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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 757 records · Page 42
All sourcesNASA NTRSDOE OSTI
DOE OSTI2017

Acceleration of diffusive molecular dynamics simulations through mean field approximation and subcycling time integration

Highlights: • Two numerical methods are presented to accelerate DMD simulations. • The methods greatly reduce the computation time without much loss of accuracy. • The accelerated DMD model is capable of simulating long-term mass transport at atomistic length scales. • Demonstrated for the propagation of an atomically sharp phase boundary over 30 seconds. - Abstract: Diffusive Molecular Dynamics (DMD) is a class of recently developed computational models for the simulation of long-term diffusive mass transport at atomistic length scales. Compared to previous atomistic models, e.g., transition state theory based accelerated molecular dynamics, DMD allows the use of larger time-step sizes, but has a higher computational complexity at each time-step due to the need to solve a nonlinear optimization problem at every time-step. This paper presents two numerical methods to accelerate DMD based simulations. First, we show that when a many-body potential function, e.g., embedded atom method (EAM), is employed, the cost of DMD is dominated by the computation of the mean of the potential function and its derivatives, which are high-dimensional random variables. To reduce the cost, we explore both first- and second-order mean field approximations. Specifically, we show that the first-order approximation, which uses a point estimate to calculate the mean, can reduce the cost by two to three orders of magnitude, but may introduce relatively large error in the solution. We show that adding an approximate second-order correction term can significantly reduce the error without much increase in computational cost. Second, we show that DMD can be significantly accelerated through subcycling time integration, as the cost of integrating the empirical diffusion equation is much lower than that of the optimization solver. To assess the DMD model and the numerical approximation methods, we present two groups of numerical experiments that simulate the diffusion of hydrogen in palladium nanoparticles. In particular, we show that the computational framework is capable of capturing the propagation of an atomically sharp phase boundary over a time window of more than 30 seconds. The effects of the proposed numerical methods on solution accuracy and computation time are also assessed quantitatively.

36 MATERIALS SCIENCE↗
DOE OSTI2022

Physics-Informed Sparse Gaussian Process for Probabilistic Stability Analysis of Large-Scale Power System with Dynamic PVs and Loads

This work proposes a physics-informed sparse Gaussian process (SGP) for probabilistic stability assessment of large-scale power systems in the presence of uncertain dynamic PVs and loads. The differential and algebraic equations considering uncertainties from dynamic PVs and loads are reformulated to a nonlinear mapping relationship that allows the application of SGP. Thanks to the nonparametric characteristic of Gaussian process, the proposed framework does not require distributions of uncertain inputs and this distinguishes it from existing approaches. As the original Gaussian process is not scalable to large-scale systems with high dimensional uncertain inputs, this paper develops the SGP with a stochastic variational inference technique. It leads to approximately two orders of complex reduction. A data pre-processing step is also introduced to tackle the coexistence of stable and unstable cases by sample clustering and constructing separate SGPs. The probabilistic transient stability index is analyzed to assess system stability under different uncertain dynamics loads and PVs. Comparisons are performed with the sampling-based, the polynomial chaos expansion-based, and traditional Gaussian process-based methods on the modified IEEE 118-bus and Texas 2000-bus systems under various scenarios, including different levels of uncertainties and the existence of nonlinear correlations among dynamic PVs. The impacts of data quality and quantity issues are also investigated. It is shown that the proposed SGP achieves significantly improved computational efficiency while maintaining high accuracy with a limited number of data.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2026

Taylor approximation variance reduction for approximation errors in PDE-constrained Bayesian inverse problems

