Piecewise Parabolic Method Research Articles

Modeling Soft X‐Ray Emissions at the Dayside Magnetopause

AbstractIn this study, we simulate the Solar Wind Charge Exchange (SWCX) soft X‐ray emissions at dayside magnetosheath and cusps by using magnetohydrodynamic (MHD) and LAtmos TEst Particle (LaTeP) models. MHD models are unable to resolve the particle kinetic effects, such as the different behaviors of ions with different q/m, or distinguish the magnetospheric plasma from the solar wind plasma. We investigate these effects with the LaTeP model. As the LaTeP model does not self‐compute magnetic and electric field, the magnetic and electric field data obtained from Open Geospace General Circulation Model (OpenGGCM) and Lagrangian version of the piecewise parabolic method (PPMLR) MHD model are used as the input to LaTeP model. The soft X‐ray emissivity maps simulated from pure OpenGGCM and PPMLR MHD approaches and from LaTeP‐OpenGGCM and LaTeP‐PPMLR approaches are presented and compared. The results indicate that the LaTeP model can well resolve the kinetic effects and can be used to investigate the individual spectral characteristics. Therefore, the LaTeP model is a complementary approach for simulating the X‐ray emissions near the dayside magnetopause. We also calculate the ratio of integrated OVII/OVIII line intensities, produced by charge exchange of O7+ ions and O8+ ions, respectively. We find a relatively higher ratio at the bow shock compared to the surrounding areas, suggesting that this ratio can be an effective parameter to identify the bow shock location.

An improved version of the piecewise parabolic method advection scheme: description and performance assessment in a bidimensional test case with stiff chemistry in toyCTM v1.0.1

Abstract. This study presents a novel method to estimate the performance of advection schemes in numerical experiments along with a semi-realistic, non-linear, stiff chemical system. This method is based on the examination of the “signature function”, an invariant of the advection equation. Apart from exposing this concept in a particular numerical test case, we show that a new numerical scheme based on a combination of the piecewise parabolic method (PPM) with the flux adjustments of Walcek outperforms both the PPM and the Walcek schemes for inert tracer advection as well as for advection of chemically active species. From a fundamental point of view, we think that our evaluation method, based on the invariance of the signature function under the effect of advection, offers a new way to evaluate objectively the performance of advection schemes in the presence of active chemistry. More immediately, we show that the new PPM + W (“piecewise parabolic method + Walcek”) advection scheme offers chemistry-transport modellers an alternative, high-performance scheme designed for Cartesian-grid Eulerian chemistry-transport models, with improved performance over the classical PPM scheme. The computational cost of PPM + W is not higher than that of PPM. With improved accuracy and controlled computational cost, this new scheme may find applications in other fields such as ocean models or atmospheric circulation models.

GPU-HADVPPM V1.0: a high-efficiency parallel GPU design of the piecewise parabolic method (PPM) for horizontal advection in an air quality model (CAMx V6.10)

Abstract. With semiconductor technology gradually approaching its physical and thermal limits, graphics processing units (GPUs) are becoming an attractive solution for many scientific applications due to their high performance. This paper presents an application of GPU accelerators in an air quality model. We demonstrate an approach that runs a piecewise parabolic method (PPM) solver of horizontal advection (HADVPPM) for the air quality model CAMx on GPU clusters. Specifically, we first convert the HADVPPM to a new Compute Unified Device Architecture C (CUDA C) code to make it computable on the GPU (GPU-HADVPPM). Then, a series of optimization measures are taken, including reducing the CPU–GPU communication frequency, increasing the data size computation on the GPU, optimizing the GPU memory access, and using thread and block indices to improve the overall computing performance of the CAMx model coupled with GPU-HADVPPM (named the CAMx-CUDA model). Finally, a heterogeneous, hybrid programming paradigm is presented and utilized with GPU-HADVPPM on the GPU clusters with a message passing interface (MPI) and CUDA. The offline experimental results show that running GPU-HADVPPM on one NVIDIA Tesla K40m and an NVIDIA Tesla V100 GPU can achieve up to a 845.4× and 1113.6× acceleration. By implementing a series of optimization schemes, the CAMx-CUDA model results in a 29.0× and 128.4× improvement in computational efficiency by using a GPU accelerator card on a K40m and V100 cluster, respectively. In terms of the single-module computational efficiency of GPU-HADVPPM, it can achieve 1.3× and 18.8× speedup on an NVIDIA Tesla K40m GPU and NVIDIA Tesla V100 GPU, respectively. The multi-GPU acceleration algorithm enables a 4.5× speedup with eight CPU cores and eight GPU accelerators on a V100 cluster.

