High-Order Numerical Methods for 2D Parabolic Problems in Single and Composite Domains

Ludvigsson, Gustav; Steffen, K.; Sticko, Simon; Wang, Siyang; Xia, Qing; Epshteyn, Yekaterina; Kreiss, Gunilla

doi:10.1007/s10915-017-0637-y

Cited by 11 publications

(11 citation statements)

References 81 publications

(254 reference statements)

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“…We have chosen the finite difference method for the spatial discretization because of the simple geometric feature for one dimensional problems. Other numerical methods, such as the finite element method, the finite volume method, and the difference potential method can however also be used [25].…”

Section: Solving the Concentration Distributionmentioning

confidence: 99%

Bridging physics-based and equivalent circuit models for lithium-ion batteries

Geng

Wang

Lacey

et al. 2021

Electrochimica Acta

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Section: Solving the Concentration Distributionmentioning

confidence: 99%

Bridging physics-based and equivalent circuit models for lithium-ion batteries

Geng

Wang

Lacey

et al. 2021

Electrochimica Acta

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“…Equations (30) and (31) were nonpositive so that (28) satisfied the energy estimate. Using the SMF, the energy analysis could be simplified to a large extent as a generic system.…”

Section: Continuous Energy Analysismentioning

confidence: 99%

“…e SBP-SAT method can develop strictly stable proof using the energy method, leading to robustness in spatial and time domains. Some review papers of SBP operators can be found in [25,26] and examples of SBP-SAT can be found in [27][28][29][30].…”

Section: Introductionmentioning

confidence: 99%

Multiblock SBP-SAT Methodology of Symmetric Matrix Form of Elastic Wave Equations on Curvilinear Grids

Sun

Yang

Jiang

et al. 2020

Shock and Vibration

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A stable and accurate finite-difference discretization of first-order elastic wave equations is derived in this work. To simplify the origin and proof of the formulas, a symmetric matrix form (SMF) for elastic wave equations is presented. The curve domain is discretized using summation-by-parts (SBP) operators, and the boundary conditions are weakly enforced using the simultaneous-approximation-term (SAT) technique, which gave rise to a provably stable high-order SBP-SAT method via the energy method. In addition, SMF can be extended to wave equations of different types (SH wave and P-SV wave) and dimensions, which can simplify the boundary derivation process and improve its applicability. Application of this approximation can divide the domain into a multiblock context for calculation, and the interface boundary conditions of blocks can also be used to simulate cracks and other structures. Several numerical simulation examples, including actual elevation within the area of Lushan, China, are presented, which verifies the viability of the framework present in this paper. The applicability of simulating elastic wave propagation and the application potential in the seismic numerical simulation of this method are also revealed.

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“…For the time being, we will consider the PKS chemotaxis model in a spherical domain, but the proposed methods can be extended to more general domains in 3D (and the main ideas of the algorithms stay the same). We employ a finite-volume-finite-difference scheme as the underlying discretization of the model (1) in space, combined with the idea of Difference Potentials Method ( [41] and some very recent work [3,13,14,34], etc. ), that provides flexibility to handle irregular domains accurately and efficiently by the use of simple Cartesian meshes.…”

Section: An Algorithm Based On Dpmmentioning

confidence: 99%

“…Therefore, to construct a unique solution to BEP (19), we need to supply the BEP (19) with zero Neumann boundary conditions for the density and concentration. To impose boundary conditions efficiently into BEP, we will introduce the extension operator (21) below, similarly to [3,13,14,34], etc.…”

Section: Construction Of the Difference Potentialsmentioning

confidence: 99%

Efficient Numerical Algorithms Based on Difference Potentials for Chemotaxis Systems in 3D

Epshteyn

Xia

2019

J Sci Comput

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In this work, we propose efficient and accurate numerical algorithms based on Difference Potentials Method for numerical solution of chemotaxis systems and related models in 3D. The developed algorithms handle 3D irregular geometry with the use of only Cartesian meshes and employ Fast Poisson Solvers. In addition, to further enhance computational efficiency of the methods, we design a Difference-Potentialsbased domain decomposition approach which allows mesh adaptivity and easy parallelization of the algorithm in space. Extensive numerical experiments are presented to illustrate the accuracy, efficiency and robustness of the developed numerical algorithms.In this work we develop efficient and accurate numerical algorithms based on Difference Potentials Method (DPM) for the Patlak-Keller-Segel (PKS) chemotaxis model and related problems in 3D. The proposed methods handle irregular geometry with the use of only Cartesian meshes, employ Fast Poisson Solvers, allow easy parallelization and adaptivity in space.Chemotaxis refers to mechanisms by which cellular motion occurs in response to an external stimulus, for instance a chemical one. Chemotaxis is an essential process in many medical and biological applications, including cell aggregation and pattern formation mechanisms and tumor growth. Modeling of chemotaxis dates back to the pioneering work by Patlak, Keller and Segel [28,29,38]. The PKS chemotaxis model consists of coupled convection-diffusion and reaction-diffusion partial differential equations:subject to zero Neumann boundary conditions and the initial conditions on the cell density ρ(x, y, z, t) and on the chemoattractant concentration c(x, y, z, t). In the system (1), the coefficient χ is a chemotactic sensitivity constant, γ ρ and γ c are the reaction coefficients. The parameter α is equal to either 1 or 0, which corresponds to the "parabolic-parabolic" or reduced "parabolic-elliptic" coupling, respectively. In this work, we will focus on the PKS chemotaxis model (1) with α = γ ρ = γ c = 1, but the developed methods are not restricted by this assumption and can be easily extended to more general chemotaxis systems and related models.In the past several years, chemotaxis models have been extensively studied (see, e.g., [19,20,21,22, 39] and references below). A known property of many chemotaxis systems is their ability to represent a concentration phenomenon that is mathematically described by fast growth of solutions in small neighborhoods of concentration points/curves. The solutions may blow up or may exhibit a very singular behavior. This blowup represents a mathematical description of a cell concentration phenomenon that occurs in real biological systems, see, e.g., [1,5,6,7,11,12,36,40].Capturing blowing up or spiky solutions is a challenging task numerically, but at the same time design of robust numerical algorithms is crucial for the modeling and analysis of chemotaxis mechanisms. Let us briefly review some of the recent numerical methods that have been proposed for the "parabolic-parabolic" cou...

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High-Order Numerical Methods for 2D Parabolic Problems in Single and Composite Domains

Cited by 11 publications

References 81 publications

Bridging physics-based and equivalent circuit models for lithium-ion batteries

Bridging physics-based and equivalent circuit models for lithium-ion batteries

Multiblock SBP-SAT Methodology of Symmetric Matrix Form of Elastic Wave Equations on Curvilinear Grids

Efficient Numerical Algorithms Based on Difference Potentials for Chemotaxis Systems in 3D

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