Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “TIME DEPENDENCE”

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.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 559 records · Page 31
All sourcesNASA NTRSDOE OSTI
DOE OSTI2026

Characterizing skyrmion flow phases with principal component analysis

Principal component analysis (PCA) is a powerful method that can identify patterns in large, complex data sets by constructing low-dimensional order parameters from higher-dimensional feature vectors. There are increasing efforts to use space-and-time-dependent PCA to detect transitions in nonequilibrium systems that are difficult to characterize with equilibrium methods. Here, we demonstrate that feature vectors incorporating the position and velocity information of driven skyrmions moving through random disorder permit PCA to resolve different types of disordered skyrmion motion as a function of driving force and the ratio of the Magnus force to the dissipation. Since the Magnus force creates gyroscopic motion and a finite Hall angle, skyrmions can exhibit a greater range of flow phases than what is observed in overdamped driven systems with quenched disorder. We show that in addition to identifying previously known skyrmion flow phases, PCA detects several additional phases, including different types of channel flow, moving fluids, and partially ordered states. Guided by the PCA analysis, we further characterize the disordered flow phases to elucidate the different microscopic dynamics and show that the changes in the PCA-derived order parameters can be connected to features in bulk transport measures, including the transverse and longitudinal velocity-force curves, differential conductivity, topological defect density, and changes in the skyrmion Hall angle as a function of drive. We discuss how asymmetric feature vectors can be used to improve the resolution of the PCA analysis, and how this technique can be extended to find disordered phases in other nonequilibrium systems with time-dependent dynamics.

36 MATERIALS SCIENCE↗
DOE OSTI2023

Thixotropy, antithixotropy, and viscoelasticity in hysteresis

Thixotropy, antithixotropy, and viscoelasticity are three types of time-dependent dynamics that involve fundamentally different underlying physical processes. Here, we show that the three dynamics exhibit different signatures in hysteresis by examining the fingerprints of the simplest thixotropic kinetic model, a new antithixotropic model that we introduce here, and the Giesekus model. We start by showing that a consistent protocol to generate hysteresis loops is a discrete shear-rate controlled ramp that begins and ends at high shear rates, rather than at low shear rates. Using this protocol, we identify two distinguishing features in the resulting stress versus shear rate loops. The first is the direction of the hysteresis loops: clockwise for thixotropy, but counterclockwise for viscoelasticity and antithixotropy. A second feature is achieved at high ramping rates where all responses lose hysteresis: the viscoelastic response shows a stress plateau at low shear rates due to lack of stress relaxation, whereas the thixotropic and antithixotropic responses are purely viscous with minimal shear thinning or thickening. We establish further evidence for these signatures by experimentally measuring the hysteresis of Laponite suspensions, carbon black suspensions, and poly(ethylene oxide) solutions, each representing a historically accepted example of each class of material behavior. The signatures measured in experiments are consistent with those predicted by the three models. This study reveals different fingerprints in hysteresis loops associated with thixotropy, antithixotropy, and viscoelasticity, which may be helpful in distinguishing the three time-dependent responses.

Mechanics↗
DOE OSTI2025

Physics-Informed Machine Learning Model for Ceramic Matrix Composite Creep

A physics-informed recurrent neural network (RNN) based surrogate model is developed to emulate the nonlinear, time-dependent constitutive behavior of ceramic matrix composites (CMCs) driven by matrix damage and constituent creep at the microscale. Physics-informed constraints are introduced into the surrogate model through regularization to ground the prediction in physics and improve its predictive capabilities. Training data is generated using the high-fidelity generalized method of cells (HFGMC) approach which calls appropriate creep and damage models for each of the constituents. This coupling permits simulating the nonlinear behavior of CMCs based on constituent response at the microscale along with microstructural features such as fiber and porosity volume fraction and fiber radius. The microscale repeating unit cell is loaded under creep fatigue conditions to replicate the material loading experienced in a turbine engine. Therefore, the RNN-based surrogate model is tasked with predicting, as a function of variable input stress sequence, temperature, and microstructural features, the resulting strain history response while satisfying physical constraints related to creep rate, isochoric inelastic deformation, and strain energy density. The trained surrogate model is shown to effectively match the strain history over quantified distributions of microstructural features and relevant loading regimes and temperatures. Neural network based surrogate models can offer efficient alternatives to running computationally intensive multiscale material models to simulate the nonlinear response of large structural models. Therefore, the presented work provides evidence towards the feasibility of developing, training, and running such models for CMCs with complex microstructures, nonlinear time-dependent material response, and under non-monotonic loading conditions.