In numerous applications, surrogate models are used as a replacement for accurate parameter-to-observable mappings when solving large-scale inverse problems governed by partial differential equations (PDEs). The surrogate model may be a computationally cheaper alternative to the accurate parameter-to-observable mappings and/or may ignore additional unknowns or sources of uncertainty. The Bayesian approximation error (BAE) approach provides a means to account for the induced uncertainties and approximation errors, i.e. the errors between the accurate parameter-to-observable mapping and the surrogate. The statistics of these errors are, however, in general unknown a priori, and are thus calculated using Monte Carlo sampling. Although the sampling is typically carried out offline, i.e. before considering the data, the process can still represent a computational bottleneck. In this work, we develop a scalable computational approach for reducing the costs associated with the sampling stage of the BAE approach. Specifically, we consider the Taylor expansion of the accurate and surrogate forward models with respect to the uncertain parameter fields either as a control variate for variance reduction or as a means to directly and efficiently approximate the mean and covariance of the approximation errors. We propose efficient methods for evaluating the expressions for the mean and covariance of the Taylor approximations based on linear(-ized) PDE solves. Furthermore, the proposed approach is independent of the dimension of the uncertain parameter, depending instead on the intrinsic dimension of the data, ensuring scalability to high-dimensional problems. The potential benefits of the proposed approach are demonstrated for two high-dimensional inverse problems governed by PDE examples, namely for the estimation of a distributed Robin boundary coefficient in a linear diffusion problem, and for a coefficient estimation problem governed by a nonlinear diffusion problem.

Bayesian approximation error↗
DOE OSTI2024

A simulation study of the ability to detect power distribution perturbations in the texas A&M TRIGA reactor with self-powered neutron detectors

Given the variety of ways that nuclear reactor core power may be perturbed, reactor operators and developers are keen on understanding the accuracy and convergence time during which perturbations in reactor power distribution may be synthesized (i.e., inferred) from an array of in-core radiation detectors. A simulation study was conducted as described herein using a highly detailed model of the Texas A&M Training, Research, Isotopes, General Atomics Reactor, in which an array of self-powered neutron detectors (SPNDs) was considered for input to the power synthesis methodology. The core power synthesis is conducted using a point-based iterative method with an iterative loop built in to ensure working equation consistency. The forward problem of SPND response to simulated perturbations in reactor power was solved for Gaussian peak-type perturbations in the reactor power distribution. These perturbations varied in variance, amplitude, and core location to assess their impact on synthesis error and to determine the number of iterations required for convergence. A relation between the unique resolvability limit and perturbation width was identified such that the maximum synthesis error increased rapidly when the peak width went beneath this limit (a width approximating half the reactor’s fuel pin-to-pin pitch); this resolvability limit is specific to the SPND configuration and fuel segmentation considered herein. The synthesis error increased linearly with perturbation peak amplitude, whereas the convergence time increased nonlinearly. Perturbations located closer to the center of the core were synthesized more accurately, albeit with a higher number of required iterations. These findings provide a qualitative and quantitative understanding of the accuracy and speed at which different types of spatial power perturbations can be resolved in light-water reactors.

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗
DOE OSTI2018

Cr 2.37 Ga 3 Se 8 : A Quasi-Two-Dimensional Magnetic Semiconductor

We present a novel magnetic semiconductor, Cr 2.37 Ga 3 Se 8 , synthesized by partially replacing magnetic Cr 3+ in antiferromagnetic Cr 5+δ Se 8 with nonmagnetic Ga 3+ . The crystal structure of Cr 2.37 Ga 3 Se 8 was determined by both powder and single-crystal X-ray diffraction. The title compound crystallizes in a monoclinic structure with space group C2/m (No. 12). In Cr 2.37 Ga 3 Se 8 , the Cr atoms are surrounded by 6 Se atoms and form filled octahedral clusters, while Ga atoms are centered in the Se 4 tetrahedral clusters. The two kinds of clusters pack alternatingly along the c-axis, which results in a quasi-two-dimensional layered structure. The magnetization (M) measurement shows the development of short-range ferromagnetic coupling below the Curie-Weiss (CW) temperature θ CW ~ 92 K, evidenced by the nonlinear field dependence of M. However, the magnetic susceptibility exhibits a peak at low fields at ~18 K, indicating the existence of an antiferromagnetic interaction as well. Electronic structure calculations using the WIEN2k program in the local spin density approximation indicate that the magnetism arises exclusively from local moments of the Cr atoms. Here, the electrical resistivity measurement of the Cr 2.37 Ga 3 Se 8 sample confirms that this material is a semiconductor with the band gap ~0.26 eV. Meanwhile, the experimental band gap (~0.26 eV) is close to the theoretical prediction using the WIEN2k program (~0.35 eV).