The dataset of the manuscript "GPU-HADVPPM V1.0: high-efficient parallel GPU design of the Piecewise Parabolic Method (PPM) for horizontal advection in air quality model (CAMx V6.10)"

The dataset of the manuscript "GPU-HADVPPM V1.0: high-efficient parallel GPU design of the Piecewise Parabolic Method (PPM) for horizontal advection in air quality model (CAMx V6.10)"

A Method for Threshold Setting and False Alarm Probability Evaluation for Radar Detectors

A threshold setting and false alarm probability evaluation method is proposed for radar detectors for which an analytical relationship between thresholds and false alarm probabilities is impossible or difficult to obtain due to complicated detection schemes or non-Gaussian noise and clutter. Importance sampling is performed to increase the sampling efficiency of low-probability samples. Then, the weighted piecewise parabolic (P2) interpolation method is used to rapidly plot the relationship curve between the threshold and the false alarm of the detector, with a low storage requirement and high accuracy. The proposed method effectively enhances the detector threshold computing scale and addresses the statistical verification of the false alarm probability for radar detectors with a low false alarm rate.

A 3D Magnetospheric CT Reconstruction Method Based on 3D GAN and Supplementary Limited‐Angle 2D Soft X‐Ray Images

AbstractSoft X‐ray imaging is a new method used to observe the structure of the Earth's magnetosphere. Reconstruction of the Earth's three‐dimensional (3‐D) magnetospheric structure from 2‐D soft X‐ray images is an important research topic in astronomical imaging and exploration. In this study, computed tomography (CT) is investigated for application to the planned Solar wind Magnetosphere Ionosphere Link Explorer (SMILE), where resulting images are collected at varying small angles. The restricted geometry of SMILE results in sparse projection data and reduced reconstruction quality. As such, a new algorithm is proposed, based on cone‐beam computed tomography (CBCT) and utilizing 3‐D generative adversarial networks (GAN) to supplement the 2‐D soft X‐ray images and increase the angular coverage of the projections. A complement optimization backpropagation network (COBN) is then used to reconstruct high‐quality 3‐D X‐ray emissivity profiles in the magnetosphere. A series of validation experiments were conducted in which the reconstructions were compared with synthetic results generated by the PPMLR‐MHD (an extended LagRangian version of the Piecewise Parabolic Method –MagnetoHydroDynamic) model under varying solar wind conditions. Results showed the supplemented profiles were more complete and accurate than the direct reconstructions, as measured by the peak signal‐to‐noise ratio (PSNR) and structural similarity (SSIM). This suggests the proposed technique could be a viable approach to observing magnetosphere structure and variability.

A New Parallel Code Based on a Simple Piecewise Parabolic Method for Numerical Modeling of Colliding Flows in Relativistic Hydrodynamics

A new parallel code based on models of special relativistic hydrodynamics is presented for describing interacting flows. A new highly accurate numerical method is considered and verified. A parallel implementation of the method by means of Coarray Fortran technology and its efficiency are described in detail. The code scalability is 92% on a cluster with Intel Xeon 6248R NKS-1P with 192 Coarray Fortran images. Different interacting relativistic flows are considered as astrophysical applications.