ceramic matrix composites↗
DOE OSTI2023

Revealing quasi-excitations in the low-density homogeneous electron gas with model exchange–correlation kernels

Time-dependent density functional theory within the linear response regime provides a solid mathematical framework to capture excitations. The accuracy of the theory, however, largely depends on the approximations for the exchange–correlation (xc) kernels. Away from the long-wavelength (or q = 0 short wave-vector) and zero-frequency (ω = 0) limit, the correlation contribution to the kernel becomes more relevant and dominant over exchange. The dielectric function, in principle, can encompass xc effects relevant to describe low-density physics. Furthermore, besides collective plasmon excitations, the dielectric function can reveal collective electron–hole excitations, often dubbed “ghost excitons.” Besides collective excitons, the physics of the low-density regime is rich, as exemplified by a static charge-density wave that was recently found for r s > 69, and was shown to be associated with softening of the plasmon mode. These excitations are seen to be present in much higher density 2D homogeneous electron gases of r s ≳ 4. Here, in this work, we perform a thorough analysis with xc model kernels for excitations of various nature. The uniform electron gas, as a useful model of real metallic systems, is used as a platform for our analysis. We highlight the relevance of exact constraints as we display and explain screening and excitations in the low-density region.

36 MATERIALS SCIENCE↗
DOE OSTI2025

An improved guess for the variational calculation of charge-transfer excitations in large systems

Ab initio quantum-chemical methods that perform well for computing the electronic ground state are not straightforwardly transferable to electronically excited states, particularly in large molecular systems. Wave function theory offers high accuracy, but is often prohibitively expensive. Methods based on time-dependent density functional theory (TD-DFT) are crucially sensitive to the chosen exchange-correlation functional (XCF) parameterization, and system-specific tuning protocols were therefore proposed to address the method's robustness. Methods based on the variational relaxation of the excited-state electron density showcased promising results for the calculation of charge-transfer excitations, but the complex shape of the electronic hypersurface makes convergence to a specific excited state much more difficult than for the ground state when standard variational techniques are applied. We address the latter aspect by providing suitable initial guesses, which we obtain by two separate constrained algorithms. Combined with the squared-gradient minimization algorithm for all-electrons relaxation in a freeze-and-release scheme (FRZ-SGM), we demonstrate that orbital-optimized density functional theory (OO-DFT) calculations can reliably converge to the charge-transfer states of interest even for large molecular systems. We test the FRZ-SGM method on a phenothiazine-anthraquinone CT excitation in a supramolecular Pd(II) coordination cage complex as a function of the cage conformation. This compound has been studied experimentally prior to our work. We compare this freeze-and-release scheme to two XCF reparameterizations, which were recently proposed as low-cost TD-DFT-based alternatives to variational methods. Two dye-semiconductor complexes, which were previously investigated in the context of photovoltaic applications, serve as a second example to investigate the convergence and stability of the FRZ-SGM approach. Our results demonstrate that FRZ-SGM provides reliable convergence for charge-transfer excited states and avoids variational collapse to lower-lying electronic states, whereas time-dependent DFT calculations with an adequate tuning procedure for the range-separation parameter provide a computationally efficient initial estimate of the corresponding energies, with a computational cost comparable to that of configuration-interaction singles (CIS) calculations.

Bogo, Nicola↗
DOE OSTI2025

Real-time inference and extrapolation with Time-Conditioned UNet: Applications in hypersonic flows, incompressible flows, and global temperature forecasting

Neural Operators are fast and accurate surrogates for nonlinear mappings between functional spaces within training domains. Extrapolation beyond the training domain remains a grand challenge across all application areas. We present Time-Conditioned UNet (TC-UNet) as an operator learning method to solve time-dependent PDEs continuously in time without any temporal discretization, including in extrapolation scenarios. TC-UNet incorporates the temporal evolution of the PDE into its architecture by combining a parameter conditioning approach with the attention mechanism from the Transformer architecture. After training, TC-UNet makes real-time inferences on an arbitrary temporal grid. We demonstrate its extrapolation capability on a climate problem by estimating the global temperature for several years and also for inviscid hypersonic flow around a double cone. We propose different training strategies involving temporal bundling and sub-sampling. We demonstrate performance improvements for several benchmarks, performing extrapolation for long time intervals and zero-shot super-resolution time.

