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Efficient Solvers for the Implicit Time Integration of Matrix-Free High-Order Methods.

紀錄類型:	書目-電子資源 : Monograph/item
正題名/作者:	Efficient Solvers for the Implicit Time Integration of Matrix-Free High-Order Methods./
作者:	Franco, Michael.
面頁冊數:	1 online resource (105 pages)
附註:	Source: Dissertations Abstracts International, Volume: 84-05, Section: B.
Contained By:	Dissertations Abstracts International84-05B.
標題:	Applied mathematics. -
電子資源:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=29256120click for full text (PQDT)
ISBN:	9798352951392

Efficient Solvers for the Implicit Time Integration of Matrix-Free High-Order Methods.
Franco, Michael.

Efficient Solvers for the Implicit Time Integration of Matrix-Free High-Order Methods. - 1 online resource (105 pages)

Source: Dissertations Abstracts International, Volume: 84-05, Section: B.

Thesis (Ph.D.)--University of California, Berkeley, 2022.

Includes bibliographical references

In this thesis, we develop and study efficient solvers for high-order Galerkin methods applied to fluid flow problems. Many flow problems necessitate implicit time-integration schemes in order to be practical. Implicit-in-time discretizations require the solution of nonlinear algebraic systems each time step, which are often in-turn solved by linear solvers. Therefore, the performance of implicit-in-time solvers is largely determined by the performance of the underlying linear solvers.One approach to create efficient methods is to work with matrix-free operators. Because assembling the underlying discretization matrix can be prohibitively expensive in terms of computational complexity and memory, matrix-free operators are an attractive alternative. These operators replace the matrix-vector products with on-the-fly sum-factorization evaluations of the discretized differential operators instead. Indeed, their high arithmetic intensity makes these operators particularly well suited for modern graphics processing units (GPU) and GPU-accelerated architectures.These matrix-free operators are particularly challenging to precondition, however, because they by design do not allow access to the underlying matrix entries. We create a suite of efficient matrix-free preconditioners for a range of fluid flow problems that are robust with respect to polynomial degree and mesh size. The main building block solver extends sparse, low-order refined preconditioners with parallel subspace corrections. This work tackles Poisson problems, saddle-point Stokes systems, and the incompressible Navier-Stokes equations in two and three spatial dimensions.A different set of problems exhibit geometrically localized stiffness, where convergence rates are degraded in a localized subregion of the mesh. Generic preconditioners do not perform well across the entire domain because of mesh size, mesh anisotropy, highly variable coefficients, or more challenging physics in the subregion. Therefore, we seek to save costs by utilizing cheap preconditioners for most of the mesh and only focus our effort on the less expensive subregion problem. Our iterative subregion correction preconditioners correct naive preconditioners with an adaptive inner subregion iteration to reduce the number of costly global iterations. This work demonstrates performance on basic convection-diffusion problems, high Reynolds number compressible flow problems, and a 30◦ angle of attack problem with massively separated flow.

Electronic reproduction.
Ann Arbor, Mich. :
ProQuest,
2023

Mode of access: World Wide Web

ISBN: 9798352951392Subjects--Topical Terms:

2122814
Applied mathematics.
Subjects--Index Terms:

Computational fluid dynamicsIndex Terms--Genre/Form:

542853
Electronic books.

Efficient Solvers for the Implicit Time Integration of Matrix-Free High-Order Methods.
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504 $a Includes bibliographical references
520 $a In this thesis, we develop and study efficient solvers for high-order Galerkin methods applied to fluid flow problems. Many flow problems necessitate implicit time-integration schemes in order to be practical. Implicit-in-time discretizations require the solution of nonlinear algebraic systems each time step, which are often in-turn solved by linear solvers. Therefore, the performance of implicit-in-time solvers is largely determined by the performance of the underlying linear solvers.One approach to create efficient methods is to work with matrix-free operators. Because assembling the underlying discretization matrix can be prohibitively expensive in terms of computational complexity and memory, matrix-free operators are an attractive alternative. These operators replace the matrix-vector products with on-the-fly sum-factorization evaluations of the discretized differential operators instead. Indeed, their high arithmetic intensity makes these operators particularly well suited for modern graphics processing units (GPU) and GPU-accelerated architectures.These matrix-free operators are particularly challenging to precondition, however, because they by design do not allow access to the underlying matrix entries. We create a suite of efficient matrix-free preconditioners for a range of fluid flow problems that are robust with respect to polynomial degree and mesh size. The main building block solver extends sparse, low-order refined preconditioners with parallel subspace corrections. This work tackles Poisson problems, saddle-point Stokes systems, and the incompressible Navier-Stokes equations in two and three spatial dimensions.A different set of problems exhibit geometrically localized stiffness, where convergence rates are degraded in a localized subregion of the mesh. Generic preconditioners do not perform well across the entire domain because of mesh size, mesh anisotropy, highly variable coefficients, or more challenging physics in the subregion. Therefore, we seek to save costs by utilizing cheap preconditioners for most of the mesh and only focus our effort on the less expensive subregion problem. Our iterative subregion correction preconditioners correct naive preconditioners with an adaptive inner subregion iteration to reduce the number of costly global iterations. This work demonstrates performance on basic convection-diffusion problems, high Reynolds number compressible flow problems, and a 30◦ angle of attack problem with massively separated flow.
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