to 3 times faster than with the Navier–Stokes method dedicated to cartesian grids, and (3) the use of the “time to solution” metric demonstrates that the relevance of one method over the other is closely linked to the underlying physics and the intended error target. In light of these results, decision aids are provided to assist in selecting the most efficient method for a given application.

Higher Order CPML for Leapfrog Complying-Divergence Implicit FDTD Method and Its Numerical Properties

The leapfrog complying-divergence implicit finite-difference time-domain (CDI-FDTD) method is an implicit scheme that is unconditionally stable involving the simplest and most efficient leapfrog update procedures with minimum floating-point operations (flops). Moreover, the method possesses complying divergence satisfying Gauss’s law, while other methods, including the leapfrog alternating direction implicit (ADI)-FDTD counterpart, do not. These desirable features make the leapfrog CDI-FDTD useful for electromagnetic simulations when the unconditional stability is to be exploited. In this article, a higher order convolutional perfectly matched layer (CPML) for leapfrog CDI-FDTD is developed and implemented for simulating open space problems. To formulate the CPML for leapfrog CDI-FDTD, we have carefully considered the order of CPML update along with proper use of main and auxiliary fields to retain the unconditional stability and complying divergence. The proposed CPML for leapfrog CDI-FDTD remains stable beyond the Courant time step as ascertained through von Neumann analysis. By comparing the flops and computation time, the leapfrog CDI-FDTD is shown to be most efficient among common implicit FDTD methods with CPML. The formulation of numerical divergence for higher order CPML is also presented. Simulation results show that the proposed higher order CPML has many advantages of simplicity, complying divergence, and high absorption efficiency.

Solutions of the mobile–immobile advection–dispersion model based on the fractional operators using the Crank–Nicholson difference scheme

In this manuscript, based on the Atangana–Baleanu Caputo (ABC) fractional derivative and the Caputo fractional derivative for the mobile–immobile advection–dispersion model are considered. For the proposed model, the Crank–Nicholson difference method (C-NDM) scheme is created. Utilizing the Von-Neumann analysis technique, the stability estimate for this difference scheme is demonstrated. The numerical results were obtained for the proposed model using the (C-NDM). The numerical results were produced using the MATLAB program. Then, the obtained approximate solutions for the proposed model through these two fractional derivative operators using the (C-NDM) were compared with the exact solution. The obtained results demonstrate the viability and utility of the proposed technique, and these two fractional derivatives yield a more comparable result. This study differs from previous studies by comparing this model which is based on the Caputo and the Atangana–Baleanu Caputo fractional derivatives using the Crank–Nicholson difference technique.

Multiple relaxation time lattice Boltzmann schemes for advection-diffusion equations with application to radar image processing

Motivated by marine radar image processing, we investigate the accuracy of multiple relaxation time lattice Boltzmann schemes designed to simulate two-dimensional convection-diffusion equations. The context of application requires to deal with non-constant advection velocity. Using Taylor expansions, instead of the widely used Chapman-Enskog expansions, we show how to control the accuracy of these schemes when deriving equivalent partial differential equations. On the one hand, a third order analysis is conducted on a scheme involving a constant advection velocity and no source term. First, this analysis derives the stability region through the von Neumann analysis. Second, a numerical convergence rate of three is obtained thanks to an appropriate choice of parameters. On the other hand, non-constant advection velocity together with non-zero source term, introduce additional terms at the second order. Regarding the targeted application, these extra terms are shown to be negligible and experiments on real data show that such multiple relaxation time lattice Boltzmann schemes are relevant for marine radar denoising and enhancement.

High-Order Compact Difference Method for Solving Two- and Three-Dimensional Unsteady Convection Diffusion Reaction Equations

In this paper, a type of high-order compact (HOC) finite difference method is developed for solving two- and three-dimensional unsteady convection diffusion reaction (CDR) equations with variable coefficients. Firstly, an HOC difference scheme is derived to solve the two-dimensional (2D) unsteady CDR equation. Discretization in time is carried out by Taylor series expansion and correction of the truncation error remainder, while discretization in space is based on the fourth-order compact difference formulas. The scheme is second-order accuracy in time and fourth-order accuracy in space. The unconditional stability is obtained by the von Neumann analysis method. Then, this scheme is extended to solve the three-dimensional (3D) unsteady CDR equation. It needs only a five-point stencil for 2D problems and a seven-point stencil for 3D problems. Moreover, the present schemes can solve the nonlinear Burgers equation. Finally, numerical experiments are conducted to show the good performances of the new schemes.

