Search NASA⌕ Search

SEARCH · Search NASA

Results for “diffusion equation”

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 433 records · Page 24
All sourcesNASA NTRSDOE OSTI
DOE OSTI2019

Evolution of Microstructure and Creep Behavior in an Fe-Ni-Cr-Nb-C Alloy during Service in Hydrocarbon Cracker Tubes

Heat resistant, cast austenitic Fe-Ni-Cr-Nb-C alloy tubes are used by the petrochemical industry to transport hydrocarbons at temperatures between 850 and 1050 °C. Exposure to carbon from the process stream and nitrogen from the air causes severe carburization and nitridation in the tube walls, which influences creep resistance and degrades their ductility. A better understanding of these processes would enable their service life to be extended. To this end, we report the results of an experimental and computational study of the changes in microstructure and creep behavior in a representative alloy during service. The microstructure of the fresh material is compared to that of 8- and 10-year-old tubes. The results confirm the presence of M{sub 23}C{sub 6}, M{sub 7}C{sub 3}, and M{sub 2}(CN) phases reported in prior studies, and also reveal the nucleation and growth of the carbo-nitride phase M{sub 6}(CN). We show that the spatial variations of carbide and nitride particle volume fractions in aged tubes can be predicted by a simple reaction–diffusion equation. In addition, the uniaxial compressive creep response of the fresh and aged tubes was obtained as functions of temperature and strain rate. The materials deform by power-law creep, with strain rate versus stress relation as ε-dot =Aσ{sup n}. Aging reduces the stress exponent from n = 8 in fresh tubes to n = 5 in aged tubes, and also reduces the pre-exponential coefficient A. Tests on model alloys with composition identical to that of the matrix combined with finite element simulations of creep in model microstructures reveal that Cr depletion is responsible for the change in stress exponent, while the change in A is caused by the increase in particle volume fraction. By combining the microstructure-scale simulations of creep with the predictions of the reaction–diffusion model of microstructure evolution, the creep behavior of tubes left in service beyond 10 years is estimated.

36 MATERIALS SCIENCE↗
DOE OSTI2017

A Least-Squares-Based Weak Galerkin Finite Element Method for Second Order Elliptic Equations

Here, in this article, we introduce a least-squares-based weak Galerkin finite element method for the second order elliptic equation. This new method is shown to provide very accurate numerical approximations for both the primal and the flux variables. In contrast to other existing least-squares finite element methods, this new method allows us to use discontinuous approximating functions on finite element partitions consisting of arbitrary polygon/polyhedron shapes. We also develop a Schur complement algorithm for the resulting discretization problem by eliminating all the unknowns that represent the solution information in the interior of each element. Optimal order error estimates for both the primal and the flux variables are established. An extensive set of numerical experiments are conducted to demonstrate the robustness, reliability, flexibility, and accuracy of the least-squares-based weak Galerkin finite element method. Finally, the numerical examples cover a wide range of applied problems, including singularly perturbed reaction-diffusion equations and the flow of fluid in porous media with strong anisotropy and heterogeneity.

nite element methods↗
DOE OSTI2026

An efficient explicit implementation of a near-optimal quantum algorithm for simulating linear dissipative differential equations

We propose an efficient block-encoding technique for the implementation of the Linear Combination of Hamiltonian Simulations (LCHS) for simulating dissipative initial-value problems. This algorithm approximates a target nonunitary operator as a weighted sum of Hamiltonian evolutions, thereby emulating a dissipative problem by mixing various time scales. We introduce an efficient encoding of the LCHS into a quantum circuit based on a simple coordinate transformation that turns the dependence on the summation index into a trigonometric function. Classically, this method is equivalent to the use of a highly accurate Fejér-Clenshaw-Curtis quadrature formula. Quantumly, this significantly simplifies block-encoding of a dissipative problem and allows one to perform an exponential number of Hamiltonian simulations by a single Quantum Signal Processing (QSP) circuit. The resulting LCHS circuit has high success probability and the selector scales logarithmically with the number of terms in the LCHS sum and linearly with time. Careful analysis of error convergence proves that this method is more efficient than other LCHS circuits that have recently appeared in the literature. We verify the quantum circuit and its scaling by simulating it on a digital emulator of fault-tolerant quantum computers and, as a test problem, solve the advection-diffusion equation. The proposed algorithm can be used for simulating a wide class of nonunitary initial-value problems including the Liouville equation with added dissipation and linear embeddings of nonlinear systems, such as the Koopman-von Neumann and Carleman embeddings.