Quokka: a code for two-moment AMR radiation hydrodynamics on GPUs

ABSTRACT We present quokka, a new subcycling-in-time, block-structured adaptive mesh refinement (AMR) radiation hydrodynamics (RHD) code optimized for graphics processing units (GPUs). quokka solves the equations of HD with the piecewise parabolic method (PPM) in a method-of-lines formulation, and handles radiative transfer via the variable Eddington tensor (VET) radiation moment equations with a local closure. We use the amrex library to handle the AM management. In order to maximize GPU performance, we combine explicit-in-time evolution of the radiation moment equations with the reduced speed-of-light approximation. We show results for a wide range of test problems for HD, radiation, and coupled RHD. On uniform grids in 3D on a single GPU, our code achieves >250 million hydrodynamic updates per second and almost 40 million radiation hydrodynamic updates per second. For RHD problems on uniform grids in 3D, our code scales from 4 to 256 GPUs with an efficiency of 76 per cent. The code is publicly released under an open-source license on GitHub.

The Effect of Momentum Loss on Detonation Failure at Very Low Activation Energy

In this paper, the behavior of detonation waves in a non-ideal environment has been studied. Modeling of detonation has been performed based on one-dimensional Euler equations (momentum conservation) by considering friction as the momentum loss source term in the equation with a single-step Arrhenius law as the chemical kinetics model. Piecewise parabolic method(PPM) has been used to simulate the flow and solve the Euler equations. The shock front conservative tracking algorithm was used to have the finer mesh (Adaptive mesh refinement AMR) at the wave front location. The non-ideal environment is an environment in which external factors, such as friction, cause the detonation behavior to deviate from the ideal behavior. Therefore, the innovation of the present work is modeling detonation in these non-ideal conditions for mixtures with very low activation energy to detect its failure mechanism. The effect of momentum loss on the detonation behavior has been parametrically studied at very low activation energy (in which the detonation behavior is completely regular, here, 8). Depending on the level of mixture activation energy, the detonation has its failure mechanism. It is concluded that the failure mechanisms of the detonation in this study are the mechanisms of pressure drop and chemical reaction rate reduction. The un-burnt packet mechanism is not involved in it. The detonation wave, regardless of the amount of mixture activation energy, fails anyway as the momentum loss exceeds a critical limit.

A Piecewise Parabolic Method for Estimating Key Parameters in a Three-dimensional Internal Tidal Adjoint Model

A higher-order and conservative Piecewise Parabolic Method was proposed to estimate the open boundary conditions (OBCs) in a 3D internal tidal adjoint model. The interpolation quality was verified by twin experiments. The results showed that the PPM can obtain smoother OBCs than the results obtained by Cressman interpolation. In real experiments, compared with the CI results, the PPM results can reduce the magnitude of misfit vector (m) between the simulated results and the observations of the tidal stations by 5 percent. Meanwhile, the PPM results can simulate the M2 tidal constituent in the northern South China Sea (SCS) more accurate than the Cressman results.

Parabolic Methods for One Dimensional Advection-Diffusion Type Equation and Application to Burger Equation

In this paper, the piecewise parabolic method is presented for solving the one-dimensional advection-diffusion type equation and its application to the burger equation. First, the given solution domain is discretized by using a uniform Discretization grid point. Next by applying the integration in terms of spatial variable, we discretized the given advection-diffusion type equation and converting it into the system of first-order ordinary differential equation in terms of temporarily variable. Next, by using Taylor series expansion we discretized the obtained system of ordinary differential equation and obtain the central finite difference equation. Then using this difference equation, the given advection-diffusion type equation is solved by using the parabolic method at each specified grid point. To validate the applicability of the proposed method, four model examples are considered and solved at each specific grid point on its solution domain. The stability and convergent analysis of the present method is worked by supported the theoretical and mathematical statements and the accuracy of the solution is obtained. The accuracy of the present method has been shown in the sense of root mean square error norm L2 and maximum absolute error norm L∞ and the local behavior of the solution is captured exactly. Numerical, exact solutions and behavior of maximum absolute error between them have been presented in terms of graphs and the corresponding root means square error norm L2 and maximum absolute error norm L∞ presented in tables. The present method approximates the exact solution very well and it is quite efficient and practically well suited for solving advection-diffusion type equation. The numerical result presented in tables and graphs indicates that the approximate solution is in good agreement with the exact solution. Hence the proposed method is accruable to solve the advection-diffusion type equation.

