skip to main content


Title: Reduced basis approximations of the solutions to spectral fractional diffusion problems
Abstract We consider the numerical approximation of the spectral fractional diffusion problem based on the so called Balakrishnan representation. The latter consists of an improper integral approximated via quadratures. At each quadrature point, a reaction-diffusion problem must be approximated and is the method bottle neck. In this work, we propose to reduce the computational cost using a reduced basis strategy allowing for a fast evaluation of the reaction-diffusion problems. The reduced basis does not depend on the fractional power s for 0 < s min ≤ s ≤ s max < 1. It is built offline once for all and used online irrespectively of the fractional power. We analyze the reduced basis strategy and show its exponential convergence. The analytical results are illustrated with insightful numerical experiments.  more » « less
Award ID(s):
1817691
NSF-PAR ID:
10169046
Author(s) / Creator(s):
; ;
Date Published:
Journal Name:
Journal of Numerical Mathematics
Volume:
0
Issue:
0
ISSN:
1570-2820
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. Abstract

    This paper considers an inverse problem for a reaction diffusion equation from overposed final time data. Specifically, we assume that the reaction termis known but modified by a space-dependent coefficientto obtain. Thus the strength of the reaction can vary with location. The inverse problem is to recover this coefficient. Our technique is to use iterative Newton-type methods although we also use and analyse higher order schemes of Halley type. We show that such schemes are well defined and prove convergence results. Our assumption about the diffusion process is also more general in that we will extend the traditional parabolic equation paradigm to include the subdiffusion case based on non-local fractional order operators in time. The final section of the paper shows numerical reconstructions based on the above methods and compares our methodology to previous work based on the linear model withas well as to the nonlinear case. We also show the interdependence between effective reconstruction ofqand the coupling between the value of the final time of measurement and the subdiffusion parameter.

     
    more » « less
  2. Abstract We consider an evolution equation involving the fractional powers, of order s ∈ (0, 1), of a symmetric and uniformly elliptic second order operator and Caputo fractional time derivative of order γ ∈ (1, 2]. Since it has been shown useful for the design of numerical techniques for related problems, we also consider a quasi–stationary elliptic problem that comes from the realization of the spatial fractional diffusion as the Dirichlet-to-Neumann map for a nonuniformly elliptic problem posed on a semi–infinite cylinder. We provide existence and uniqueness results together with energy estimates for both problems. In addition, we derive regularity estimates both in time and space; the time–regularity results show that the usual assumptions made in the numerical analysis literature are problematic. 
    more » « less
  3. Abstract

    This study presents a new methodology for identifying near‐optimal sensor locations for contaminant source tracing in river networks. We define an optimal sensor placement as one that enables the best overall reconstruction of contaminant concentrations from observed data. To establish a physical basis for the problem, we first derive a linear time‐invariant (LTI) model for riverine contaminant transport using the one‐dimensional advection‐reaction‐diffusion equation. We then formulate an optimization problem to find the sensor placement that maximizes theobservabilityof the modeled system and identify two heuristics for efficiently achieving this goal. By evaluating each sensor placement strategy on its ability to reconstruct initial contaminant loads from observed outputs, we find that the best sensor placement is obtained by maximizing the rank of the LTI system's Observability Gramian. This sensor placement strategy enables the best overall reconstruction of both magnitudes and distributions of nonpoint‐source contaminants. Our methodology will enable researchers to build sensor networks that better interpolate pollutant loads in ungauged locations, improve contaminant source identification, and inform more effective pollution control strategies.