Novikau, I [Lawrence Livermore National Laboratory↗
NASA NTRS2013

A Green's Function Approach to Simulate DNA Damage by the Indirect Effect

The DNA damage is of fundamental importance in the understanding of the effects of ionizing radiation. DNA is damaged by the direct effect of radiation (e.g. direct ionization) and by indirect effect (e.g. damage by.OH radicals created by the radiolysis of water). Despite years of research, many questions on the DNA damage by ionizing radiation remains. In the recent years, the Green's functions of the diffusion equation (GFDE) have been used extensively in biochemistry [1], notably to simulate biochemical networks in time and space [2]. In our future work on DNA damage, we wish to use an approach based on the GFDE to refine existing models on the indirect effect of ionizing radiation on DNA. To do so, we will use the code RITRACKS [3] developed at the NASA Johnson Space Center to simulate the radiation track structure and calculate the position of radiolytic species after irradiation. We have also recently developed an efficient Monte‐Carlo sampling algorithm for the GFDE of reversible reactions with an intermediate state [4], which can be modified and adapted to simulate DNA damage by free radicals. To do so, we will use the known reaction rate constants between radicals (OH, eaq, H,...) and the DNA bases, sugars and phosphates and use the sampling algorithms to simulate the diffusion of free radicals and chemical reactions with DNA. These techniques should help the understanding of the contribution of the indirect effect in the formation of DNA damage and double‐strand breaks.

Plante, Ianik↗
DOE OSTI2016

Arbitrary Order Mixed Mimetic Finite Differences Method with Nodal Degrees of Freedom

In this work we consider a modification to an arbitrary order mixed mimetic finite difference method (MFD) for a diffusion equation on general polygonal meshes [1]. The modification is based on moving some degrees of freedom (DoF) for a flux variable from edges to vertices. We showed that for a non-degenerate element this transformation is locally equivalent, i.e. there is a one-to-one map between the new and the old DoF. Globally, on the other hand, this transformation leads to a reduction of the total number of degrees of freedom (by up to 40%) and additional continuity of the discrete flux.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1999

High Order Approximations for Compressible Fluid Dynamics on Unstructured and Cartesian Meshes

The development of high-order accurate numerical discretization techniques for irregular domains and meshes is often cited as one of the remaining challenges facing the field of computational fluid dynamics. In structural mechanics, the advantages of high-order finite element approximation are widely recognized. This is especially true when high-order element approximation is combined with element refinement (h-p refinement). In computational fluid dynamics, high-order discretization methods are infrequently used in the computation of compressible fluid flow. The hyperbolic nature of the governing equations and the presence of solution discontinuities makes high-order accuracy difficult to achieve. Consequently, second-order accurate methods are still predominately used in industrial applications even though evidence suggests that high-order methods may offer a way to significantly improve the resolution and accuracy for these calculations. To address this important topic, a special course was jointly organized by the Applied Vehicle Technology Panel of NATO's Research and Technology Organization (RTO), the von Karman Institute for Fluid Dynamics, and the Numerical Aerospace Simulation Division at the NASA Ames Research Center. The NATO RTO sponsored course entitled "Higher Order Discretization Methods in Computational Fluid Dynamics" was held September 14-18, 1998 at the von Karman Institute for Fluid Dynamics in Belgium and September 21-25, 1998 at the NASA Ames Research Center in the United States. During this special course, lecturers from Europe and the United States gave a series of comprehensive lectures on advanced topics related to the high-order numerical discretization of partial differential equations with primary emphasis given to computational fluid dynamics (CFD). Additional consideration was given to topics in computational physics such as the high-order discretization of the Hamilton-Jacobi, Helmholtz, and elasticity equations. This volume consists of five articles prepared by the special course lecturers. These articles should be of particular relevance to those readers with an interest in numerical discretization techniques which generalize to very high-order accuracy. The articles of Professors Abgrall and Shu consider the mathematical formulation of high-order accurate finite volume schemes utilizing essentially non-oscillatory (ENO) and weighted essentially non-oscillatory (WENO) reconstruction together with upwind flux evaluation. These formulations are particularly effective in computing numerical solutions of conservation laws containing solution discontinuities. Careful attention is given by the authors to implementational issues and techniques for improving the overall efficiency of these methods. The article of Professor Cockburn discusses the discontinuous Galerkin finite element method. This method naturally extends to high-order accuracy and has an interpretation as a finite volume method. Cockburn addresses two important issues associated with the discontinuous Galerkin method: controlling spurious extrema near solution discontinuities via "limiting" and the extension to second order advective-diffusive equations (joint work with Shu). The articles of Dr. Henderson and Professor Schwab consider the mathematical formulation and implementation of the h-p finite element methods using hierarchical basis functions and adaptive mesh refinement. These methods are particularly useful in computing high-order accurate solutions containing perturbative layers and corner singularities. Additional flexibility is obtained using a mortar FEM technique whereby nonconforming elements are interfaced together. Numerous examples are given by Henderson applying the h-p FEM method to the simulation of turbulence and turbulence transition.