On the properties of high-order least-squares finite-volume schemes

High-order finite-volume schemes based on polynomial least-squares methods are an active research topic for the discretization of hyperbolic equations as they allow to obtain high-order spatial discretization schemes in arbitrary meshes. However, few studies have analyzed their performance in good-quality/near-to-uniform meshes, which are commonly used as a meshing strategy in zones where turbulent effects are important. In this paper, the theoretical numerical properties of commonly used least-squares (LSQ) k-exact high-order finite volume schemes are studied in one-dimensional and in several two-dimensional meshes (with some remarks regarding their properties in three-dimensional meshes). These results are compared to those obtained using fully-constrained polynomial reconstructions only compatible with structured meshes. The numerical properties of the schemes are investigated through the von Neumann analysis methodology applied to the one-dimensional and two-dimensional finite volume formulation, including temporal discretization errors. This analysis is also extended to non-uniform and unstructured two-dimensional meshes. At last, the schemes are tested with several numerical experiments using the linear advection, the Euler equations and the Navier-Stokes equations. The analysis of both studies yields similar conclusions regarding the numerical errors and stability of the different studied schemes showing that the high-order least-squares finite volume schemes yield stable and robust results across different uniform and non-uniform unstructured meshes. However, their performance is heavily degraded in simulations with low mesh resolution compared to schemes specially catered to structured meshes. On the other hand, the latter schemes lack stability and robustness in general structured meshes and its formulation cannot be straightforwardly extended to unstructured meshes. Moreover, this work shows that the use of weighted-LSQ can drastically improve the results of LSQ schemes in under-resolved simulations.

Performance and accuracy of hybridized flux reconstruction schemes

We present a unified framework for the hybridization of flux reconstruction (FR) schemes. Specifically, we discuss the hybridized flux reconstruction (HFR) and interior-embedded flux reconstruction (EFR) methods and analyze their performance and accuracy for a range of correction functions on quadrilateral elements. HFR uses discontinuous traces on the faces of the elements, and EFR considers globally continuous trace functions. We perform two-dimensional von Neumann analysis to characterize the numerical error of EFR methods in relation to HFR and FR. HFR and FR are equivalent for linear advection, but EFR is only equivalent to FR one-dimensional advection directions. It is observed that EFR methods generally introduce additional numerical error for lower-order simulations, but behave closer to FR methods for higher orders. In addition, the behavior for different correction functions follows the same trend as the conventional FR formulation. Verification with linear and nonlinear numerical examples showed the expected order of accuracy of p+1 was obtained for all formulations, where p is the polynomial degree of the solution. With appropriate choices of correction functions, EFR methods can be more accurate than other HFR and FR methods. Furthermore, it is observed that the choice of correction function has an impact on the performance of the implicit solver by a factor of up to 2 for FR methods for an airfoil-vortex interaction problem, but had a smaller influence on the hybridized forms. For this problem, we observed speedups factors of up to 4.3 and 6.7 for HFR and EFR, respectively, when compared with FR. Hence, hybridized methods are a viable approach to reducing the computational cost of implicit FR solvers.

Two-stage explicit schemes based on numerical approximations of convection-diffusion equations

A wide family of finite difference methods for the convection-diffusion problems based on an explicit two-stage scheme and a seemingly implicit method are presented. In this paper, to have a greater stability region while maintaining the second-order accuracy, a family of methods that combines the MacCormack and Saul'vey schemes is proposed. The stability analysis of the combined methods is investigated using the von Neumann approach. In each case, it is found that it is the convection term that limits the stability of the scheme. Based on the von Neumann analysis, valuable stability limits in terms of mesh parameters for maintaining accurate results are determined in an analytic manner and demonstrated through computer simulations. Two model problems consisting of linear advection-diffusion and nonlinear viscous Burgers equation are given to illustrate some properties of the present technique, such as stability and the ability to propagate discontinuities.

