Direct solution of partial difference equations by tensor product methods

Lynch, Robert E.; Rice, John R.; Thomas, Donald

doi:10.1007/bf01386067

Cited by 260 publications

(150 citation statements)

References 7 publications

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“…However, this cost can be significantly reduced by using a discrete version of "separation of variables" -the matrix decomposition/diagonalization method [15,8,24]. To this end, we consider the following generalized eigenvalue problems:…”

Section: Efficient Implementationmentioning

confidence: 99%

“…In particular, we shall extend most of the solution techniques in [24] for usual PDEs to fractional PDEs. More precisely, for separable fractional PDEs, we shall apply the matrix diagonalization methods (i.e., discrete separation of variables) [15,8,24] to reduce a (discrete) multi-dimensional problem to a sequence of (discrete ) one-dimensional problems or to a diagonal system with a total cost of just a few N d+1 flops (d is the space dimension); for non-separable fractional PDEs, we shall apply a preconditioned BICGSTAB iterative method using (i) a related fractional separable problem with constant-coefficients as preconditioned, and (ii) a fast matrix-free algorithm for the matrix-vector multiplication, so that the total cost is still O(N d+1 ). Thus, the cost of a spectral method for fractional PDEs is essentially the same as that for usual PDEs.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Efficient spectral–Galerkin methods for fractional partial differential equations with variable coefficients

Mao

Shen

2016

Journal of Computational Physics

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Section: Efficient Implementationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Efficient spectral–Galerkin methods for fractional partial differential equations with variable coefficients

Mao

Shen

2016

Journal of Computational Physics

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“…We exploit this in the application of our solvers and preconditioners as discussed in this section. In particular, we show how to generalize the FDM described in [11], to non-symmetric discrete convection-diffusion systems with constant wind w. We then examine the use of FDM applied to (2.11) on a single element.…”

Section: Fast Diagonalization Methods (Fdm)mentioning

confidence: 99%

Fast iterative solver for convection‐diffusion systems with spectral elements

Lott

Elman

2011

Numerical Methods Partial

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Abstract. We introduce a solver and preconditioning technique based on Domain Decomposition and the Fast Diagonalization Method that can be applied to tensor product based discretizations of the steady convection-diffusion equation. The method is based on iterative substructuring where fast diagonalization is used to efficiently eliminate the interior degrees of freedom and subsidiary subdomain solves. We demonstrate the effectiveness of this method in numerical simulations using a spectral element discretization.Key words. Convection-Diffusion, Domain Decomposition, Preconditioning, Spectral Element Method 1. Introduction. Numerical simulation of fluid flow allows for improved prediction and design of natural and engineered systems such as those involving water, oil, or blood. The interplay between inertial and viscous forces in a fluid flow dictates the length scale where energy is transferred, thus determining the resolution required to capture flow information accurately. This resolution requirement poses computational challenges in situations where the convective nature of the flow dominates diffusive effects. In such flows, convection and diffusion occur on disparate scales, causing sharp flow features that require fine numerical grid resolution. This leads to a large system of equations which is often solved using an iterative method. Exacerbating the challenge of solving a large linear system, the discrete convection-diffusion operator is non-symmetric and poorly conditioned. This leads to slow convergence of iterative solvers. In total, as convection dominates the flow the discrete fluid model becomes exceedingly challenging to solve.In recent years, the spectral element method has gained popularity as a technique for numerical simulation of fluids [9], [18]. This is due in part to the method's high-order accuracy, which produces solutions with low dissipation and low dispersion with relatively few degrees of freedom. Also important is the inherent computational efficiency gained through the use of a hierarchical grid structure based on unstructured macro-elements with fine tensor-structured interiors. This structure has enabled the development of efficient multi-level solvers and preconditioners based on Fast Diagonalization and Domain Decomposition [8], [10], [21]. Application of these techniques, however, has been restricted to symmetric systems.One way to apply such methods to non-symmetric systems is through use of timesplitting techniques, which split the system into symmetric and non-symmetric components. For convection-diffusion systems, the standard method for performing steady and unsteady flow simulations with spectral elements is operator integration factor splitting (OIFS) [12], which requires time integration even in steady flow simulations. Using this standard approach, convection and diffusion are treated separately; convection components are tackled explicitly using a sequence of small time steps that satisfy a CFL condition, and diffusive components are treated implicitly with larger t...

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“…Hence, (3.12) can be efficiently solved by using the matrix decomposition method [15,27] at a cost of…”

Section: Space Discretization-spectral-galerkin Methodsmentioning

confidence: 99%

On a Wind-Driven, Double-Gyre, Quasi-Geostrophic Ocean Model: Numerical Simulations and Structural Analysis

Shen

Medjo

Wang

1999

Journal of Computational Physics

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The main objectives of this paper are to adapt an efficient and accurate spectralprojection method for a wind-driven, double-gyre, mid-latitude, quasi-geostrophic ocean model, and to study the double-gyre phenomenon from numerical and structural analysis points of view. A number of numerical simulations are carried out and their structural stability and structural transition/bifurcation are investigated using a new dynamical systems theory of two-dimensional incompressible flows.

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Direct solution of partial difference equations by tensor product methods

Cited by 260 publications

References 7 publications

Efficient spectral–Galerkin methods for fractional partial differential equations with variable coefficients

Efficient spectral–Galerkin methods for fractional partial differential equations with variable coefficients

Fast iterative solver for convection‐diffusion systems with spectral elements

On a Wind-Driven, Double-Gyre, Quasi-Geostrophic Ocean Model: Numerical Simulations and Structural Analysis

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