Efficient approximation of the exponential operator for discrete 2D advection–diffusion problems

Bergamaschi, Luca; Caliari, Marco; Vianello, Marco

doi:10.1002/nla.288

Cited by 31 publications

(55 citation statements)

References 26 publications

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“…Therefore, the assumption ρ(B) < 1 that is required for the Taylor series to converge is not overly restrictive. (Applications where Krylov and Chebyshev iterations are effective typically have Re(λ(B)) very large and negative [2,12].) Second, the eigenvalue symmetry means that a disk centered on 0, enclosing λ(B), will tend to be a much better approximation of λ(B) than in the general, non-Hamiltonian, case.…”

Section: Other Inexact Newton Iterations (See [10 13 14])mentioning

confidence: 99%

“…In the inexact Newton method an (e.g., Krylov or Chebyshev) iterative method is applied [2,4,5]; in the Newton-chord method [18] the Jacobian itself is approximated to simplify the solution step. In the Jacobian-free-Newton-Krylov method [12] the Jacobian-vector multiplications are approximated by finite differences, so the Jacobian itself is never formed.…”

Section: Introductionmentioning

confidence: 99%

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A New Implementation of Symplectic Runge–Kutta Methods

McLachlan¹

2007

SIAM J. Sci. Comput.

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Abstract. We propose a "Newton-Taylor" iteration for solving the implicit equations of symplectic Runge-Kutta methods, using the Jacobian of the vector field and matrix-vector multiplications whose extra cost for certain structured problems is negligible. The structure of Hamiltonian ODEs allows this very simple iteration to be effective. The iteration reduces the number of vector field evaluations almost to that of Newton's method, often only one or two per time step, making symplectic Runge-Kutta methods more efficient even at relatively large time steps.Key words. Runge-Kutta, implicit, symplectic integrators, inexact Newton methods AMS subject classifications. 65L06, 65P10 DOI. 10.1137/06065338X1. Introduction. Symplectic integrators based on splitting, such as the leapfrog method for separable Hamiltonian systems, are fast and simple and give very good results for long simulations [8]. They are widely, almost universally, used in applications from accelerator physics to molecular dynamics to celestial mechanics [16]. However, the only symplectic integrators for general Hamiltonian systems are implicit, such as the Gaussian Runge-Kutta methods [11], which has tended to limit their popularity.(The implicit equations must be solved to within round-off error, to preserve symplecticity.) Hairer, Lubich, and Wanner [8] have studied the implementation issues and found the simple ("standard") iteration, (4) below, superior to Newton's method; for very small error tolerances it can even compete with high-order methods based on splitting and composition. However, for the relatively low orders and large time steps often used in long symplectic integrations, the convergence of the standard iteration deteriorates, making the method more expensive, which defeats the purpose of using a large time step.For other applications of implicit methods, there are methods in which the linear equations defining the Newton step are solved only approximately. In the inexact Newton method an (e.g., Krylov or Chebyshev) iterative method is applied [2,4,5]; in the Newton-chord method [18] the Jacobian itself is approximated to simplify the solution step. In the Jacobian-free-Newton-Krylov method [12] the Jacobian-vector multiplications are approximated by finite differences, so the Jacobian itself is never formed.The application to Hamiltonian systems has a number of special features that lead us to propose a very simple inexact Newton method that, for many structured problems, costs only a constant factor close to 1 times the cost of the standard iteration, yet has a convergence rate very close to Newton's method. For the midpoint rule, for example, the method may require only one or two evaluations of the vector field per time step.

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Section: Other Inexact Newton Iterations (See [10 13 14])mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A New Implementation of Symplectic Runge–Kutta Methods

McLachlan¹

2007

SIAM J. Sci. Comput.

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“…In fact the approach can extended to many other situations where a vector of the form f (A)v is to be computed. The problem of approximating f (A)v has been extensively studied, see, e.g., [28,23,4,16,15], though the attention was primarily focussed on the the case when f is analytic (e.g., f (t) = exp(t)). Problems which involve non continous functions, such as the step function, or the sign function can also be important.…”

Section: Computing F (A)vmentioning

confidence: 99%

Filtered Conjugate Residual‐type Algorithms with Applications

Saad¹

2006

SIAM J. Matrix Anal. & Appl.