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2022

The roles of kinematic constraint and diffusion in non-equilibrium solid state phase transformations of Ti-6Al-4V

A solid state phase transformation of Ti-6Al-4V was studied using high speed in situ x-ray diffraction measurements made during rapid cooling of a cold metal transfer arc weld bead deposited onto a water cooled substrate. Analysis of body centered cubic (BCC) and hexagonal close packed (HCP) lattices revealed an abrupt, nonlinear shift in the lattice parameters of both phases just after the HCP phase had nucleated. Postmortem transmission electron microscopy confirmed that V diffusion was mostly suppressed during cooling. Together, these results indicate that at this cooling rate of approximately 10 4 K/s, which is representative of cooling rates of many additive manufacturing and welding processes, kinematic coherency of the BCC–HCP interfaces gives rise to the anomalous lattice expansion and contraction behaviors of both phases during the initial nucleation and growth stages of (mostly) martensitic transformation from BCC to HCP; the role of diffusion in such lattice anomalies is shown to be minimal.

36 MATERIALS SCIENCE↗
DOE OSTI2017

Stability of new exact solutions of the nonlinear Schrödinger equation in a Pöschl–Teller external potential

Here, we discuss the stability properties of the solutions of the general nonlinear Schrödinger equation in 1 + 1 dimensions in an external potential derivable from a parity-time ($ \newcommand{\PT}{\mathcal{PT}} \PT$ ) symmetric superpotential $W(x)$ that we considered earlier in Kevrekidis et al (2015 Phys. Rev. E 92 042901). In particular we consider the nonlinear partial differential equation $ \{i \, \partial_t + \partial_x^2 - V(x) + g \vert \psi(x, t) \vert ^{2\kappa} \} \, \psi(x, t) = 0 \>, $ for arbitrary nonlinearity parameter κ, where $g= \pm1$ and V is the well known Pöschl–Teller potential which we allow to be repulsive as well as attractive. Using energy landscape methods, linear stability analysis as well as a time dependent variational approximation, we derive consistent analytic results for the domains of instability of these new exact solutions as a function of the strength of the external potential and κ. For the repulsive potential we show that there is a translational instability which can be understood in terms of the energy landscape as a function of a stretching parameter and a translation parameter being a saddle near the exact solution. In this case, numerical simulations show that if we start with the exact solution, the initial wave function breaks into two pieces traveling in opposite directions. If we explore the slightly perturbed solution situations, a 1% change in initial conditions can change significantly the details of how the wave function breaks into two separate pieces. For the attractive potential, changing the initial conditions by 1% modifies the domain of stability only slightly. For the case of the attractive potential and negative g perturbed solutions merely oscillate with the oscillation frequencies predicted by the variational approximation.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2017

Thermal instability of a reconnecting current layer as a trigger for solar flares

The stability of small perturbations of a reconnecting current layer (CL) in a plasma with a strong magnetic field has been investigated in the approximation of dissipative magnetohydrodynamics. The case where the wavevector of the perturbations is parallel to the electric current in the CL has been considered. The suppression of plasma heat conduction by a magnetic field perturbation inside the CL is shown to be responsible for the instability. At the linear stage of instability development, the perturbations grow with the characteristic radiative plasma cooling time calculated in the approximation of an optically thin plasma with cosmic abundances of elements. The formation of a periodic structure of cold and hot magnetic flux tubes, viz., filaments, located across the direction of the electric current, should be expected at the nonlinear stage of the instability in the CL. The proposed mechanism of the thermal CL instability can explain the sequential brightening (ignition) in the arcades of magnetic loops in solar flares.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2026

Measurement and feedback-driven adaptive dynamics in the classical and quantum kicked top