Deep learning↗
DOE OSTI2022

Thickness- and Twist-Angle-Dependent Interlayer Excitons in Metal Monochalcogenide Heterostructures

Interlayer excitons, or bound electron-hole pairs whose constituent quasiparticles are located in distinct stacked semiconducting layers, are being intensively studied in heterobilayers of two dimensional semiconductors. They owe their existence to an intrinsic type-II band alignment between both layers that convert these into p-n junctions. Here, we unveil a pronounced interlayer exciton (IX) in heterobilayers of metal monochalcogenides, namely gamma-InSe on epsilon-GaSe, whose pronounced emission is adjustable just by varying their thicknesses given their number of layers dependent direct bandgaps. Time-dependent photoluminescense spectroscopy unveils considerably longer interlayer exciton lifetimes with respect to intralayer ones, thus confirming their nature. The linear Stark effect yields a bound electron-hole pair whose separation d is just (3.6 ± 0.1) Å with d being very close to d Se = 3.4 Å which is the calculated interfacial Se separation. The envelope of IX is twist angle dependent and describable by superimposed emissions that are nearly equally spaced in energy, as if quantized due to localization induced by the small moiré periodicity. Furthermore, these heterostacks are characterized by extremely flat interfacial valence bands making them prime candidates for the observation of magnetism or other correlated electronic phases upon carrier doping.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2022

Optimizing the accuracy of viscoelastic characterization with AFM force–distance experiments in the time and frequency domains

Atomic Force Microscopy (AFM) force-distance (FD) experiments have emerged as an attractive alternative to traditional micro-rheology measurement techniques owing to their versatility of use in materials of a wide range of mechanical properties. Here, we show that the range of time dependent behaviour which can reliably be resolved from the typical method of FD inversion (fitting constitutive FD relations to FD data) is inherently restricted by the experimental parameters: sampling frequency, experiment length, and strain rate. Specifically, we demonstrate that violating these restrictions can result in errors in the values of the parameters of the complex modulus. In the case of complex materials, such as cells, whose behaviour is not specifically understood a priori, the physical sensibility of these parameters cannot be assessed and may lead to falsely attributing a physical phenomenon to an artifact of the violation of these restrictions. We use arguments from information theory to understand the nature of these inconsistencies as well as devise limits on the range of mechanical parameters which can be reliably obtained from FD experiments. The results further demonstrate that the nature of these restrictions depends on the domain (time or frequency) used in the inversion process, with the time domain being far more restrictive than the frequency domain. Lastly, we demonstrate how to use these restrictions to better design FD experiments to target specific timescales of a material's behaviour through our analysis of a polydimethylsiloxane (PDMS) polymer sample.

information theory↗
DOE OSTI2024

Endophyte mediated Populus trichocarpa water use efficiency is dependent on time of day and plant water status

Endophytes are potential partners for improving the resource use efficiency of bioenergy feedstock systems such as short rotation coppiced Populus. Endophytes isolated from members of the Salicaceae family have broad host compatibility and can improve water use efficiency (WUE) through decreases in stomatal conductance. However, the literature is inconsistent with regards to the environmental conditions and temporal patterns of these benefits. This study investigates how endophyte mediated changes in Populus trichocarpa (Torr. and Gray) ‘Nisqually-1’ stomatal conductance and WUE shift with time and scale in response to water-deficit stress. Leaf gas exchange and above-ground productivity were used to evaluate the carbon and water balances of glasshouse grown plants in response to endophyte inoculation and water-deficit. Differences in stomatal conductance between control and inoculated plants were more pronounced (39.7 % decrease, Welch’s two-sample t(14.34 dfadj) = -2.358, p-value = 0.033) under water-deficit conditions in the late morning during a period of higher light intensity. The decrease in stomatal conductance accompanied a substantial increase in intrinsic WUE (iWUE) for water-deficit inoculated plants. However, increases in iWUE did not result in improvements in aboveground productivity or shoot biomass WUE (WUEsb) for water-deficit inoculated plants. This decoupling between iWUE and aboveground productivity may be an indicator of assimilate allocation to microbial metabolism as an additional carbon sink or a shift in carbon allocation towards belowground biomass. Future work should take a whole-plant approach that accounts for diurnal patterns in incident irradiance to evaluate the impact of endophyte inoculation on host WUE and stress tolerance.