Barth, Timothy↗
DOE OSTI2019

Approximate Bayesian Model Inversion for PDEs with Heterogeneous and State-Dependent Coefficients

We present two approximate Bayesian inference methods for parameter estimation in partial differential equation (PDE) models with space-dependent and state-dependent parameters. We demonstrate that these methods provide accurate and cost-effective alternatives to Markov Chain Monte Carlo simulation. We assume a parameterized Gaussian prior on the unknown functions, and approximate the posterior density by a parameterized multivariate Gaussian density. The parameters of the prior and posterior are estimated from sparse observations of the PDE model's states and the unknown functions themselves by maximizing the evidence lower bound (ELBO), a lower bound on the log marginal likelihood of the observations. The first method, Laplace-EM, employs the expectation maximization algorithm to maximize the ELBO, with a Laplace approximation of the posterior on the E-step, and minimization of a Kullback-Leibler divergence on the M-step. The second method, DSVI-EB, employs the doubly stochastic variational inference (DSVI) algorithm, in which the ELBO is maximized via gradient-based stochastic optimization, with nosiy gradients computed via simple Monte Carlo sampling and Gaussian backpropagation. We apply these methods to identifying diffusion coefficients in linear and nonlinear diffusion equations, and we find that both methods provide accurate estimates of posterior densities and the hyperparameters of Gaussian priors. While the Laplace-EM method is more accurate, it requires computing Hessians of the physics model. The DSVI-EB method is found to be less accurate but only requires gradients of the physics model.

approximate Bayesian inference, model inversion, v↗
DOE OSTI2024

Preserving Superconvergence of Spectral Elements for Curved Domains

Spectral element methods (SEM), extensions of finite element methods (FEM), have emerged as significant techniques for solving partial differential equations in physics and engineering. SEM can potentially deliver superior accuracy due to the potential superconvergence in nodal solutions for well-shaped tensor-product elements. However, the accuracy of SEM often degrades in complex geometries due to geometric inaccuracies near curved boundaries and the loss of superconvergence with simplicial or non-tensor-product elements. To overcome the first issue, we propose using geometric refinement, which both refines the mesh near high-curvature regions and increases the degree of geometric basis functions. We show that when using mixed-element meshes with tensor-product elements in the interior of the domain, curvature-based geometric refinement near boundaries can improve the accuracy of the interior elements by reducing pollution errors and preserving the superconvergence in nodal solutions. To address the second issue, we introduce ApSEM, a post-processing technique using the adaptive extended stencil finite element method (AES-FEM) to recover the accuracy near the curved boundaries. The combination of curvature-based geometric refinement and accurate post-processing offers an effective and easier-to-implement alternative to methods reliant on exact geometries. We demonstrate our techniques by solving the convection-diffusion equation in 2D and 3D and show up to two orders of magnitude of improvement in the solution accuracy, even when the elements are poorly shaped near boundaries. We also show the efficiency of ApSEM as it can recover superconvergence in nodal solutions without drastically increasing the computational cost.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Preserving Superconvergence of Spectral Elements for Curved Domains via $h$ and $p$-Geometric Refinement

