High-order numerical solution of the Helmholtz equation for domains with reentrant corners

Magura, S.; Petropavlovsky, S. V.; Tsynkov, Semyon; Turkel, Eli

doi:10.1016/j.apnum.2017.02.013

Cited by 18 publications

(10 citation statements)

References 36 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Note that, any point p ∈ γ ∩ ζ contributes two values in the density u i+1 γ ∨ζ : one from density u i+1 γ and the other from density u i+1 ζ ( = 1, 2). Next, we follow [35] and take the average of the two values corresponding to point p and assign the average to be the "effective" density at point p. The "effective" density will be used in computing the Difference Potentials P N + (γ ∨ζ ) u γ ∨ζ . Also, note that we impose two equations at point p ∈ γ ∩ ζ in the BEPs (45): one corresponds to u i+1 γ and the other for u i+1 ζ ( = 1, 2).…”

Section: Beps For Case 2: Z ∩ γmentioning

confidence: 99%

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

Epshteyn

Xia

2019

J Sci Comput

View full text Add to dashboard Cite

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...

show abstract

Section: Beps For Case 2: Z ∩ γmentioning

confidence: 99%

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

Epshteyn

Xia

2019

J Sci Comput

View full text Add to dashboard Cite

show abstract

“…For example, one can consider piecewise-smooth, locally-supported basis functions (defined on Γ) as the part of the Extension Operator. For example, [52] use this approach to design a high-order accurate numerical method for the Helmholtz equation, in a geometry with a reentrant corner. Furthermore, [80,81] combine the DPM together with the XFEM, and design a DPM for linear elasticity in a non-Lipschitz domain (with a cut).…”

Section: Dpmmentioning

confidence: 99%

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

et al. 2018

View full text Add to dashboard Cite

In this work, we discuss and compare three methods for the numerical approximation of constant-and variable-coefficient diffusion equations in both single and composite domains with possible discontinuity in the solution/flux at interfaces, considering (i) the Cut Finite Element Method; (ii) the Difference Potentials Method; and (iii) the summationby-parts Finite Difference Method. First we give a brief introduction for each of the three methods. Next, we propose benchmark problems, and consider numerical tests-with respect to accuracy and convergence-for linear parabolic problems on a single domain, and continue with similar tests for linear parabolic problems on a composite domain (with the interface defined either explicitly or implicitly). Lastly, a comparative discussion of the methods and numerical results will be given.

show abstract

“…3. Define a suitable system of basis functions along the boundary to approximate the boundary density Γ as in (19).…”

Section: Summary Of Computational Proceduresmentioning

confidence: 99%

“…For the first time, the DPM was applied in non-Lipschitz domains for the solution of the Poisson and Chaplygin equations in other works. [16][17][18] In the work of Magura et al, 19 the Helmholtz equation was solved for a problem with a singularity, which exists due to a nonsmooth boundary. In addition, the work of Britt et al 20 is worth noting in which a discontinuity in the boundary conditions was considered.…”

mentioning

confidence: 99%

Developments of the method of difference potentials for linear elastic fracture mechanics problems

Woodward

Utyuzhnikov

Massin

2018

Numerical Meth Engineering

View full text Add to dashboard Cite

Recently, the method of difference potentials has been extended to linear elastic fracture mechanics. The solution was calculated on a grid boundary belonging to the domain of an auxiliary problem, which must be solved multiple times. Singular enrichment functions, such as those used within the extended finite element method, were introduced to improve the approximation near the crack tip leading to near-optimal convergence rates. Now, the method is further developed by significantly reducing the computation time. This is achieved via the implementation of a system of basis functions introduced along the physical boundary of the problem. The basis functions form an approximation of the trace of the solution at the physical boundary. This method has been proven efficient for the solution of problems on regular (Lipschitz) domains. By introducing the singularity into the finite element space, the approximation of the crack can be realised by regular functions. Near-optimal convergence rates are then achieved for the enriched formulation. A solution algorithm using the fast Fourier transform is provided with the aim of further increasing the efficiency of the method. KEYWORDSextended finite element method, fast Fourier transform, fracture, method of difference potentials, rate of convergence INTRODUCTIONThe method of difference potentials (DPM) was originally developed by Ryaben'kii 1 and Ryaben'kii and Tsynkov 2 and has been applied to boundary value problems (BVPs) in fluid dynamics, 3 acoustics, 4-6 and many other applications. 1 The methodology is based on the same boundary integral equation as that used in boundary element methods. 7-11 Within the boundary element method, this boundary integral equation is numerically integrated given the knowledge of a suitable Green's function. The DPM avoids any need for Green's functions by embedding the problem within a well-defined auxiliary problem. The auxiliary problem is then solved numerically, avoiding the solution of the integral equation.A boundary system of equations must be constructed to calculate appropriate grid densities at the embedded boundary. This requires the auxiliary problem to be solved a multiple number of times, where the number of solutions required increases with required accuracy. Definition of the auxiliary problem can be flexible provided that it is well defined and completely encloses the embedded domain and its boundary. The auxiliary problem is therefore chosen as to facilitate the most efficient solution method. Previously, a square domain with a regular grid has been used along with the use of finite difference methods, which can be particularly efficient for regular problems such as this. In this paper, an adjusted Int J Numer Methods Eng. 2018;115:75-98.wileyonlinelibrary.com/journal/nme

show abstract

High-order numerical solution of the Helmholtz equation for domains with reentrant corners

Cited by 18 publications

References 36 publications

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

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

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

Developments of the method of difference potentials for linear elastic fracture mechanics problems

Contact Info

Product

Resources

About