59 BASIC BIOLOGICAL SCIENCES↗
DOE OSTI2022

A non-hierarchical correlation discrete variable representation

The correlation discrete variable representation (CDVR) facilitates (multi-layer) multi-configurational time-dependent Hartree (MCTDH) calculations with general potentials. It employs a layered grid representation to efficiently evaluate all potential matrix elements appearing in the MCTDH equations of motion. The original CDVR approach and its multi-layer extension show a hierarchical structure: the size of the grids employed at the different layers increases when moving from an upper layer to a lower one. In this work, a non-hierarchical CDVR approach, which uses identically structured quadratures at all layers of the MCTDH wavefunction representation, is introduced. Here, the non-hierarchical CDVR approach crucially reduces the number of grid points required, compared to the hierarchical CDVR, shows superior scaling properties, and yields identical results for all three representations showing the same topology. Numerical tests studying the photodissociation of NOCl and the vibrational states of CH 3 demonstrate the accuracy of the non-hierarchical CDVR approach.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2025

Efficient Modeling of Structural, Electronic, and Optical Properties of Silver and Gold Metal Nanoclusters and Alloys Using Optimized SCC-DFTB Parameters

Computation of optical properties using conventional time-dependent density functional theory (TD-DFT) is time-consuming and memory-intensive. In this study, we investigate the accuracy and efficiency of the density functional tight binding (DFTB) framework with newly optimized Slater–Koster (SK) parameters for modeling the structural, electronic properties, and absorption spectra of silver and gold nanoclusters and their alloys. Our investigation of the ground state (GS) properties demonstrates that the newly developed GS-SK parameters enable DFTB to closely approximate DFT-calculated bond lengths for octahedron, tetrahedron, icosahedra, and truncated octahedron with sizes Ag n /Au n (n = 19, 20, 38, 55), nanoclusters and Ag 20 /Au 20 nanoalloys, with a maximum deviation of approximately 0.15 Å. Formation energy results indicate that the GS-SK parameters can closely estimate changes in formation energies with alloy composition, and the comparison of electronic structures for Ag 20 , Au 20 , and AgAu alloy nanoclusters using the DFTB approximation reveals good agreement in the projected density of states (DOS) profiles and energy levels. A second set of SK parameters, ES-SK, has been developed to describe excited state (ES) properties, including the absorption spectra of silver octahedron Ag 19 , tetrahedral Ag n (n = 20, 56, 84), truncated octahedron Ag 38 , and icosahedra Ag 55 closed-shell clusters and their gold and alloy counterparts over a broad range of alloy compositions. This parametrization uses TD-DFTB calculations and fine-tunes the d and p eigenvalues by comparing them to reference absorption spectra from first-principles TD-DFT. This enables the generation of absorption spectra that closely match the reference spectra when plasmon excitation is dominant, as demonstrated by studying the plasmonic properties of icosahedral Ag n and Au n (n = 309 and 561) nanoparticles. This includes the rapid loss in plasmon quality when Au partially replaces Ag in alloy clusters. Furthermore, these results provide a foundation for addressing computational bottlenecks in plasmonics and with new prospects for applications in the quantum plasmonics for bimetallic alloys.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2021

Ultrafast formation of domain walls of a charge density wave in SmTe 3

We study ultrafast x-ray diffraction on the charge density wave (CDW) of SmTe 3 using an x-ray free-electron laser. The high momentum and time resolution afforded by the x-ray laser enabled capturing fine wave-vector and time-dependent features of the CDW that originate from fast (in time) and sharp (in real space) variations of the CDW lattice distortion, which we attribute to an inversion of the order parameter. These domain inversions occur near the surface and are caused by the short penetration depth of the near-infrared pump with the wavelength centered at 800 nm, resulting in CDW domain walls perpendicular to the sample surface. These domain walls break the CDW long-range order on the scale of the x-ray probe depth, controlled experimentally by the x-ray incidence angle and suppress the diffraction intensity of the CDW for times much longer than the ~1–ps recovery of the electronic gap observed in time and angle-resolved photoemission spectroscopy. We model the spatial and temporal dependences of the order parameter using a simple Ginzburg-Landau model with all the parameters obtained from the published literature. We find reasonable agreement between the calculated and the measured diffraction across the momentum, time, fluence, and incidence angle dependence without adjusting any parameters. In conclusion, we reconstruct the spatial and temporal dependences of the lattice order parameter and find that at long times, depending on the pump fluence, multiple domain walls remain at distances of a few nanometers from the surface.