Spectral element methods (SEM), extensions of finite element methods (FEM), have emerged as significant techniques for solving partial differential equations in physics and engineering. SEM can potentially deliver superior accuracy due to the potential superconvergence in nodal solutions for well-shaped tensor-product elements. However, the accuracy of SEM often degrades in complex geometries due to geometric inaccuracies near curved boundaries and the loss of superconvergence with simplicial or non-tensor-product elements. To overcome the first issue, we propose using h- and p-geometric refinement, which refines the mesh near high-curvature regions and increases the degree of geometric basis functions, respectively. We show that when using mixed-element meshes with tensor-product elements in the interior of the domain, curvature-based geometric refinement near boundaries can improve the accuracy of the interior elements by reducing pollution errors and preserving the superconvergence in nodal solutions. To address the second issue, we introduce a post-processing technique using the adaptive extended stencil finite element method (AES-FEM) to recover the accuracy near the curved boundaries. The combination of curvature-based geometric refinement and accurate post-processing offers an effective and easier-to-implement alternative to methods reliant on exact geometries. We demonstrate our techniques by solving the convection-diffusion equation in 2D and show up to two orders of magnitude of improvement in the solution accuracy, even when the elements are poorly shaped near boundaries.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Classification of machine learning frameworks for data-driven thermal fluid models

Thermal fluid processes are inherently multi-physics and multi-scale, involving mass-momentum-energy transport phenomena at multiple scales. Thermal fluid simulation (TFS) is based on solving conservative equations, for which – except for “first-principles” direct numerical simulation – closure relations (CRs) are required to provide microscopic interactions or so-called sub-grid-scale physics. In practice, TFS is realized through reduced-order modeling, and its CRs as low-fidelity models can be informed by observations and data from relevant and adequately evaluated experiments and high-fidelity simulations. This paper is focused on data-driven TFS models, specifically on their development using machine learning (ML). Five ML frameworks are introduced including physics-separated ML (PSML or Type I ML), physics-evaluated ML (PEML or Type II ML), physics-integrated ML (PIML or Type III ML), physics-recovered (PRML or Type IV ML), and physics-discovered ML (PDML or Type V ML). The frameworks vary in their performance for different applications depending on the level of knowledge of governing physics, source, type, amount and quality of available data for training. Notably, outlined for the first time in this paper, Type III models present stringent requirements on modeling, substantial computing resources for training, and high potential in extracting value from “big data” in thermal fluid research. Here the current paper demonstrates and investigates ML frameworks in three examples. First, we utilize the heat diffusion equation with a nonlinear conductivity model formulated by convolutional neural networks (CNNs) and feedforward neural networks (FNNs) to illustrate the applications of Type I, Type II, Type III, and Type V ML. The results indicate a preference for Type II ML under deficient data support. Type III ML can effectively utilize field data, potentially generating more robust predictions than Type I and Type II ML. CNN-based closures exhibit more predictability than FNN-based closures, but CNN-based closures require more training data to obtain accurate predictions. Second, we illustrate how to employ Type I ML and Type II ML frameworks for data-driven turbulence modeling using reference works. Third, we demonstrate Type I ML by building a deep FNN-based slip closure for two-phase flow modeling. The results show that deep FNN-based closures exhibit a bounded error in the prediction domain.

42 ENGINEERING↗
DOE OSTI2022

Feedback between radiation and transport in photovoltaics

By building on the generalized Hovel model, this work develops an analytical solution to unify photon recycling, light trapping, and carrier transport in solar cells. Enhanced transport due to photon recycling diffusion is considered. While photon recycling diffusion is often negligible, it can significantly support transport in cases where charge diffusion is low but radiative efficiency is high. Next, the diffusion equation is solved to determine carrier collection for planar and textured cells. Results elucidate novel features in the external quantum efficiency (EQE) curve that occur for textured cells with high absorption and limited transport. Then, the effect of limited transport on the radiative recombination of solar cells is investigated in detail. While the traditional treatment of detailed balance uses the applied voltage to calculate a cell's radiative recombination, the present approach uses the spatially resolved quasi-Fermi-level splitting (QFLS). Further, the discrepancy between the models is shown to be inconsequential to performance for cells with low radiative recombination (Si) or with high mobilities (lowly doped GaAs). However, for cells with high radiative recombination and low mobility (highly doped GaAs, polycrystalline CdTe, and thin-film perovskites), incorporating variation in QFLS in the determination of radiative recombination can become necessary for accurate photovoltaic modeling.

