Directional Splitting Research Articles

High order accurate schemes for Euler and Navier–Stokes equations on staggered Cartesian grids

We present here a new class of staggered schemes for solving the compressible 1D Euler equations in internal energy formulation on uniform grids. The schemes are applicable to arbitrary equation of states and can be extended to high order of accuracy in both time and space on smooth flows. High order accuracy in time is reached thanks to Cauchy–Kovalevskaya procedure. Modifications on the initial schemes are performed to give sufficient conditions for stability on 1D wave equations. Results obtained for wave equations are extended to 1D Euler equations and then to 2D compressible Navier–Stokes equations using directional splitting methods. Results on the conservation of total energy are given, proper shock capturing is observed experimentally. Numerical results are provided up to 4th-order accuracy in 1D and 2D.

Splitting Methods for Rotations: Application to Vlasov Equations

In this work, a splitting strategy is introduced to approximate two-dimensional rotation motions. Unlike standard approaches based on directional splitting which usually lead to a wrong angular velocity and then to large error, the splitting studied here turns out to be exact in time. Combined with spectral methods, the so-obtained numerical method is able to capture the solution to the associated partial differential equation with a very high accuracy. A complete numerical analysis of this method is given in this work. Then, the method is used to design highly accurate time integrators for Vlasov type equations: the Vlasov-Maxwell system and the Vlasov-HMF model. Finally , several numerical illustrations and comparisons with methods from the literature are discussed.

Negative Refraction and Partition in Acoustic Valley Materials of a Square Lattice

Acoustic valley materials, usually constructed in a hexagonal lattice, provide an additional degree of freedom (so-called valleys in their band structures) for manipulating sound waves. This study realizes such a material in a $s\phantom{\rule{0}{0ex}}q\phantom{\rule{0}{0ex}}u\phantom{\rule{0}{0ex}}a\phantom{\rule{0}{0ex}}r\phantom{\rule{0}{0ex}}e$ lattice, in which the topological transport and negative refraction of acoustic kink waves are observed in experiment. At the intersection of channels, a counterintuitive transport phenomenon is seen: The kink waves tend to choose the path that contains a sharp turn. These findings may be useful for directional vocalization, noise reduction, and beam splitting.

Fluctuation splitting Riemann solver for a non-conservative modeling of shear shallow water flow

In this paper we propose a fluctuation splitting finite volume scheme for a non-conservative modeling of shear shallow water flow (SSWF). This model was originally proposed by Teshukov (2007) in [14] and was extended to include modeling of friction by Gavrilyuk et al. (2018) in [7]. The directional splitting scheme proposed by Gavrilyuk et al. (2018) in [7] is tricky to apply on unstructured grids. Our scheme is based on the physical splitting in which we separate the characteristic waves of the model to form two different hyperbolic sub-systems. The fluctuations associated with each sub-systems are computed by developing Riemann solvers for these sub-systems in a local coordinate system. These fluctuations enable us to develop a Godunov-type scheme that can be easily applied on mixed/unstructured grids. While the equation of energy conservation is solved along with the SSWF model in [7], in this paper we solve only SSWF model equations.We develop a cell-centered finite volume code to validate the proposed scheme with the help of some numerical tests. As expected, the scheme shows first order convergence. The numerical simulation of 1D roll waves shows a good agreement with the experimental results. The numerical simulations of 2D roll waves show similar transverse wave structures as observed in [7].

Mesoscopic-based finite volume solutions for waterhammer flows

ABSTRACTIt is becoming evident that high resolution finite volume (FV) numerical schemes for multi-dimensional waterhammer problems are needed for the development of an accurate transient-based condition assessment of pipelines. As a prerequisite, FV methods for one-dimensional waterhammer flows need to be developed. Such models can then be applied to multi-dimensional problems via directional splitting. In this paper, two FV schemes (one uses Bhatnagar–Gross–Krook Boltzmann (BGK) and the other uses kinetic flux vector splitting (KFVS) in the flux approximation) for one-dimensional waterhammer problems are formulated and applied. It is found that the KFVS and BGK schemes correctly capture discontinuity fronts in a classical reservoir-pipe-valve system. An oscillation-free collision time formulation has been proposed, tested and found to be robust. The stability of the proposed schemes is guaranteed when . Comparison between the BGK, KFVS, fixed-grid MOC, and first and second order Godunov schemes reveals that the second-order Godunov performs best, followed closely by the BGK scheme. However, the BGK should not be quickly dismissed since it is known that its real power becomes evident when dealing with complex physics, multi-scale and multi-dimensional problems.