The Supercomputing Simulation of Instability and Shock Waves in Gas Giant

The research of the gas giant atmosphere behavior during its interaction with a stellar wind is given in paper. The various case of interaction of atmosphere with the dense gas current are considered. The numerical model based on gravitational hydrodynamics equations was used. The combination of operator splitting approach, Godunov’s and HLL methods, and the piecewise parabolic method on local stencil is used for the solution of equations. The computing experiments were made on the supercomputer equipped with Intel Xeon Phi accelerators.

Modeling Ultrasound Propagation in the Moving Brain: Applications to Shear Shock Waves and Traumatic Brain Injury.

Traumatic brain injury (TBI) studies on the living human brain are experimentally infeasible due to ethical reasons and the elastic properties of the brain degrade rapidly postmortem. We present a simulation approach that models ultrasound propagation in the human brain, while it is moving due to the complex shear shock wave deformation from a traumatic impact. Finite difference simulations can model ultrasound propagation in complex media such as human tissue. Recently, we have shown that the fullwave finite difference approach can also be used to represent displacements that are much smaller than the grid size, such as the motion encountered in shear wave propagation from ultrasound elastography. However, this subresolution displacement model, called impedance flow, was only implemented and validated for acoustical media composed of randomly distributed scatterers. Herein, we propose a generalization of the impedance flow method that describes the continuous subresolution motion of structured acoustical maps, and in particular of acoustical maps of the human brain. It is shown that the average error in simulating subresolution displacements using impedance flow is small when compared to the acoustical wavelength ( λ /1702). The method is then applied to acoustical maps of the human brain with a motion that is imposed by the propagation of a shear shock wave. This motion is determined numerically with a custom piecewise parabolic method that is calibrated to ex vivo observations of shear shocks in the porcine brain. Then the fullwave simulation tool is used to model transmit-receive imaging sequences based on an L7-4 imaging transducer. The simulated radio frequency data are beamformed using a conventional delay-and-sum method and a normalized cross-correlation method designed for shock wave tracking is used to determine the tissue motion. This overall process is an in silico reproduction of the experiments that were previously performed to observe shear shock waves in fresh porcine brain. It is shown that the proposed generalized impedance flow method accurately captures the shear wave motion in terms of the wave profile, shock front characteristics, odd harmonic spectrum generation, and acceleration at the shear shock front. We expect that this approach will lead to improvements in image sequence design that takes into account the aberration and multiple reflections from the brain and in the design of tracking algorithms that can more accurately capture the complex brain motion that occurs during a traumatic impact. These methods of modeling ultrasound propagation in moving media can also be applied to other displacements, such as those generated by shear wave elastography or blood flow.

A new code for the numerical simulation of relativistic flows on supercomputers by means of a low-dissipation scheme

A new code to simulate special relativistic hydrodynamic flows on supercomputer architectures with distributed memory is described. The code is based on a combination of Godunov’s method and a piecewise parabolic method with a local stencil. This approach has good conservation properties, correctly reproduces shock waves, and ensures high accuracy on smooth solutions and low dissipation on discontinuities. Only a local computation stencil is needed for the piecewise parabolic reconstruction of the solution. The code scalability is 94% on a cluster, Intel Xeon X5670 NKS-30T, with 768 cores. The results of code verification using a relativistic jet problem and computational experiments on the evolution of a galactic jet are presented. Program summaryProgram title: RHydroBox3DCPC Library link to program files:http://dx.doi.org/10.17632/xskchgynfy.1Licensing provisions: GNU General Public License 3Programming language: C++Nature of problem: The special relativistic hydrodynamic flows.Solution method: The combination of Godunov’s method and a piecewise parabolic method on a local stencil.Additional comments including restrictions and unusual features: External library – MPI.

A Low-Dissipation Numerical Scheme Based on a Piecewise Parabolic Method on a Local Stencil for Mathematical Modeling of Relativistic Hydrodynamic Flows

A low-dissipation numerical method based on a combination of Godunov’s method and a piecewise parabolic method on a local stencil is presented. The construction of the method is described in detail. The method is tested using a one-dimensional problem of breakdown of a discontinuity. The results of a numerical simulation of collision of two relativistic gas spheres are given.

