Numerical Study of the Two-Species Vlasov-Amp\`{e}re System: Energy-Conserving Schemes and the Current-Driven Ion-Acoustic Instability

In this paper, we propose energy-conserving Eulerian solvers for the two-species Vlasov-Amp\`{e}re (VA) system and apply the methods to simulate current-driven ion-acoustic instability. The algorithm is generalized from our previous work for the single-species VA system and Vlasov-Maxwell (VM) system. The main feature of the schemes is their ability to preserve the total particle number and total energy on the fully discrete level regardless of mesh size. Those are desired properties of numerical schemes especially for long time simulations with under-resolved mesh. The conservation is realized by explicit and implicit energy-conserving temporal discretizations, and the discontinuous Galerkin (DG) spatial discretizations. We benchmarked our algorithms on a test example to check the one-species limit, and the current-driven ion-acoustic instability. To simulate the current-driven ion-acoustic instability, a slight modification for the implicit method is necessary to fully decouple the split equations. This is achieved by a Gauss-Seidel type iteration technique. Numerical results verified the conservation and performance of our methods

Energy-conserving discontinuous Galerkin methods for the Vlasov-Amp\`{e}re system

In this paper, we propose energy-conserving numerical schemes for the Vlasov-Amp\`{e}re (VA) systems. The VA system is a model used to describe the evolution of probability density function of charged particles under self consistent electric field in plasmas. It conserves many physical quantities, including the total energy which is comprised of the kinetic and electric energy. Unlike the total particle number conservation, the total energy conservation is challenging to achieve. For simulations in longer time ranges, negligence of this fact could cause unphysical results, such as plasma self heating or cooling. In this paper, we develop the first Eulerian solvers that can preserve fully discrete total energy conservation. The main components of our solvers include explicit or implicit energy-conserving temporal discretizations, an energy-conserving operator splitting for the VA equation and discontinuous Galerkin finite element methods for the spatial discretizations. We validate our schemes by rigorous derivations and benchmark numerical examples such as Landau damping, two-stream instability and bump-on-tail instability

A multi-fidelity machine learning based semi-Lagrangian finite volume scheme for linear transport equations and the nonlinear Vlasov-Poisson system

Machine-learning (ML) based discretization has been developed to simulate complex partial differential equations (PDEs) with tremendous success across various fields. These learned PDE solvers can effectively resolve the underlying solution structures of interest and achieve a level of accuracy which often requires an order-of-magnitude finer grid for a conventional numerical method using polynomial-based approximations. In a previous work in [13], we introduced a learned finite volume discretization that further incorporates the semi-Lagrangian (SL) mechanism, enabling larger CFL numbers for stability. However, the efficiency and effectiveness of such methodology heavily rely on the availability of abundant high-resolution training data, which can be prohibitively expensive to obtain. To address this challenge, in this paper, we propose a novel multi-fidelity ML-based SL method for transport equations. This method leverages a combination of a small amount of high-fidelity data and sufficient but cheaper low-fidelity data. The approach is designed based on a composite convolutional neural network architecture that explore the inherent correlation between high-fidelity and low-fidelity data. The proposed method demonstrates the capability to achieve a reasonable level of accuracy, particularly in scenarios where a single-fidelity model fails to generalize effectively. We further extend the method to the nonlinear Vlasov-Poisson system by employing high order Runge-Kutta exponential integrators. A collection of numerical tests are provided to validate the efficiency and accuracy of the proposed method

A learned conservative semi-Lagrangian finite volume scheme for transport simulations

Semi-Lagrangian (SL) schemes are known as a major numerical tool for solving transport equations with many advantages and have been widely deployed in the fields of computational fluid dynamics, plasma physics modeling, numerical weather prediction, among others. In this work, we develop a novel machine learning-assisted approach to accelerate the conventional SL finite volume (FV) schemes. The proposed scheme avoids the expensive tracking of upstream cells but attempts to learn the SL discretization from the data by incorporating specific inductive biases in the neural network, significantly simplifying the algorithm implementation and leading to improved efficiency. In addition, the method delivers sharp shock transitions and a level of accuracy that would typically require a much finer grid with traditional transport solvers. Numerical tests demonstrate the effectiveness and efficiency of the proposed method.Comment: 24 page

Evaluation of eight-style Tai chi on cognitive function in patients with cognitive impairment of cerebral small vessel disease: study protocol for a randomised controlled trial

