A semi-circulant preconditioner for the convection-diffusion equation

Hemmingsson, Lina

doi:10.1007/s002110050390

Cited by 16 publications

(17 citation statements)

References 16 publications

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“…In this paper, we present a number of related variants of Elman's BFB T preconditioner which are based on fast transform approximations of the advection-di usion operator F, which have recently been worked on by the ÿrst author [16]. There it is shown that use of these semi-circulant matrices as preconditioners for advection-di usion problems leads to almost optimal solution algorithms (with computation complexity O(N log 2 N ) for a discrete problem of size N ) for uni-directional advection.…”

Section: Introductionmentioning

confidence: 99%

A nearly optimal preconditioner for the Navier–Stokes equations

Hemmingsson-Frändén

Wathen

2001

Numerical Linear Algebra App

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We present a preconditioner for the linearized Navier-Stokes equations which is based on the combination of a fast transform approximation of an advection di usion problem together with the recently introduced 'BFB T ' preconditioner of Elman (SIAM Journal of Scientiÿc Computing, 1999; 20:1299-1316. The resulting preconditioner when combined with an appropriate Krylov subspace iteration method yields the solution in a number of iterations which appears to be independent of the Reynolds number provided a mesh PÃ eclet number restriction holds, and depends only mildly on the mesh size. The preconditioner is particularly appropriate for problems involving a primary ow direction.

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Section: Introductionmentioning

confidence: 99%

A nearly optimal preconditioner for the Navier–Stokes equations

Hemmingsson-Frändén

Wathen

2001

Numerical Linear Algebra App

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“…Such matrices arise in, for example, partial differential equations [12,24] and image deblurring [14]. To see this, let B n = S n ⊗ T m where S n ∈ R n×n and T m ∈ R m×m are Toeplitz matrices and ⊗ represents a Kronecker product.…”

Section: Extension To Block Matricesmentioning

confidence: 99%

A Preconditioned MINRES Method for Nonsymmetric Toeplitz Matrices

Pestana¹,

Wathen²

2015

SIAM J. Matrix Anal. & Appl.

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Abstract. Circulant preconditioning for symmetric Toeplitz linear systems is well-established; theoretical guarantees of fast convergence for the conjugate gradient method are descriptive of the convergence seen in computations. This has led to robust and highly efficient solvers based on use of the fast Fourier transform exactly as originally envisaged in Gil Strang's 'Proposal for Toeplitz Matrix Calculations' (Studies in Applied Mathematics, 74, pp. 171-176, 1986.). For nonsymmetric systems, the lack of generally descriptive convergence theory for most iterative methods of Krylov type has provided a barrier to such a comprehensive guarantee, though several methods have been proposed and some analysis of performance with the normal equations is available.In this paper, by the simple device of reordering, we rigorously establish a circulant preconditioned short recurrence Krylov subspace iterative method of minimum residual type for nonsymmetric (and possibly highly nonnormal) Toeplitz systems. Convergence estimates similar to those in the symmetric case are established.

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“…Various preconditioning methods have been developed in the past in the context of different applications. For instance, the semi-circulant preconditioner for convection-diffusion equations [15]; the ILU and characteristic Gauss-Seidel preconditioners for the finite volume discretization of hyperbolic problems [18]; the circulant preconditioner for Hermitian systems [3]. These methods are, however, for compact discretization schemes with special matrix structures.…”

Section: L(u)mentioning

confidence: 99%

“…In our computations, in each Newton step we effectively have to solve (15) for the specific example (17).…”

Section: Formulation and Linearizationmentioning

confidence: 99%

Preconditioning for a Class of Spectral Differentiation Matrices

2005

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We propose an efficient preconditioning technique for the numerical solution of first-order partial differential equations (PDEs). This study has been motivated by the computation of an invariant torus of a system of ordinary differential equations. We find the torus by discretizing a nonlinear first-order PDE with a full two-dimensional Fourier spectral method and by applying Newton's method. This leads to large nonsymmetric linear algebraic systems. The sparsity pattern of these systems makes the use of direct solvers prohibitively expensive. Commonly used iterative methods, e.g., GMRes, BiCGStab and CGNR (Conjugate Gradient applied to the normal equations), are quite slow to converge. Our preconditioner is derived from the solution of a PDE with constant coefficients; it has a fast implementation based on the Fast Fourier Transform (FFT). It effectively increases the clustering of the spectrum, and speeds up convergence significantly. We demonstrate the performance of the preconditioner in a number of linear PDEs and the nonlinear PDE arising from the Van der Pol oscillator.

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A semi-circulant preconditioner for the convection-diffusion equation

Cited by 16 publications

References 16 publications

A nearly optimal preconditioner for the Navier–Stokes equations

A nearly optimal preconditioner for the Navier–Stokes equations

A Preconditioned MINRES Method for Nonsymmetric Toeplitz Matrices

Preconditioning for a Class of Spectral Differentiation Matrices

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