Directional splitting of Gaussian density in non‐linear random variable transformation

Transformation of a random variable is a common need in a design of many algorithms in signal processing, automatic control, and fault detection. Typically, the design is tied to an assumption on a probability density function of the random variable, often in the form of the Gaussian distribution. The assumption may be, however, difficult to be met in algorithms involving non-linear transformation of the random variable. This paper focuses on techniques capable to ensure validity of the Gaussian assumption of the non-linearly transformed Gaussian variable by approximating the to-be-transformed random variable distribution by a Gaussian mixture (GM) distribution. The stress is laid on an analysis and selection of design parameters of the approximate GM distribution to minimise the error imposed by the non-linear transformation such as the location and number of the GM terms. A special attention is devoted to the definition of the novel GM splitting directions based on the measures of non-Gaussianity. The proposed splitting directions are analysed and illustrated in numerical simulations.

High-order staggered schemes for compressible hydrodynamics. Weak consistency and numerical validation

Staggered grids schemes, formulated in internal energy, are commonly used for CFD applications in industrial context. Here, we prove the consistency of a class of high-order Lagrange-Remap staggered schemes for solving the Euler equations in 1D and 2D on Cartesian grids. The main result of the paper is that using an a posteriori internal energy corrector, the Lagrangian schemes are proved to be conservative in mass, momentum and total energy and to be weakly consistent with the 1D Lagrangian formulation of the Euler equations. Extension in 2D is done using directional splitting methods and face-staggering. Numerical examples in both 1D and 2D illustrate the accuracy, the convergence and the robustness of the schemes.

A moving boundary flux stabilization method for Cartesian cut-cell grids using directional operator splitting

An explicit moving boundary method for the numerical solution of time-dependent hyperbolic conservation laws on grids produced by the intersection of complex geometries with a regular Cartesian grid is presented. As it employs directional operator splitting, implementation of the scheme is rather straightforward. Extending the method for static walls from Klein et al. (2009) [37], the scheme calculates fluxes needed for a conservative update of the near-wall cut-cells as linear combinations of “standard fluxes” from a one-dimensional extended stencil. Here the standard fluxes are those obtained without regard to the small sub-cell problem, and the linear combination weights involve detailed information regarding the cut-cell geometry. This linear combination of standard fluxes stabilizes the updates such that the time-step yielding marginal stability for arbitrarily small cut-cells is of the same order as that for regular cells. Moreover, it renders the approach compatible with a wide range of existing numerical flux-approximation methods. The scheme is extended here to time dependent rigid boundaries by reformulating the linear combination weights of the stabilizing flux stencil to account for the time dependence of cut-cell volume and interface area fractions. The two-dimensional tests discussed include advection in a channel oriented at an oblique angle to the Cartesian computational mesh, cylinders with circular and triangular cross-section passing through a stationary shock wave, a piston moving through an open-ended shock tube, and the flow around an oscillating NACA 0012 aerofoil profile.

Grid refinement for aeroacoustics in the lattice Boltzmann method: A directional splitting approach.

This study focuses on grid refinement techniques for the direct simulation of aeroacoustics, when using weakly compressible lattice Boltzmann models, such as the D3Q19 athermal velocity set. When it comes to direct noise computation, very small errors on the density or pressure field may have great negative consequences. Even strong acoustic density fluctuations have indeed a clearly lower amplitude than the hydrodynamic ones. This work deals with such very weak spurious fluctuations that emerge when a vortical structure crosses a refinement interface, which may contaminate the resulting aeroacoustic field. We show through an extensive literature review that, within the framework described above, this issue has never been addressed before. To tackle this problem, we develop an alternative algorithm and compare its behavior to a classical one, which fits our in-house vertex-centered data structure. Our main idea relies on a directional splitting of the continuous discrete velocity Boltzmann equation, followed by an integration over specific characteristics. This method can be seen as a specific coupling between finite difference and lattice Boltzmann, locally on the interface between the two grids. The method is assessed considering two cases: an acoustic pulse and a convected vortex. We show how very small errors on the density field arise and propagate throughout the domain when a vortical flow crosses the refinement interface. We also show that an increased free stream Mach number (but still within the weakly compressible regime) strongly deteriorates the situation, although the magnitude of the errors may remain negligible for purely aerodynamic studies. A drastically reduced level of error for the near-field spurious noise is obtained with our approach, especially for under-resolved simulations, a situation that is crucial for industrial applications. Thus, the vortex case is proved useful for aeroacoustic validations of any grid refinement algorithm.

