High-order finite difference schemes for the solution of second-order BVPs

AbstractWe introduce new methods in the class of boundary value methods (BVMs) to solve boundary value problems (BVPs) for a second-order ODE. These formulae correspond to the high-order generalizations of classical finite difference schemes for the first and second derivatives. In this research, we carry out the analysis of the conditioning and of the time-reversal symmetry of the discrete solution for a linear convection–diffusion ODE problem. We present numerical examples emphasizing the good convergence behavior of the new schemes. Finally, we show how these methods can be applied in several space dimensions on a uniform mesh

Parameter functional dependence in an electrochemical model: theoretical and computational issues

In this article we study the simplest parameter dependent ODE model that allows to describe the electrochemical impedance as a curve $Z(ω)=X(ω)+iY(ω), ω ∈ [ω0, ωf]$ in the complex plane. The parameters of the original ODE having a straightforward physical meaning appear in $Z(ω)$ combined in a highly nonlinear form. Usually, a nonlinear least squares procedure is applied to identify these parameters by fitting experimental impedance data and, as shown in [2], this can yield an ill-posed and ill-conditioned problem. In fact, several sets of different parameters,called Numerical Global Minima (NGM) can be identified that produce undistinguishable fitting curves.In this paper, we show that: 1) ill- posedness can be avoided by working in a different parameter space, where the new parameters have a physical meaning that is different from the traditional one but nevertheless exhibit a clear relationship with them, and a unique optimal set can be identified; 2) there exist curves of NGMs in the original space

Turing patterns in a 3D morpho-chemical bulk-surface reaction-diffusion system for battery modeling

In this paper we introduce a bulk-surface reaction-diffusion (BSRD) model in three space dimensions that extends the DIB morphochemical model to account for the electrolyte contribution in the application, in order to study structure formation during discharge-charge processes in batteries. Here we propose to approximate the model by the Bulk-Surface Virtual Element Method on a tailor-made mesh that proves to be competitive with fast bespoke methods for PDEs on Cartesian grids. We present a selection of numerical simulations that accurately match the classical morphologies found in experiments. Finally, we compare the Turing patterns obtained by the coupled 3D BS-DIB model with those obtained with the original 2D version.Comment: 25 pages, 11 figures, 1 tabl

Model-reduction techniques for {PDE} models with Turing type electrochemical phase formation dynamics

Next-generation battery research will heavily rely on physico-chemical models, combined with deep learning methods and high-throughput and quantitative analysis of experimental datasets, encoding spectral information in space and time. These tasks will require highly efficient computational approaches, to yield rapidly accurate approximations of the models. This paper explores the capabilities of a representative range of model reduction techniques to face this problem in the case of a well-assessed electrochemical phase-formation model. We consider the Proper Orthogonal Decomposition (POD) with a Galerkin projection and the Dynamic Mode Decomposition (DMD) techniques to deal first of all with a semi-linear heat equation 2D in space as a test problem. As an application, we show that it is possible to save computational time by applying POD-Galerkin for different choices of the parameters without recalculating the snapshot matrix. Finally, we consider two reaction–diffusion (RD) PDE systems with Turing-type dynamics: the well-known Schnackenberg model and the DIB model for electrochemical phase formation. We show that their reduced models obtained by POD and DMD with suitable low-dimensional projections are able to approximate carefully both the Turing patterns at the steady state and the reactivity dynamics in the transient regime. Finally, for the DIB model we show that POD-Galerkin applied for different choices of parameters, by calculating once the snapshot matrices, is able to find reduced Turing patterns of different morphology

Turing pattern formation on the sphere for a morphochemical reaction-diffusion model for electrodeposition

The present paper deals with the pattern formation properties of a specific morpho- electrochemical reaction-diffusion model on a sphere. The physico-chemical background to this study is the morphological control of material electrodeposited onto spherical parti- cles. The particular experimental case of interest refers to the optimization of novel metal- air flow batteries and addresses the electrodeposition of zinc onto inert spherical supports. Morphological control in this step of the high-energy battery operation is crucial to the energetic efficiency of the recharge process and to the durability of the whole energy- storage device. To rationalise this technological challenge within a mathematical modeling perspective, we consider the reaction-diffusion system for metal electrodeposition intro- duced in [Bozzini et al., J. Solid State Electr.17, 467–479 (2013)] and extend its study to spherical domains. Conditions are derived for the occurrence of the Turing instability phe- nomenon and the steady patterns emerging at the onset of Turing instability are investi- gated. The reaction-diffusion system on spherical domains is solved numerically by means of the Lumped Surface Finite Element Method (LSFEM) in space combined with the IMEX Euler method in time. The effect on pattern formation of variations in the domain size is investigated both qualitatively, by means of systematic numerical simulations, and quan- titatively by introducing suitable indicators that allow to assign each pattern to a given morphological class. An experimental validation of the obtained results is finally presented for the case of zinc electrodeposition from alkaline zincate solutions onto copper spheres