36 MATERIALS SCIENCE↗
DOE OSTI2020

Predicting Saturated Hydraulic Conductivity over Time for Degrading Saltstone Vault Concrete - 20376

At the Savannah River Site (SRS), low-level liquid tank waste is mixed with cementitious materials to produce a solid, monolithic waste form referred to as saltstone. The liquid saltstone mixture is pumped into massive concrete structures referred to as saltstone disposal units (SDUs) for curing and long-term storage. SDUs are vaults made of high performance concrete forming a barrier between saltstone and the surrounding environment. Predicting the long-term performance of SDU vault concrete, as well as the saltstone material itself, is critical to understanding the potential releases of radionuclides to the surrounding environment as described in the Saltstone Disposal Facility (SDF) performance assessment (PA). The hydraulic conductivity profile for SDU vault concrete was conservatively assumed to degrade linearly from the initial saturated hydraulic conductivity of the intact concrete over a long period of time. The initial hydraulic properties of saturated SDU vault concrete have been measured, and the final hydraulic properties of degraded vault concrete are assumed to be the same as the porous backfill material that will surround the vault after closure. The assumed final condition of the vault concrete is also conservative in that it provides a greater estimate of water penetration through the vault than would generally be expected. Therefore, insights that provide more realistic but conservative estimates as to how degradation processes may change the hydraulic properties of the SDU vault concrete, the time dependent relationship of the vault concrete hydraulic properties to degradation (and thus time), and the final hydraulic properties of the vault concrete provides useful additions to the SDF PA. This paper focuses on the assumption that the saltstone vault concrete saturated hydraulic conductivity would decrease linearly as a function of degradation time; this assumption resulted from a worst-case, weighted (by relative layer thickness), arithmetic average of intact and backfill material hydraulic conductivities representing flow parallel to a layered system. This research indicates that the effective saturated hydraulic conductivity of the degraded SDU vault concrete should be based on the relative thickness-weighted geometric mean of the intact and degraded layers hydraulic conductivities. This alternative, which is supported extensively in the relevant literature and evaluated in this paper, leads to a more realistic and defensible estimate of the effective saturated hydraulic conductivity in degraded vault concrete while remaining conservative. The impact of using the more realistic estimate of the effective saturated hydraulic conductivity for the degraded concrete decreases the effective saturated hydraulic conductivity by up to several orders of magnitude for early times (i.e., when the SDU vault concrete is assumed degraded via sulfate attack and carbonation). This revised estimate of the effective saturated hydraulic conductivity has been implemented in the recently revised SDF PA. (authors)

12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗
DOE OSTI2020

The Coordination Behaviour of Cu I Photosensitizers Bearing Multidentate Ligands Investigated by X-ray Absorption Spectroscopy

A systematic series of four novel homo- and heteroleptic CuI photosensitizers based on tetradentate 1,10-phenanthroline ligands of the type X^N^N^X containing two additional donor moieties in the 2,9-position (X=SMe or OMe) were designed. Their solid-state structures were assessed by X-ray diffraction. Cyclic voltammetry, UV-vis absorption, emission and X-ray absorption spectroscopy were then used to determine their electrochemical, photophysical and structural features in solution. Following, time-resolved X-ray absorption spectroscopy in the picosecond time scale, coupled with time-dependent density functional theory calculations, provided in-depth information on the excited state electron configurations. For the first time, a significant shortening of the Cu–X distance and a change in the coordination mode to a pentacoordinated geometry is shown in the excited states of the two homoleptic complexes. These findings are important with respect to a precise understanding of the excited state structures and a further stabilization of this type of photosensitizers.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2021

Time evolution of ML-MCTDH wavefunctions. I. Gauge conditions, basis functions, and singularities

We derive a family of equations-of-motion (EOMs) for evolving multi-layer multiconfiguration time-dependent Hartree (ML-MCTDH) wavefunctions that, unlike the standard ML-MCTDH EOMs, never require the evaluation of the inverse of singular matrices. All members of this family of EOMs make use of alternative static gauge conditions than those used for standard ML-MCTDH. These alternative conditions result in an expansion of the wavefunction in terms of a set of potentially arbitrary orthonormal functions, rather than in terms of a set of non-orthonormal and potentially linearly dependent functions, as is the case for standard ML-MCTDH. We show that the EOMs used in the projector splitting integrator (PSI) and the invariant EOM approaches are two special cases of this family obtained from different choices for the dynamic gauge condition, with the invariant EOMs making use of a choice that introduces potentially unbounded operators into the EOMs. As a consequence, all arguments for the existence of parallelizable integration schemes for the invariant EOMs can also be applied to the PSI EOMs.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2021