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In a number of applications, certain computations to be done with a given matrix are performed by replacing this matrix by its best low rank approximation. This has the effect of yielding the most relevant part of the desired solution while discarding noise and redundancies. One such application is that of regularization where the righthand side of the original linear system is noisy or inaccurate while the coefficient matrix is very ill-conditioned. Solving such linear systems accurately is counter-productive as the noise tends to be amplified. A common remedy is to compute the pseudo-inverse solution in which the inverses of the smallest singular values are replaced by zeros or small quantities. A similar procedure is also used in methods related to Principal Component Analysis, such as in Latent Semantic Indexing in information retrieval. Here the low-rank approximation to the original matrix is used to analyze similarities with a given query vector. This paper presents a few conjugate-gradient like methods to provide solutions to these two types of problems by iterative procedures which utilize only matrix-vector products.

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“…These methods seems to work better when a good preconditioner for the system matrix is not available for the implicit methods, which is the case when it is desirable not to load the system matrix due to memory requirements, and only the matrix vector product by this matrix is available. Moreover, other studies suggest that polynomial interpolation for the exponential matrix, such as the Real Leja points Method [68], converge as fast as Krylov methods without the memory requirements to save all the vectors defining the Krylov subspace. A Chebyshev approximation for the exponential of a matrix [69,70] is included for completeness, and also higher order exponential integrators based on the Magnus expansion with a commutator free formulation [71,72,73] are tested against a classical second order exponential integrator.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…A well known method to approximate e hA v is based on the Chebyshev polynomial expansion (see for instance [69,68]). …”

Section: Chebyshev Polynomialsmentioning

confidence: 99%

Approximation of the Neutron Diffusion Equation on Hexagonal Geometries

Pintor¹

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SummaryThe neutron diffusion equation describes the neutron population in a nuclear reactor core. This work deals with this model for nuclear reactors with hexagonal geometries. First, the stationary neutron diffusion equation is studied. This is a differential eigenvalue problem, called Lambda modes problem. To solve the Lambda modes problem, different methods have been compared in one-dimensional geometries, resulting as the best one the Spectral Element Method. The operators are then discretized using this scheme in two-and three-dimensional geometries, and the resulting algebraic eigenvalue problem is solved with the implicit restarted Arnoldi method.Once the solution for the steady state neutron distribution is obtained, it is used as initial condition for the time integration neutron diffusion equation. Initially, a one step backard Euler method is used to discretize this equation in time. The transients to test the behaviour of the method are based on moving the control rods of the reactor, simulating an accident where a control rod is ejected and a scram is intialized to control the power evolution. An unphysical behaviour appears when a node is partially rodded, the rod cusping effect, which is corrected by weighting the cross sections with the flux of the previous time step. Good results are obtained for the power evolution and the rod cusping effect is corrected for different transients. To solve the algebraic systems arising in the backward method, a Krylov method is used, and different preconditioning strategies are tested. The first one consists of using the block structure obtained by the energy groups to solve the system by blocks, and different acceleration techniques for the block iterative scheme and a preconditioner using this block structure are proposed. Also, a spectral preconditioner, which makes use of the information in a Krylov subspace to preconditionate the next system is studied. Second and fourth order exponential methods are also proposed to integrate the time dependent neutron diffusion equation, where the exponential of the system matrix has to be multiplied by a vector. These schemes allow us to work without building explicitly the system matrix, and different methods are compared to calculate the product of the system matrix by a vector, such as a Krylov method, a Chebyshev approximation method, and a Leja Points method.Some situations arise in which a set of modes of a nuclear reactor core have to be updated, as in perturbative calculations or in the use of modal methods. Updating this set of modes can be very expensive when using the Arnoldi method, and for this reason several methods based on a Newton Iteration are proposed, such as the Modified Block Newton method, a One Sided Block Newton method and a Two Sided Block Newton method. As an alternative strategy, a reduced order model to update a set of modes based on the Proper Generalized Decomposition is also proposed. This method obtains an approximated solution as a sum of separable functions over the whole domain, reduc...

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Efficient approximation of the exponential operator for discrete 2D advection–diffusion problems

Cited by 31 publications

References 26 publications

A New Implementation of Symplectic Runge–Kutta Methods

A New Implementation of Symplectic Runge–Kutta Methods

Filtered Conjugate Residual‐type Algorithms with Applications

Approximation of the Neutron Diffusion Equation on Hexagonal Geometries

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