Curl Constraint-Preserving Reconstruction and the Guidance it Gives for Mimetic Scheme Design

Several important PDE systems, like magnetohydrodynamics and computational electrodynamics, are known to support involutions where the divergence of a vector field evolves in divergence-free or divergence constraint-preserving fashion. Recently, new classes of PDE systems have emerged for hyperelasticity, compressible multiphase flows, so-called first-order reductions of the Einstein field equations, or a novel first-order hyperbolic reformulation of Schrödinger’s equation, to name a few, where the involution in the PDE supports curl-free or curl constraint-preserving evolution of a vector field. We study the problem of curl constraint-preserving reconstruction as it pertains to the design of mimetic finite volume (FV) WENO-like schemes for PDEs that support a curl-preserving involution. (Some insights into discontinuous Galerkin (DG) schemes are also drawn, though that is not the prime focus of this paper.) This is done for two- and three-dimensional structured mesh problems where we deliver closed form expressions for the reconstruction. The importance of multidimensional Riemann solvers in facilitating the design of such schemes is also documented. In two dimensions, a von Neumann analysis of structure-preserving WENO-like schemes that mimetically satisfy the curl constraints, is also presented. It shows the tremendous value of higher order WENO-like schemes in minimizing dissipation and dispersion for this class of problems. Numerical results are also presented to show that the edge-centered curl-preserving (ECCP) schemes meet their design accuracy. This paper is the first paper that invents non-linearly hybridized curl-preserving reconstruction and integrates it with higher order Godunov philosophy. By its very design, this paper is, therefore, intended to be forward-looking and to set the stage for future work on curl involution-constrained PDEs.

An optimized second order numerical scheme applied to the non-linear Fisher’s reaction-diffusion equation

Four one-parameter θ -family of simple and seemingly implicit finite difference schemes are proposed to obtain an accurate approximate solution for the non-linear Fisher model problem arising in population dynamics. The underlying methods are based upon the Saul’yev’s asymmetrical methods and the method of lagging. It is shown that, in general, the proposed schemes are consistent with the first and second-order accuracy in time and in space respectively. However, for two families, temporally and spatially second-order accuracy is achieved if one uses a special value of the weighting parameter. Hence, based on these families, a two-parameter (Θ,θ) -family of implicit finite difference approximation is introduced. By optimizing these parameters, we show that the latter scheme is second order accurate in both time and space. In fact, by selecting Θ,θ = 1/2, the stability of the method is established in the sense of von Neumann analysis. Both the accuracy and the convergence rates of the proposed discretized schemes are verified through numerical simulations and comparisons with available exact solutions and alternative existing numerical values are also made. Numerical results show the present methods and in particular the implicit approach are superior to the known methods in terms of accuracy.

Recursive finite-difference Lattice Boltzmann schemes

The motivation of this study is twofold. First, a recursive mathematical formulation of the discrete-velocity Boltzmann equation (DVBE) under the Bhatnagar-Gross-Krook (BGK) approximation is introduced. This formulation allows us to formally express the solution of the DVBE as an infinite sum over successive particle derivatives of the distributions associated with local equilibrium. A Chapman-Enskog multiple-scales expansion shows that this sum can be safely truncated beyond the second order if the Navier-Stokes level of description is requested. Therefore, the distribution functions depend only on the first and second-order derivatives of the related equilibrium distributions. This alternative equation to the DVBE defines a basis to design kinetic schemes for the evolution of the distribution functions based solely on flow variables that are sufficient to define local equilibrium. Second, a family of mass-conserving numerical schemes is introduced from this kinetic equation by discretizing the particle derivatives by backward finite differences. Interestingly, a so-called “simplified Lattice Boltzmann method” introduced by Chen et al. in 2017 can be recast in this family. Numerical simulations highlight a level of numerical dissipation that is generally higher than the level obtained with a standard Lattice Boltzmann scheme, as expected by approximating derivatives by finite differences. Nevertheless, we show by using a von Neumann analysis that it is possible to parametrize our scheme, according to the relaxation coefficient of the DVBE, to reduce significantly its numerical dissipation and improve its spectral properties.We believe that this modeling can also be of interest to connect macroscopic and kinetic representations, e.g. when dealing with initial and boundary conditions or in hybrid simulations matching Navier-Stokes and Lattice Boltzmann schemes.