14 SOLAR ENERGY↗
NASA NTRS2004

Characteristics of Electromagnetic Pulse Propagation in Metal

It is well known that the solution of the diffusion equation for an electromagnetic field with a time harmonic term, e(sup iwt), is in the form of a traveling wave whose amplitude attenuates over distance into a conducting medium. As the attenuation is an increasing function of frequency, the high frequency components attenuate more rapidly than those of low ones upon entering a well conducting object. At the same time, the phase velocity of an individual component is also an increasing function of frequency causing a broadening of the pulse traveling inside a conductor. In the results of our previous study of numerical simulations, the problem of using a gaussian input pulse was immediately clear. First, having the dominant frequency components distributed around zero, the movement of the peak was not well defined. Second, with the amplitude of fourier components varying slowly over a wide range, the dispersion-induced blurring of the peak position was seen to be severe. For the present study, we have used a gaussian modulated single frequency sinusoidal wave, i. e., the carrier, as an input pulse in an effort to improve the issues related to the unclear movement of peak and dispersion as described above. This was based on the following two anticipated advantages: First, the packet moves in a conductor at the group velocity calculated at the carrier frequency, which means it is well controllable. Second, the amplitude of frequency components other than that of the carrier can be almost negligible, such that the effect of dispersion can be significantly reduced. A series of experiments of transmitting electromagnetic pulses through aluminum plates of various thickness was performed to test the validity of the above points. The results of numerical simulation based on wave propagation are discussed with respect to the experimental results. Finally, a simple simulation was performed based on diffusion of a continuous sine wave input and the results are compared with those of a single frequency sinusoidal wave observed over time at difference locations inside a conductor.

Namkung, M.↗
DOE OSTI2017

A novel approach to evaluate soil heat flux calculation: An analytical review of nine methods

Abstract There are no direct methods to evaluate calculated soil heat flux (SHF) at the surface ( G 0 ). Instead, validation and cross evaluation of methods for calculating G 0 usually rely on the conventional calorimetric method or the degree of the surface energy balance closure. However, there is uncertainty in the calorimetric method itself, and factors apart from G 0 also contribute to nonclosure of the surface energy balance. Here we used a novel approach to evaluate nine different methods for calculating SHF, including the calorimetric method and methods based on analytical solutions of the heat diffusion equation. The SHF ( G z ) measured by a self‐calibrating SHF plate at a depth of z = 5 cm below the surface (hereafter G m _5 cm ) was deployed as a reference. Each SHF calculation method was assessed by comparing the calculated G z at the same depth (hereafter G c _5 cm ) with G m _5 cm . The calorimetric method and simple measurement method performed best in determining G c _5 cm but still underestimated G m _5 cm by 19% during the daytime. Possible causes for this underestimation include errors and uncertainties in SHF measurements and soil thermal properties, as well as the phase lag between G c _5 cm and G m _5 cm . Our results indicate that the calorimetric method achieves the most accurate SHF estimates if self‐calibrating SHF plates are deployed at two depths (e.g., 5 cm and 10 cm), soil temperature and water content measurements are made in a few depths between the two plates, and soil thermal properties are accurately quantified.

Gao, Zhongming↗
DOE OSTI2018

An Extended Trajectory Mechanics Approach for Calculating the Path of a Pressure Transient: Derivation and Illustration

Following an approach used in quantum dynamics, an exponential representation of the hydraulic head transforms the diffusion equation governing pressure propagation into an equivalent set of ordinary differential equations. Using a reservoir simulator to determine one set of dependent variables leaves a reduced set of equations for the path of a pressure transient. Unlike the current approach for computing the path of a transient, based on a high-frequency asymptotic solution, the trajectories resulting from this new formulation are valid for arbitrary spatial variations in aquifer properties. For a medium containing interfaces and layers with sharp boundaries, the trajectory mechanics approach produces paths that are compatible with travel time fields produced by a numerical simulator, while the asymptotic solution produces paths that bend too strongly into high permeability regions. The breakdown of the conventional asymptotic solution, due to the presence of sharp boundaries, has implications for model parameter sensitivity calculations and the solution of the inverse problem. For example, near an abrupt boundary, trajectories based on the asymptotic approach deviate significantly from regions of high sensitivity observed in numerical computations. In contrast, paths based on the new trajectory mechanics approach coincide with regions of maximum sensitivity to permeability changes.