In classical dynamical systems, stochastic feedback can stabilize otherwise unstable periodic orbits, giving rise to distinct controlled and uncontrolled phases as the rate of control application is varied. In this work, we apply these control protocols in classical, semiclassical, and quantum regimes to the kicked top, a paradigmatic model of quantum chaos. The quantum kicked top, modeled as the dynamics of a spin-S object, naturally interpolates between these regimes with the spin size S acting as an effective Planck constant. We show that the dynamics of the kicked top in classical, semiclassical, and fully quantum limits can all be controlled using stochastic feedback protocols. Comparing the full quantum dynamics to a truncated Wigner approximation that captures quantum noise but neglects interference beyond the Ehrenfest time, we find that low-moment observables are largely accounted for semiclassically, while the remaining discrepancy in higher moments is consistent with contributions from interference and possibly nonlinearities in rare trajectories that explore the compact phase space. We also find rapid purification in the numerics studied for all rates of control considered, suggesting that control quenches the top's ability to encode a qubit of quantum information even in the uncontrolled phase.

Prasad, Mahaveer [Tata Inst.; Singapore U. Tech. D↗
DOE OSTI2015

RE-EXAMINATION OF THE EXPECTED GAMMA-RAY EMISSION OF SUPERNOVA REMNANT SN 1987A

A nonlinear kinetic theory, combining cosmic-ray (CR) acceleration in supernova remnants (SNRs) with their gas dynamics, is used to re-examine the nonthermal properties of the remnant of SN 1987A for an extended evolutionary period of 5–50 year. This spherically symmetric model is approximately applied to the different features of the SNR, consisting of (i) a blue supergiant wind and bubble, and (ii) of the swept-up red supergiant (RSG) wind structures in the form of an H ii region, an equatorial ring (ER), and an hourglass region. The RSG wind involves a mass loss rate that decreases significantly with elevation above and below the equatorial plane. The model adapts recent three-dimensional hydrodynamical simulations by Potter et al. in 2014 that use a significantlysmaller ionized mass of the ER than assumed in the earlier studies by the present authors. The SNR shock recently swept up the ER, which is the densest region in the immediate circumstellar environment. Therefore, the expected gamma-ray energy flux density at TeV energies in the current epoch has already reached its maximal value of ∼10{sup −13} erg cm{sup −2} s{sup −1}. This flux should decrease by a factor of about two over the next 10 years.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2016

Quantization and anomalous structures in the conductance of Si/SiGe quantum point contacts

Quantum point contacts (QPCs) are fabricated on modulation-doped Si/SiGe heterostructures and ballistic transport is studied at low temperatures. We observe quantized conductance with subband separations up to 4 meV and anomalies in the first conductance plateau at 4e{sup 2}/h. At a temperature of T = 22 mK in the linear transport regime, a weak anomalous kink structure arises close to 0.5(4e{sup 2}/h), which develops into a distinct plateau-like structure as temperature is raised up to T = 4 K. Under magnetic field parallel to the wire up to B = 14 T, the anomaly evolves into the Zeeman spin-split level at 0.5(4e{sup 2}/h), resembling the '0.7 anomaly' in GaAs/AlGaAs QPCs. Additionally, a zero-bias anomaly (ZBA) is observed in nonlinear transport spectroscopy. At T = 22 mK, a parallel magnetic field splits the ZBA peak up into two peaks. At B = 0, elevated temperatures lead to similar splitting, which differs from the behavior of ZBAs in GaAs/AlGaAs QPCs. Under finite dc bias, the differential resistance exhibits additional plateaus approximately at 0.8(4e{sup 2}/h) and 0.2(4e{sup 2}/h) known as '0.85 anomaly' and '0.25 anomaly' in GaAs/AlGaAs QPCs. Unlike the first regular plateau at 4e{sup 2}/h, the 0.2(4e{sup 2}/h) plateau is insensitive to dc bias voltage up to at least V{sub DS} = 80 mV, in-plane magnetic fields up to B = 15 T, and to elevated temperatures up to T = 25 K. We interpret this effect as due to pinching off one of the reservoirs close to the QPC. We do not see any indication of lifting of the valley degeneracy in our samples.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2021