High-order compact LOD methods for solving high-dimensional advection equations

In this paper, by using the local one-dimensional (LOD) method, Taylor series expansion and correction for the third derivatives in the truncation error remainder, two high-order compact LOD schemes are established for solving the two- and three- dimensional advection equations, respectively. They have the fourth-order accuracy in both time and space. By the von Neumann analysis method, it shows that the two schemes are unconditionally stable. Besides, the consistency and convergence of them are also proved. Finally, numerical experiments are given to confirm the accuracy and efficiency of the present schemes.

Application of approximate dispersion-diffusion analyses to under-resolved Burgers turbulence using high resolution WENO and UWC schemes

This paper presents a space-time approximate diffusion-dispersion analysis of high-order, finite volume Upwind Central (UWC) and Weighted Essentially Non-Oscillatory (WENO) schemes. We perform a thorough study of the numerical errors to find a-priori guidelines for the computation of under-resolved turbulent flows. In particular, we study the 3-rd, 5-th and 7-th order UWC and WENO reconstructions in space, and 3-rd and 4-th order Runge-Kutta time integrators. To do so, we use the approximate von Neumann analysis for non-linear schemes introduced by Pirozzoli. Moreover, we apply the “1% rule” for the dispersion-diffusion curves proposed by Moura et al. [41] to determine the range of wavenumbers that are accurately resolved by each scheme. The dispersion-diffusion errors estimated from these analyses agree with the numerical results for the forced Burgers' turbulence problem, which we use as a benchmark. The cut-off wavenumbers defined by the “1% rule” are evidenced to serve as a good estimator of the beginning of the dissipation region of the energy cascade and they are shown to be associated to a similar level of dissipation, with independence of the scheme.Finally, we show that WENO schemes are more diffusive than UWC schemes, leading to stable simulations at the price of more dissipative results. It is concluded both UWC and WENO schemes may be suitable schemes for iLES turbulence modeling, given their numerical dissipation level acting at the appropriate wavenumbers.

Flux reconstruction using Jacobi correction functions in discontinuous spectral element method

A flux reconstruction approach is implemented in the spectral difference (SD) framework, which we refer to it as discontinuous spectral element method (DSEM), to improve the accuracy and stability of numerical simulations. A new class of correction functions is developed based on a weighted orthogonality condition, which leads to the introduction of Jacobi correction functions. Jacobi correction functions can construct the well-known DSEM and staggered-grid version of nodal discontinuous Galerkin spectral element method (DGSEM) as well as a broad range of other high-order numerical schemes with a variety of numerical characteristics, such as high numerical dissipation to suppress aliasing-driven errors, super accuracy, and solution boundedness in shock prediction. Three single-parameter families of Jacobi correction functions are recognized, and a von Neumann analysis is performed to acquire their wave propagation properties. The order of accuracy and stability criterion of a broad range of schemes are calculated, and the most super-accurate and stable numerical schemes are identified and later validated through a set of numerical experiments on a one-dimensional advection equation. The solution unboundedness issue of the high-order DSEM scheme is discussed through simulating the Burgers equation. An accuracy test performed on the Burgers equation revealed that the new staggered-grid flux reconstruction approach could acquire a higher accuracy than that of the energy stable flux reconstruction framework on a collocated grid. The two-dimensional extension of the new framework is also examined by simulation of a non-linear Euler system of equations for the isentropic vortex in free-stream flow. The staggered-grid flux reconstruction using the Jacobi correction function can precisely reconstruct the DSEM approach for non-linear two-dimensional hyperbolic equations for all polynomial orders.