42 ENGINEERING↗
DOE OSTI2022

Modelling of edge plasma dynamics with active wall boundary conditions

A self-consistent 2D model is presented for transport in boundary plasma and plasma-facing material walls. Plasma dynamics in the domain is represented by a 2D collisional plasma fluid model in the edge-plasma code UEDGE, and transport of hydrogen and heat in the wall is represented by a system of reaction–diffusion equations in the 1D wall code FACE. To account for variation of parameters along the wall, in the coupled model multiple instances of the FACE code run in parallel. Here, the coupled model provides a tool for investigating a range of dynamic plasma–material interactions phenomena in 2D. For demonstration of its capability, one application of particular interest is the role of active wall in tokamak strike point sweeping proposed for mitigation of divertor heat loads. In the present study, the coupled calculations are applied to investigation of the impact of heat and hydrogen transport in the material wall on the divertor plasma and target heat load during sweeping of the target strike point for parameters of a high-power tokamak.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2019

Kinetic Simulations of Cu Doping in Chlorinated CdSeTe PV Absorbers

This paper reports on 1D kinetic modeling of p-type doping formation in Cu-doped chlorinated CdSeTe photovoltaic (PV) absorbers with graded Se profiles. Following the recent progress in kinetic simulations of the defect chemistry in stoichiometric CdTe films, this work extends simulation capabilities to a domain of semiconductor alloy films with graded stoichiometries. The defect formation energies and ionization levels, as well as the diffusion and reaction barriers used in the reaction-diffusion equations are calculated from the first principles. A new formalism is employed to account for the uncertainty of the defect formation energies in semiconductor alloys caused by unknown local surrounding. Finally, the developed methodology is used to model a practical fabrication process of p-type doping formation in graded CdSeTe absorbers and to study the effect of alloy stoichiometry on the doping activation.

14 SOLAR ENERGY↗
DOE OSTI2020

Learning compact physics‐aware delayed photocurrent models using dynamic mode decomposition

Abstract Radiation‐induced photocurrent in semiconductor devices can be simulated using complex physics‐based models, which are accurate, but computationally expensive. This presents a challenge for implementing device characteristics in high‐level circuit simulations where it is computationally infeasible to evaluate detailed models for multiple individual circuit elements. In this work we demonstrate a procedure for learning compact delayed photocurrent models that are efficient enough to implement in large‐scale circuit simulations, but remain faithful to the underlying physics. Our approach utilizes dynamic mode decomposition (DMD), a system identification technique for learning reduced‐order discrete‐time dynamical systems from time series data based on singular value decomposition. To obtain physics‐aware device models, we simulate the excess carrier density induced by radiation pulses by solving numerically the ambipolar diffusion equation, then use the simulated internal state as training data for the DMD algorithm. Our results show that the significantly reduced‐order delayed photocurrent models obtained via this method accurately approximate the dynamics of the internal excess carrier density—which can be used to calculate the induced current at the device boundaries—while remaining compact enough to incorporate into larger circuit simulations.

Hanson, Joshua↗
DOE OSTI2020

Nonoverlapping Localized Exponential Time Differencing Methods for Diffusion Problems

In this paper, we propose nonoverlapping localized exponential time differencing (ETD) methods for diffusion problems. The model time-dependent diffusion equation is first reformulated on subdomains based on the nonoverlapping domain decomposition, in which Neumann boundary conditions are imposed on the interfaces for the subdomain problems and Dirichlet type conditions are enforced to form a space-time interface problem. After spatial discretization by standard central finite differences and temporal integration with the first or second order ETD methods, the fully discrete interface problem is obtained. Such an interface problem is then solved iteratively either at each time step or over the whole time interval: the former involves the solution of stationary problems in each subdomain at each iteration while the latter involves the solution of time-dependent subdomain problems at each iteration. For both approaches, we prove that localized ETD solutions conserve mass exactly and converge in time to the exact space semidiscrete solution. Numerical experiments in two dimensions are also presented to illustrate the performance of the proposed methods.

Mass conservation↗