     
    more » « less
  4. Enhancing battery energy storage capability and reducing the cost per average energy capacity is urgent to satisfy the increasing energy demand in modern society. The lithium-sulfur (Li-S) battery is especially attractive because of its high theoretical specific energy (around 2600 W h kg-1), low cost, and low toxicity.1 Despite these advantages, the practical utilization of lithium-sulfur (Li-S) batteries to date has been hindered by a series of obstacles, including low active material loading, shuttle effects, and sluggish sulfur conversion kinetics.2 The traditional 2D planer thick electrode is considered as a general approach to enhance the mass loading of the Li-S battery.3 However, the longer diffusion length of lithium ions, which resulted in high tortuosity in the compact stacking thick electrode, decreases the penetration ability of the electrolyte into the entire cathode.4 Although an effort to induce catalysts in the cathode was made to promote sulfur conversion kinetic conditions, catalysts based on transition metals suffered from the low electronic conductivity, and some elements (i.e.: Co, Mn) may even absorb and restrict polysulfides for further reaction. 5 To mitigate the issues listed above, herein we propose a novel sulfur cathode design strategy enabled by additive manufacturing and oxidative chemical vapor deposition (oCVD). 6,7 Specifically, the cathode is designed to have a hierarchal hollow structure via a stereolithography technique to increase sulfur usage. Microchannels are constructed on the tailored sulfur cathode to further fortify the wettability of the electrolyte. The as-printed cathode is then sintered at 700 °C in an N2 atmosphere in order to generate a carbon skeleton (i.e.: carbonization of resin) with intrinsic carbon defects. The intrinsic carbon defects are expected to create favorable sulfur conversion conditions with sufficient electronic conductivity. In this study, the oCVD technique is leveraged to produce a conformal coating layer to eliminate shuttle effects. Identified by scanning electron microscopy and energy-dispersive X-ray spectroscopy mapping characterizations, the oCVD PEDOT is not only covered on the surface of the cathode but also on the inner surface of the microchannels. High-resolution x-ray photoelectron spectroscopy analyses (C 1s and S 2p orbitals) between pristine and modified samples demonstrate that a high concentration of the defects has been produced on the sulfur matrix after sintering and posttreatment. In-operando XRD diffractograms show that the Li2S is generated in the oCVD PEDOT-coated sample during the charge and discharge process even with a high current density, confirming an eminent sulfur conversion kinetic condition. In addition, ICP-OES results of lithium metal anode at different states of charge (SoC) verify that the shuttle effects are excellently restricted by oCVD PEDOT. Overall, the high mass loading (> 5 mg cm-2) with an elevated sulfur utilization ratio, accelerated reaction kinetics and stabilized electrochemical process have been achieved on the sulfur cathode by implementing this innovative cathode design strategy. The results of this study demonstrate significant promises of employing pure sulfur powder with high electrochemical performance and suggest a pathway to the higher energy and power density battery. References: 1 Chen, Y. Adv Mater 33, e2003666. 2 Bhargav, A. Joule 4, 285-291. 3 Liu, S. Nano Energy 63, 103894. 4 Chu, T. Carbon Energy 3. 5 Li, Y. Matter 4, 1142-1188. 6 John P. Lock. Macromolecules 39, 4 (2006). 7 Zekoll, S. Energy & Environmental Science 11, 185-201. 
    more » « less
  5. Simulation of flow and transport in petroleum reservoirs involves solving coupled systems of advection-diffusion-reaction equations with nonlinear flux functions, diffusion coefficients, and reactions/wells. It is important to develop numerical schemes that can approximate all three processes at once, and to high order, so that the physics can be well resolved. In this paper, we propose an approach based on high order, finite volume, implicit, Weighted Essentially NonOscillatory (iWENO) schemes. The resulting schemes are locally mass conservative and, being implicit, suited to systems of advection-diffusion-reaction equations. Moreover, our approach gives unconditionally L-stable schemes for smooth solutions to the linear advection-diffusion-reaction equation in the sense of a von Neumann stability analysis. To illustrate our approach, we develop a third order iWENO scheme for the saturation equation of two-phase flow in porous media in two space dimensions. The keys to high order accuracy are to use WENO reconstruction in space (which handles shocks and steep fronts) combined with a two-stage Radau-IIA Runge-Kutta time integrator. The saturation is approximated by its averages over the mesh elements at the current time level and at two future time levels; therefore, the scheme uses two unknowns per grid block per variable, independent of the spatial dimension. This makes the scheme fairly computationally efficient, both because reconstructions make use of local information that can fit in cache memory, and because the global system has about as small a number of degrees of freedom as possible. The scheme is relatively simple to implement, high order accurate, maintains local mass conservation, applies to general computational meshes, and appears to be robust. Preliminary computational tests show the potential of the scheme to handle advection-diffusion-reaction processes on meshes of quadrilateral gridblocks, and to do so to high order accuracy using relatively long time steps. The new scheme can be viewed as a generalization of standard cell-centered finite volume (or finite difference) methods. It achieves high order in both space and time, and it incorporates WENO slope limiting. 
    more » « less