A fast integral equation method for the two-dimensional Navier-Stokes equations

Klinteberg, Ludvig af; Askham, Travis; Kropinski, Mary Catherine A.

doi:10.1016/j.jcp.2020.109353

Cited by 24 publications

(23 citation statements)

References 56 publications

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“…In a separate paper [18], we present an adaptive quadrature scheme in the spirit of [19] that lifts the previous restriction on α. This scheme is also used to solve the modified Stokes equation in [20].…”

Section: Introductionmentioning

confidence: 99%

“…However, the algorithmic development in these efforts is essential to increase the applicability of integral equation-based numerical methods which sport several attractive features, including that complex geometry naturally enters the problem and generation of an unstructured mesh is redundant, ill-conditioning associated with discretising the operators is avoided, high accuracy can be attained, and boundary data and far-field conditions are simple to incorporate. Developments for the heat equation are also related to extension from Stokes to Navier-Stokes equations [20].…”

Section: Introductionmentioning

confidence: 99%

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An integral equation–based numerical method for the forced heat equation on complex domains

2020

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Integral equation–based numerical methods are directly applicable to homogeneous elliptic PDEs and offer the ability to solve these with high accuracy and speed on complex domains. In this paper, such a method is extended to the heat equation with inhomogeneous source terms. First, the heat equation is discretised in time, then in each time step we solve a sequence of so-called modified Helmholtz equations with a parameter depending on the time step size. The modified Helmholtz equation is then split into two: a homogeneous part solved with a boundary integral method and a particular part, where the solution is obtained by evaluating a volume potential over the inhomogeneous source term over a simple domain. In this work, we introduce two components which are critical for the success of this approach: a method to efficiently compute a high-regularity extension of a function outside the domain where it is defined, and a special quadrature method to accurately evaluate singular and nearly singular integrals in the integral formulation of the modified Helmholtz equation for all time step sizes.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

An integral equation–based numerical method for the forced heat equation on complex domains

2020

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“…Here we present an efficient deep learning technique relying on the ANN for the model reduction of the Navier-Stokes Equation [3] for incompressible flow. Consider a two dimensional Navier-Stokes Equation with the continuity and Fig.…”

Section: Solving the Naiver-stokes Equationmentioning

confidence: 99%

A deep artificial neural network architecture for mesh free solutions of nonlinear boundary value problems

2021

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Seeking efficient solutions to nonlinear boundary value problems is a crucial challenge in the mathematical modelling of many physical phenomena. A well-known example of this is solving the Biharmonic equation relating to numerous problems in fluid and solid mechanics. One must note that, in general, it is challenging to solve such boundary value problems due to the higher-order partial derivatives in the differential operators. An artificial neural network is thought to be an intelligent system that learns by example. Therefore, a well-posed mathematical problem can be solved using such a system. This paper describes a mesh free method based on a suitably crafted deep neural network architecture to solve a class of well-posed nonlinear boundary value problems. We show how a suitable deep neural network architecture can be constructed and trained to satisfy the associated differential operators and the boundary conditions of the nonlinear problem. To show the accuracy of our method, we have tested the solutions arising from our method against known solutions of selected boundary value problems, e.g., comparison of the solution of Biharmonic equation arising from our convolutional neural network subject to the chosen boundary conditions with the corresponding analytical/numerical solutions. Furthermore, we demonstrate the accuracy, efficiency, and applicability of our method by solving the well known thin plate problem and the Navier-Stokes equation.

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“…as the matrix product ends up computing the approximations to the inner products, as given by ( 17) and (19). Observe that A is a matrix of size (…”

Section: Numerical Implementationmentioning

confidence: 99%

“…The smooth selection embedding method [17,18] attempts to solve the extension problem by formulating it as a Sobolev norm optimization problem. Another important contribution is the partition of unity extension approach developed in the context of boundary integral methods and applied to heat and fluid flow problems [19][20][21]. Most pertinent to the current work is the Smooth Forcing Extension (SFE) method [22], which searches for extensions to the inhomogeneous terms in a finite-dimensional space by imposing regularity constraints at the nodes of the discretized physical boundaries.…”

Section: Introductionmentioning

confidence: 99%

The Projection Extension Method: A Spectrally Accurate Technique for Complex Domains

Qadeer¹,

Nazockdast²,

Griffith³

2021

Preprint

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An essential ingredient of a spectral method is the choice of suitable bases for test and trial spaces. On complex domains, these bases are harder to devise, necessitating the use of domain partitioning techniques such as the spectral element method. In this study, we introduce the Projection Extension (PE) method, an approach that yields spectrally accurate solutions to various problems on complex geometries without requiring domain decomposition. This technique builds on the insights used by extension methodologies such as the immersed boundary smooth extension and Smooth Forcing Extension (SFE) methods that are designed to improve the order of accuracy of the immersed boundary method. In particular, it couples an accurate extension procedure, that functions on arbitrary domains regardless of connectedness or regularity, with a least squares minimization of the boundary conditions. The resulting procedure is stable under iterative application and straightforward to generalize to higher dimensions. Moreover, it rapidly and robustly yields exponentially convergent solutions to a number of challenging test problems, including elliptic, parabolic, Newtonian fluid flow, and viscoelastic problems.

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A fast integral equation method for the two-dimensional Navier-Stokes equations

Cited by 24 publications

References 56 publications

An integral equation–based numerical method for the forced heat equation on complex domains

An integral equation–based numerical method for the forced heat equation on complex domains

A deep artificial neural network architecture for mesh free solutions of nonlinear boundary value problems

The Projection Extension Method: A Spectrally Accurate Technique for Complex Domains

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