Study of a collocated Lagrange‐remap scheme for multi‐material flows adapted to HPC

SummaryThis paper describes a collocated numerical scheme for multi‐material compressible Euler equations, which attempts to suit to parallel computing constraints. Its main features are conservativity of mass, momentum, total energy and entropy production, and second order in time and space. In the context of a Eulerian Lagrange‐remap scheme on planar geometry and for rectangular meshes, we propose and compare remapping schemes using a finite volume framework. We consider directional splitting or fully multi‐dimensional remaps, and we focus on a definition of the so‐called corner fluxes. We also address the issue of the internal energy behavior when using a conservative total energy remap. It can be perturbed by the duality between kinetic energy obtained through the conservative momentum remap or implicitly through the total energy remap. Therefore, we propose a kinetic energy flux that improves the internal energy remap results in this context. Copyright © 2016 John Wiley & Sons, Ltd.

Comparison of Numerical Solvers for Anisotropic Diffusion Equations Arising in Plasma Physics

This work is devoted to the comparison of numerical schemes to approximate anisotropic diffusion problems arising in tokamak plasma physics. We focus on the spatial approximation by using finite volume method and on the time discretization. This latter point is delicate since the use of explicit integrators leads to a severe restriction on the time step. Then, implicit and semi-implicit schemes are coupled to finite volumes space discretization and are compared for some classical problems relevant for magnetically confined plasmas. It appears that the semi-implicit approaches (using ARK methods or directional splitting) turn out to be the most efficient on the numerical results, especially when nonlinear problems are studied on refined meshes, using high order methods in space.

A new directional-splitting CIP interpolation with high accuracy and low memory consumption

A new directional-splitting CIP interpolation of third-order accuracy is proposed. This scheme uses only the value and first-order derivatives of the physical field as computational variables, and thus largely reduces the memory requirement. The higher-order cross derivatives are not treated as the computational variables but calculated by recursively using the local Taylor expansions within each single mesh cell, which retains the third order accuracy even for meshes of non-uniform grid spacings. The proposed directional splitting can be easily extended to solve advection equations of any multi dimensionality.

A Three-Dimensional Conservative Coupling Method Between an Inviscid Compressible Flow and a Moving Rigid Solid

We present a conservative method for the three-dimensional coupling between an inviscid compressible flow and a moving rigid solid. We consider an inviscid Euler fluid in conservative form discretized using a high-order monotonicity-preserving finite volume method with a directional operator splitting. An immersed boundary technique is employed through the modification of the finite volume fluxes in the vicinity of the solid. The method yields exact conservation of mass, momentum, and energy of the system, and also exhibits important consistency properties, such as conservation of uniform movement of both fluid and solid as well as the absence of numerical roughness on a straight boundary. The coupling scheme evaluates the fluxes on the fluid side and the forces and torques on the solid side only once every time step, ensuring the computational efficiency of the coupling. We present numerical results assessing the robustness of the method in the case of rigid solids with large displacements.

Influence of adaptive mesh refinement and the hydro solver on shear-induced mass stripping in a minor-merger scenario

We compare two different codes for simulations of cosmological structure formation to investigate the sensitivity of hydrodynamical instabilities to numerics, in particular, the hydro solver and the application of adaptive mesh refinement (AMR). As a simple test problem, we consider an initially spherical gas cloud in a wind, which is an idealized model for the merger of a subcluster or galaxy with a big cluster. Based on an entropy criterion, we calculate the mass stripping from the subcluster as a function of time. Moreover, the turbulent velocity field is analyzed with a multi-scale filtering technique. We find remarkable differences between the commonly used PPM solver with directional splitting in the Enzo code and an unsplit variant of PPM in the Nyx code, which demonstrates that different codes can converge to systematically different solutions even when using uniform grids. For the test case of an unbound cloud, AMR simulations reproduce uniform-grid results for the mass stripping quite well, although the flow realizations can differ substantially. If the cloud is bound by a static gravitational potential, however, we find strong sensitivity to spurious fluctuations which are induced at the cutoff radius of the potential and amplified by the bow shock. This gives rise to substantial deviations between uniform-grid and AMR runs performed with Enzo, while the mass stripping in Nyx simulations of the subcluster is nearly independent of numerical resolution and AMR. Although many factors related to numerics are involved, our study indicates that unsplit solvers with advanced flux limiters help to reduce grid effects and to keep numerical noise under control, which is important for hydrodynamical instabilities and turbulent flows.