Optimization of battery swapping infrastructure for e-commerce drone delivery

Drone delivery is widely-researched to alter the current e-commerce delivery convention for providing short delivery lead times. Yet, flight range of drones, constrained by the available battery technology, sets a milestone toward realizing the sole-drone delivery. To tackle with the flight range limitation, locating automated battery swapping machines (ABSM) have been proposed and a few studies modeled the problem. Using the ABSMs, drones can be loaded with fully-charged batteries along the route to a demand location. Here, we introduce a mixed-integer nonlinear program to model the problem. The objective of the program is to optimally select ABSM locations, determine the delivery-mode choices (drone-only, truck-only, and mixed delivery) of demand locations, find drone delivery routes, and approximate the baseline requirements for the number of drones and batteries needed. The program minimizes the overall delivery system costs including: ABSM, delivery, drone ownership, battery inventory, and service congestion. A cutting-plane method is developed to find exact solutions in finite iterations. Computational experiments showed that the method quickly yields the optimal solution to instances with less than 60 ABSM candidates and 20 demand locations. A case study shows that the optimal drone delivery infrastructure can save almost 20% cost compared to the conventional truck-only delivery. Sensitivity analyses were conducted to reveal the impact of key parameters in the decision-making and found that a decrease in the ABSM and drone costs highly affect the system cost.

25 ENERGY STORAGE↗
DOE OSTI2020

An explicit scheme to enforce charge conservation in transient Particle-in-Cell simulations with Maxwell-Boltzmann electrons

The Maxwell-Boltzmann electron model is a very popular approximation of the electronic behavior in electrostatic plasmas, adopted in all cases where the electron dynamics is governed only by a balance between electric and pressure forces. In such a model, the electron dynamics is reduced to solving for a nonlinear Poisson problem together with an additional expression enforcing charge conservation. In this work we derive an expression of charge conservation which can be conveniently applied to explicit schemes to update the reference Boltzmann density, particularly useful when simulating plasma sheaths. The scheme can be equally applied to both steady-state and transient problems. Two examples are shown, a steady-state plasma sheath, and a radio-frequency magnetic presheath. Our proposed scheme allows to enforce global charge conservation locally in time, and can thus be applied to the simulation of transient phenomena.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2020

A Comparison of Interface Growth Models Applied to Rayleigh–Taylor and Richtmyer–Meshkov Instabilities

Sophisticated numerical models that contain fluid interfaces rely upon interface evolution models to approximate the transition to turbulence near interfaces, in the presence of Rayleigh–Taylor (RTI) or Richtmyer–Meshkov (RMI) instability. Semi-analytical models have been developed in recent decades to predict the interface growth from an initial state into the nonlinear regime. Two of these models are considered in this study. They are the Goncharov and the z-models. Both of these interface models have strengths and weaknesses, which are examined here. Both of them have been implemented in the xRAGE compressible flow solver, which models fluid interfaces. The flow solver provides the fluids acceleration as a body force to the interface model. The purpose of such interface model is to evolve the early times of interface position as a subgrid model within a compressible flow simulation in order to then initialize a turbulence model. In this work, the interface models are assessed and compared for their evolution of RTI and RMI. The z-model performed better than the Goncharov model for all cases that were explored.

42 ENGINEERING↗
DOE OSTI2022

Adaptive Space-Time Methods for Large Scale Optimal Design

When modeling complex physical systems with advanced dynamics, such as shocks and singularities, many classic methods for solving partial differential equations can return inaccurate or unusable results. One way to resolve these complex dynamics is through r-adaptive refinement methods, in which a fixed number of mesh points are shifted to areas of high interest. The mesh refinement map can be found through the solution of the Monge-Ampére equation, a highly nonlinear partial differential equation. Due to its nonlinearity, the numerical solution of the Monge-Ampére equation is nontrivial and has previously required computationally expensive methods. In this report, we detail our novel optimization-based, multigrid-enabled solver for a low-order finite element approximation of the Monge-Ampére equation. This fast and scalable solver makes r-adaptive meshing more readily available for problems related to large-scale optimal design. Beyond mesh adaptivity, our report discusses additional applications where our fast solver for the Monge-Ampére equation could be easily applied.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2016

Effects of molecular rotation after ionization and prior to fragmentation on observed recoil-frame photoelectron angular distributions in the dissociative photoionization of nonlinear molecules