An Algorithm for Recovering the Characteristics of the Initial State of Supernova

An optimization method for the numerical solution of the inverse problem of recovering the initial state of a supernova is proposed. The gradient of the inverse problem objective functional is constructed. The solution of the direct and adjoint problems is based on a combination of the large particle method, Godunov’s method, and piecewise parabolic method on a local stencil. Results of the verification of the proposed numerical method and the results of numerical solution of the Evrard and Sedov collapse problems are presented.

The Piecewise Parabolic Method for the Parameters Estimation in an Ocean Pollutant Transport Adjoint Model

In this paper, a higher-order and conservative Piecewise Parabolic interpolation is proposed to estimate key parameters of an ocean pollutant transport model. The Piecewise Parabolic interpolation quality is verified by groups of experiments. The results show that the Piecewise Parabolic Method (PPM) in independent point scheme (IPS) can realize better parameter estimation with fewer independent points.

Techniques for improving monotonicity in a fourth-order finite-volume algorithm solving shocks and detonations

Techniques are proposed to reduce numerical oscillations in a fourth-order, finite-volume algorithm for solving thermally-perfect, reacting fluid flows with strong discontinuities, such as shock or detonation waves. These additional mechanisms have proven necessary for multispecies flows solved at fourth-order accuracy, and contribute towards bounding the variation of the solution in the vicinity of strong discontinuities. There, oscillations can form due to strong gradients in the flow and may be further intensified by numerical procedures introduced to treat the thermally-perfect thermodynamic system and physical constraints on species mass fractions. The new techniques are designed to respect the conservative property of the base algorithm, retain fourth-order accuracy of the solution in regions of smooth flow, and cooperate with the high-order piecewise parabolic method limiter. Extensive numerical tests, ranging from multispecies mixing flows to reacting flows with detonations, are performed to verify that the new techniques meet the design criteria while effectively suppressing oscillations. The proposed techniques are applied to solve the Shu-Osher and double Mach reflection problems and a set of oblique detonation wave problems. The results demonstrate the effectiveness and robustness of the algorithm.

Mixing in turbulent compressible heated coaxial jets: A numerical study

The computation of compressible coaxial jets has moved into the field of interest not only in fundamental research but also in industrial applications, especially in chemical engineering. Numerical simulations of such flows are performed here, using a code specifically developed for gaseous turbulent flows which can also take into account chemical reactions. The coaxial jet can be regarded as a model of injection device in industrial applications; one can cite combustion and aeroacoustics technology. Three-dimensional numerical simulations in this configuration, already published by various authors in open literature are limited to incompressible isothermal flow. In our work, we have explicitly taken into account the temperature gradient effects on the dynamics and mixing mechanisms. Indeed, we have investigated a spatially developing compressible (isothermal and non-isothermal coaxial jet). The numerical model is based on time and space resolutions of compressible Navier-Stokes equations. The piecewise parabolic method (PPM) is combined with a linearized Riemann solver. This scheme adds non-linear dissipation intermittently just where and when needed in order to avoid spurious oscillations and guarantee monotonicity for the advection equation. The simulation can, therefore, be regarded as large-Eddy simulations: large scales are accurately solved with minimal viscosity and non-linear dissipation extracts energy out of the small scales in order to avoid non-physical oscillations. In order to study the mixing between Air-Air flows, we consider the mixture fraction f to track the mixing between two species seeded in the coaxial jets. A great attention is paid to the spatial-temporal evolution of the mixture fraction f, with a particular interest to probability density function. The comparison between the numerical results and experimental data is fairly good, with respect to the mean and turbulent fields. We found that the inner potential core length in the non-isothermal configuration reduced with respect to the isothermal coaxial jet due to the gradient of temperature. It was shown that in the non-isothermal coaxial jet, the temperature gradient leads to the rapid development of the inner Kelvin-Helmholtz vortices implying an efficient mixing of the species close to the exit of the computational domain.

The use of the piecewise parabolic method on a local stencil for constructing a low dissipation of a numerical solution scheme for mathematical modeling of special relativistic hydrodynamic flows

The use of the piecewise parabolic method on a local stencil for constructing a low dissipation of a numerical solution scheme for mathematical modeling of special relativistic hydrodynamic flows