Multi-borehole electro-blast method for concrete monolith splitting off

The multi-borehole method of monolith splitting off by pulsed electrical discharge is presented. The experimental data of simultaneous and step by step discharge initiation for monolith splitting off with using of the polyethylene cartridge are given. The simultaneous initiation of discharge in two boreholes with 30 cm spacing, transferred energy in each borehole about 23 kJ as well as sequential initiation of discharge with transferred energy in one borehole about 47 kJ are investigated. The concrete splitting off with the dimensions of 300×300×650 mm by six boreholes with step-by-step initiation in each borehole and with total energy of about 280 kJ as well as the concrete splitting off with dimensions of 300×300×300 mm by the four boreholes with simultaneous initiation in each two boreholes and the total energy of about 90 kJ take place. It is shown that the simultaneous multi-borehole initiation of the electrical discharge leads to the cracks formation in predefined direction and to the directional solid splitting off. The experimental results show that the simultaneous discharge initiation in comparison with step-by-step one has advantage in consumed energy.

An efficient parallel immersed boundary algorithm using a pseudo-compressible fluid solver

We propose an efficient algorithm for the immersed boundary method on distributed-memory architectures that has the computational complexity of a completely explicit method and also has excellent parallel scaling. The algorithm utilizes the pseudo-compressibility method recently proposed by Guermond and Minev that uses a directional splitting strategy to discretize the incompressible Navier–Stokes equations, thereby reducing the linear systems to a series of one-dimensional tridiagonal systems. We perform numerical simulations of several fluid–structure interaction problems in two and three dimensions and study the accuracy and convergence rates of the proposed algorithm. We also compare the proposed algorithm with other second-order projection-based fluid solvers. Lastly, the execution time and scaling properties of the proposed algorithm are investigated and compared to alternate approaches.

Improving conservation properties of a 5D gyrokinetic semi-Lagrangian code

In gyrokinetic turbulent simulations, the knowledge of some stationary states can help reducing numerical artifacts. Considering long-term simulations, the qualities of the Vlasov solver and of the radial boundary conditions have an impact on the conservation properties. In order to improve mass and energy conservation mainly, the following methods are investigated: fix the radial boundary conditions on a stationary state, use a 4D advection operator that avoids a directional splitting, interpolate with a delta-f approach. The combination of these techniques in the semi-Lagrangian code gysela leads to a net improvement of the conservation properties in 5D simulations.

Two-fluid compressible simulations on GPU cluster

In this work we propose an efficient finite volume approximation of twofluid flows. Our scheme is based on three ingredients. We first construct a conservative scheme that removes the pressure oscillations phenomenon at the interface. The construction relies on a random sampling at the interface [6, 5]. Secondly, we replace the exact Riemann solver by a faster relaxation Riemann solver with good stability properties [4]. Finally, we apply Strang directional splitting and optimized memory transpositions in order to achieve high performance on Graphics Processing Unit (GPU) or GPU cluster computations.

Measuring the magnetic field of a magnetized plasma using Raman scattering

We studied the Raman scattering in a magnetized plasma by one-dimensional particle-in-cell (PIC) simulations in non-relativistic regime. It is found from the X-mode dispersion relation that the frequency of the backward scattered wave is downshifted by an amount of upper hybrid frequency, while that of the forward scattered wave merely depends on the magnetic field. We propose such a spectral difference be used to measure simultaneously the plasma density and magnetic field of magnetized plasmas. The idea was verified by a series of PIC simulations, where we used the directional field splitting method to obtain accurate peak position of the scattered waves' frequencies. We compared the frequency shift and the growth rate of the scattering from theory and simulations to obtain reasonably good agreement between them for different external magnetic fields.

ADI splitting schemes for a fourth-order nonlinear partial differential equation from image processing

We present directional operator splitting schemes for the numerical solution of a fourth-order, nonlinear partial differential evolution equation which arises in image processing. This equation constitutes the $H^{-1}$-gradient flow of the total variation and represents a prototype of higher-order equations of similar type which are popular in imaging for denoising, deblurring and inpainting problems. The efficient numerical solution of this equation is very challenging due to the stiffness of most numerical schemes. We show that the combination of directional splitting schemes with implicit time-stepping provides a stable and computationally cheap numerical realisation of the equation.