Spectral Approximation of the Free-Space Heat Kernel

Greengard, Leslie; Lin, Patrick

doi:10.1006/acha.2000.0310

Cited by 73 publications

(90 citation statements)

References 13 publications

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“…Discretization and evaluation of the single layer potential. We first consider the evaluation of the single layer potential, following the treatment of the heat equation in [31,35,53]. There are three fundamental observations to be made.…”

Section: Discretization and Numerical Evaluation Of Layer Potentialsmentioning

confidence: 99%

“…Note that the integrand decays to zero exponentially fast when t − τ ≥ δ is bounded away from 0. In [31], it was shown that an efficient sum-of-exponentials approximation for the heat kernel exists in this case. Exactly the same idea can be applied here, with the result that N F ≈ 1/δ discrete Fourier modes are need to satisfy an estimate of the form…”

Section: Evaluation Of the History Partmentioning

confidence: 99%

“…The estimate (4.11) holds uniformly for all t − τ ≥ δ > 0 and x , y ≤ R for some fixed value of R. Remark 4.3. In practice, the construction of [31], which uses tensor product quadrature on dyadically scaled intervals can be improved through the use of generalized Gaussian quadrature [6,83,55]. The number of Fourier modes needed depends on δ, R, and the desired precision .…”

Section: Evaluation Of the History Partmentioning

confidence: 99%

“…Once the F j have been computed, however, the history modes H j (ξ j , iδ) can be seen to satisfy a simple recurrence relation [31,35,52]. More precisely, for a kth-order multistep schemes, we have…”

Section: Evaluation Of the History Partmentioning

confidence: 99%

“…Because of their ability to handle complex, moving geometries and unbounded domains, however, substantial efforts have been made to improve efficiency. There now exist asymptotically fast algorithms for heat, acoustic, electromagnetic, and elastic wave potentials [31,34,35,52,60,79,81,82].…”

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confidence: 99%

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Integral Equation Methods for Unsteady Stokes Flow in Two Dimensions

Jiang¹,

Veerapaneni²,

Greengard³

2012

SIAM J. Sci. Comput.

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Abstract. We present an integral equation formulation for the unsteady Stokes equations in two dimensions. This problem is of interest in its own right, as a model for slow viscous flow, but perhaps more importantly, as an ingredient in the solution of the full, incompressible Navier-Stokes equations. Using the unsteady Green's function, the velocity evolves analytically as a divergence-free vector field. This avoids the need for either the solution of coupled field equations (as in fully implicit PDE-based marching schemes) or the projection of the velocity field onto a divergence free field at each time step (as in operator splitting methods). In addition to discussing the analytic properties of the operators that arise in the integral formulation, we describe a family of high-order accurate numerical schemes and illustrate their performance with several examples.

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Section: Discretization and Numerical Evaluation Of Layer Potentialsmentioning

confidence: 99%

Section: Evaluation Of the History Partmentioning

confidence: 99%

Section: Evaluation Of the History Partmentioning

confidence: 99%

Section: Evaluation Of the History Partmentioning

confidence: 99%

mentioning

confidence: 99%

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Integral Equation Methods for Unsteady Stokes Flow in Two Dimensions

Jiang¹,

Veerapaneni²,

Greengard³

2012

SIAM J. Sci. Comput.

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Time‐Dependent Problems with the Boundary Integral Equation Method

Costabel¹

2004

Encyclopedia of Computational Mechanics

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Time-dependent problems that are modeled by initial-boundary value problems for parabolic or hyperbolic partial differential equations can be treated with the boundary integral equation method. The ideal situation is when the right-hand side in the partial differential equation and the initial conditions vanish, the data are given only on the boundary of the domain, the equation has constant coefficients, and the domain does not depend on time. In this situation, the transformation of the problem to a boundary integral equation follows the same well-known lines as for the case of stationary or time-harmonic problems modeled by elliptic boundary value problems. The same main advantages of the reduction to the boundary prevail: Reduction of the dimension by one, and reduction of an unbounded exterior domain to a bounded boundary.There are, however, specific difficulties due to the additional time dimension: Apart from the practical problems of increased complexity related to the higher dimension, there can appear new stability problems. In the stationary case, one often has unconditional stability for reasonable approximation methods, and this stability is closely related to variational formulations based on the ellipticity of the underlying boundary value problem. In the timedependent case, instabilities have been observed in practice, but due to the absence of ellipticity, the stability analysis is more difficult and fewer theoretical results are available.In this article, the mathematical principles governing the construction of boundary integral equation methods for time-dependent problems are presented. We describe some of the main algorithms that are used in practice and have been analyzed in the mathematical literature.

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Time‐Dependent Problems with the Boundary Integral Equation Method

Costabel

2017

Encyclopedia of Computational Mechanics Second Edition

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Time‐dependent problems that are modeled by initial‐boundary value problems for parabolic or hyperbolic partial differential equations can be treated with the boundary integral equation (BIE) method. The ideal situation is when the right‐hand side in the partial differential equation and the initial conditions vanish, the data are given only on the boundary of the domain, the equation is linear with constant coefficients, and the domain does not depend on time. In this situation, the transformation of the problem to a BIE follows the same well‐known lines as for the case of stationary or time‐harmonic problems modeled by elliptic boundary value problems. The same main advantages of the reduction to the boundary prevail: reduction of the dimension by one, and reduction of an unbounded exterior domain to a bounded boundary. There are, however, specific difficulties due to the additional time dimension. Apart from the practical problems of increased complexity related to the higher dimension, there can appear new stability problems. In the stationary case, one often has unconditional stability for reasonable approximation methods, and this stability is closely related to variational formulations based on the ellipticity of the underlying boundary value problem. In the time‐dependent case, instabilities have been observed in practice, but due to the absence of ellipticity, the stability analysis is more difficult and fewer theoretical results are available. In this chapter, the mathematical principles governing the construction of BIE methods for time‐dependent problems are presented. We describe some of the main numerical algorithms that are used in practice and have been analyzed in the mathematical literature.

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Spectral Approximation of the Free-Space Heat Kernel

Cited by 73 publications

References 13 publications

Integral Equation Methods for Unsteady Stokes Flow in Two Dimensions

Integral Equation Methods for Unsteady Stokes Flow in Two Dimensions

Time‐Dependent Problems with the Boundary Integral Equation Method

Time‐Dependent Problems with the Boundary Integral Equation Method

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