Statistical learning for predicting density–matrix-based electron dynamics

In this study, we consider the problem of learning density-dependent molecular Hamiltonian matrices from time series of electron density matrices, all in the context of Hartree–Fock theory. Prior work developed a solution to this problem for small molecular systems with density and Hamiltonian matrices of size at most 6 × 6. Here, using a battery of techniques, we scale prior methods to larger molecular systems with, for example, 29 × 29 matrices. This includes systems that either have more electrons or are expressed in large basis sets such as 6-311++G**. Scaling the method to larger systems enhances its relevance for realistic applications in chemistry and physics. To achieve this scaling, we apply dimensionality reduction, ridge regression and analytic computation of Hessians. Through the combination of these techniques, we are able to learn Hamiltonians by minimizing an objective function that encodes local propagation error. Importantly, these learned Hamiltonians can then be used to predict electron dynamics for thousands of steps: When we use our learned Hamiltonians to numerically solve the time-dependent Hartree–Fock equation, we obtain predicted dynamics that are in close quantitative agreement with ground truth dynamics. This includes field-off trajectories similar to the training data and field-on trajectories outside of the training data.

59 BASIC BIOLOGICAL SCIENCES↗
DOE OSTI2021

Bayesian inference of heterogeneous epidemic models: Application to COVID-19 spread accounting for long-term care facilities

Here we propose a high dimensional Bayesian inference framework for learning heterogeneous dynamics of a COVID-19 model, with a specific application to the dynamics and severity of COVID-19 inside and outside long-term care (LTC) facilities. We develop a heterogeneous compartmental model that accounts for the heterogeneity of the time-varying spread and severity of COVID-19 inside and outside LTC facilities, which is characterized by time-dependent stochastic processes and time-independent parameters in ~ 1500 dimensions after discretization. To infer these parameters, we use reported data on the number of confirmed, hospitalized, and deceased cases with suitable post-processing in both a deterministic inversion approach with appropriate regularization as a first step, followed by Bayesian inversion with proper prior distributions. To address the curse of dimensionality and the ill-posedness of the high-dimensional inference problem, we propose use of a dimension-independent projected Stein variational gradient descent method, and demonstrate the intrinsic low-dimensionality of the inverse problem. We present inference results with quantified uncertainties for both New Jersey and Texas, which experienced different epidemic phases and patterns. Moreover, we also present forecasting and validation results based on the empirical posterior samples of our inference for the future trajectory of COVID-19.

42 ENGINEERING↗
DOE OSTI2021

Maximum bound principle preserving integrating factor Runge–Kutta methods for semilinear parabolic equations

A large class of semilinear parabolic equations satisfy the maximum bound principle (MBP) in the sense that the time-dependent solution preserves for any time a uniform pointwise bound imposed by its initial and boundary conditions. Here, the MBP plays a crucial role in understanding the physical meaning and the well-posedness of the mathematical model. Investigation on numerical algorithms with preservation of the MBP has attracted increasingly attentions in recent years, especially for the temporal discretizations, since the violation of MBP may lead to nonphysical solutions or even blow-ups of the algorithms. In this paper, we study high-order MBP-preserving time integration schemes by means of the integrating factor Runge-Kutta (IFRK) method. Beginning with the space-discrete system of semilinear parabolic equations, we present the IFRK method in general form and derive the sufficient conditions for the method to preserve the MBP. In particular, we show that the classic four-stage, fourth-order IFRK scheme is MBP preserving for some typical semilinear systems although not strong stability preserving, which can be instantly applied to the Allen-Cahn type of equations. To our best knowledge, this is the first time to present a fourth-order linear numerical method preserving the MBP. In addition, convergence of these numerical schemes is proved theoretically and verified numerically, as well as their efficiency by simulations of 2D and 3D long-time evolutional behaviors. Numerical experiments are also carried out for a model which is not a typical gradient flow as the Allen-Cahn type of equations.

97 MATHEMATICS AND COMPUTING↗