Experimental angle-resolved photoelectron-photoion coincidence experiments measure photoelectron angular distributions (PADs) in dissociative photoionization (DPI) in the reference frame provided by the momenta of the emitted heavy fragments. By extension of the nomenclature used with DPI of diatomic molecules, we refer to such a PAD as a recoil-frame PAD (RFPAD). When the dissociation is fast compared to molecular rotational and bending motions, the emission directions of the heavy fragments can be used to determine the orientation of the bonds that are broken in the DPI at the time of the ionization, which is known as the axial-recoil approximation (ARA). When the ARA is valid, the RFPADs correspond to molecular-frame photoelectron angular distributions (MFPADs) when the momenta of a sufficient number of the heavy fragments are determined. When only two fragments are formed, the experiment cannot measure the orientation of the fragments about the recoil axes so that the resulting measured PAD is an azimuthally averaged RFPAD (AA-RFPAD). In this paper we consider how the breakdown of the ARA due to rotation will modify the observed RFPADs for DPI processes in nonlinear molecules for ionization by light of arbitrary polarization. This model is applied to the core C $1s$ DPI of ${\mathrm{CH}}_{4}$, with the results compared to experimental measurements and previous theoretical calculations done within the ARA. The published results indicate that there is a breakdown in the ARA for two-fragment events where the heavy-fragment kinetic energy release was less than 9 eV. Finally, including the breakdown of the ARA due to rotation in our calculations gives very good agreement with the experimental AA-RFPAD, leading to an estimate of upper bounds on the predissociative lifetimes as a function of the kinetic energy release of the intermediate ion states formed in the DPI process.

74 ATOMIC AND MOLECULAR PHYSICS↗
DOE OSTI2021

How Snow Aggregate Ellipsoid Shape and Orientation Variability Affects Fall Speed and Self-Aggregation Rates

Snow aggregate shapes and orientations have long been known to exhibit substantial variability. Despite this observed variability, most weather and climate prediction models use fixed power-law functions that deterministically map particle size to mass and fall speed. As such, integrated quantities like precipitation and self-aggregation rates currently ignore nonlinear effects resulting from variation in shape and orientation for aggregates of the same size. This study therefore develops an analytic framework that couples an empirically based bivariate distribution of ellipsoid shapes to classical hydrodynamic theory so as to capture an appropriate dispersion of masses, projected areas, and fall speeds for an assumed size distribution. For a fixed aggregate size, shape variations produce approximately ±0.13 m s -1 standard deviation of fall speed which increases the mass flux fall speed dispersion by more than 100% over traditional microphysics models. This increased fall speed dispersion results predominantly from shape-induced mass dispersion whereas orientation and drag dispersion play a lesser role. Shape variations can increase mass- and reflectivity-weighted fall speeds by up to 60% of traditional models whereas self-aggregation rates can increase by a factor of 100 for very small slope parameters. This implies that aggregate shape variations effectively forestall the theorized onset of fall speed distribution narrowing and subsequent quenching of the aggregation process. As a result, it is likely that secondary ice formation is necessary to prevent an ever decreasing slope parameter. The mathematical theory presented in this study is used to develop simple correction factors for snow forecast and climate models.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2021

A scalable exponential-DG approach for nonlinear conservation laws: With application to Burger and Euler equations

In this work, we propose an Exponential DG framework for partial differential equations. We decompose 7 governing equations into linear and nonlinear parts to which we apply the discontinuous Galerkin 8 (DG) spatial discretization. In particular, we construct the linear part using Jacobian that effectively 9 capture stiff characteristics in the system. The former is integrated analytically, whereas the latter 10 is approximated. This approach i) is stable with a large Courant number (Cr > 1); ii) supports 11 high-order solutions both in time and space; iii) is computationally favorable compared to IMEX 12 DG methods with no preconditioner; iv) becomes comparable to explicit RKDG methods on uniform 13 mesh and beneficial on non-uniform grid for Euler equations; v) is scalable in a modern massively 14 parallel computing architecture due to its explicit nature of exponential time integrators and com15 pact communication stencil of DG method. Numerical results demonstrate the performance of our 16 proposed methods through various examples. We also discuss the stability and convergence analysis 17 for our exponential DG scheme in the context of Burgers equation.

42 ENGINEERING↗