Introduction Cerebral small vessel disease (CSVD) is a critical factor that causes cognitive decline and progresses to vascular dementia and acute cerebrovascular events. Tai chi has been proven to improve nerve plasticity formation and directly improve cognitive function compared with other sports therapy, which has shown its unique advantages. However, more medical evidence needs to be collected in order to verify that Tai chi exercises can improve cognitive impairment due to CSVD. The main purposes of this study are to investigate the effect of Tai chi exercise on neuropsychological outcomes of patients with cognitive impairment related to CSVD and to explore its mechanism of action with neuroimaging, including functional MRI (fMRI) and event-related potential (P300).Methods and analysis The design of this study is a randomised controlled trial with two parallel groups in a 1:1 allocation ratio with allocation concealment and assessor blinding. A total of 106 participants will be enrolled and randomised to the 24-week Tai chi exercise intervention group and 24-week health education control group. Global cognitive function and the specific domains of cognition (memory, processing speed, executive function, attention and verbal learning and memory) will be assessed at baseline and 12 and 24 weeks after randomisation. At the same time, fMRI and P300 will be measured the structure and function of brain regions related to cognitive function at baseline and 24 weeks after randomisation. Recruitment is currently ongoing (recruitment began on 9 November 2020). The approximate completion date for recruitment is in April 2021, and we anticipate to complete the study by December 2021.Ethics and dissemination Ethics approval was given by the Medical Ethics Committee of the Affiliated People’s Hospital of Fujian University of Traditional Chinese Medicine (approval number: 2019-058-04). The findings will be disseminated through peer-reviewed publications and at scientific conferences.Trial registration number ChiCTR2000033176; Pre-results

Highly efficient energy-conserving moment method for the multi-dimensional Vlasov-Maxwell system

We present an energy-conserving numerical scheme to solve the Vlasov-Maxwell (VM) system based on the regularized moment method proposed in [Z. Cai, Y. Fan, and R. Li. CPAM, 2014]. The globally hyperbolic moment system is deduced for the multi-dimensional VM system under the framework of the Hermite expansions, where the expansion center and the scaling factor are set as the macroscopic velocity and local temperature, respectively. Thus, the effect of the Lorentz force term could be reduced into several ODEs about the macroscopic velocity and the moment coefficients of higher order, which could significantly reduce the computational cost of the whole system. An energy-conserving numerical scheme is proposed to solve the moment equations and the Maxwell equations, where only a linear equation system needs to be solved. Several numerical examples such as the two-stream instability, Weibel instability, and the two-dimensional Orszag Tang vortex problem are studied to validate the efficiency and excellent energy-preserving property of the numerical scheme

Energy-conserving numerical simulations of electron holes in two-species plasmas

In this paper, we apply our recently developed energy-conserving discontinuous Galerkin (DG) methods for the two-species Vlasov-Ampère system to simulate the evolution of electron holes (EHs). The EH is an important Bernstein-Greene-Kurskal (BGK) state and is constructed based on the Schamel distribution in our simulation.Even though the knowledge of steady state EHs has advanced significantly, little is known about the full dynamics of EHs that nonlinearly interact with ions in plasmas. In this paper, we simulate the full dynamics of EHs with DG finite element methods, coupled with explicit and implicit time integrators. Our methods are demonstrated to be conservative in the total energy and particle numbers for both species. By varying the mass and temperature ratios, we observe the stationary and moving EHs, as well as the break up of EHs at later times upon initial perturbation of the electron distribution. In addition, we perform a detailed numerical study for the BGK states for the nonlinear evolutions of EH simulations. Our simulation results should help to understand the dynamics of large amplitude EHs that nonlinearly interact with ions in space and laboratory plasmas

Serum magnesium levels and lung cancer risk: a meta-analysis

Abstract Background Whether serum magnesium levels were lower in patients with lung cancer than that in healthy controls is controversial. The aim of this study was to identify and synthesize all citations evaluating the relationship between serum magnesium levels and lung cancer. Methods We searched PubMed, WanFang, China National Knowledge Internet (CNKI), and SinoMed databases for relevant studies before December 31, 2017. Two authors independently selected studies, extracted data, and assessed risk of bias. Results Eleven citations comprising 707 cases with lung cancer and 7595 healthy controls were included in our study. Serum magnesium levels were not significantly lower in patients with lung cancer [summary SMD = 0.193, 95%CI = − 1.504 to 1.890] when compared to health controls, with significant heterogeneity (I 2 = 99.6%, P < 0.001) found. Negative associations were found among Asian populations [summary SMD = 0.229, 95%CI = − 1.637 to 2.094] and European populations [summary SMD = − 0.168, 95%CI = − 0.482 to 0.147]. No publication bias was found using the test of Egger and funnel plot. Conclusions Our study suggested that serum magnesium levels had no significant association on lung cancer risk