Arnoldi and Crank-Nicolson methods for integration in time of the transport equation

Pini, Giorgio; Gambolati, Giuseppe

doi:10.1002/1097-0363(20010115)35:1<25::aid-fld80>3.0.co;2-d

Cited by 10 publications

(10 citation statements)

References 20 publications

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“…[13]). In the case of the relevant ODEs system (2) (with stationary b and q), its variable step-size version writes as…”

Section: Crank-nicolson (Cn) Methodsmentioning

confidence: 99%

“…To this aim, two classes of polynomial methods are currently used. We have Krylov-like methods, which are based on the idea of projecting the operator on a "small" Krylov subspace of the matrix via the Arnoldi process, and typically involve long-term recurrences in the nonsymmetric case; see, e.g., [10,11,12], and [13,14] for other (nonstandard) Krylov-like approaches. The second class consists of methods based on polynomial interpolation or series expansion of the entire function ϕ on a suitable compact subset containing the spectrum (or in general the field of values) of the matrix (e.g.…”

Section: An Exponential Integrator Via Mass-lumpingmentioning

confidence: 97%

“…[13]), concerning application of the ReLPM exponential integrator (5) to advection-dispersion models like (1), together with the comparison with the classical variable step-size Crank-Nicolson solver. = [1, .…”

Section: Application: 2d and 3d Advection-dispersion Modelsmentioning

confidence: 99%

See 2 more Smart Citations

The ReLPM Exponential Integrator for FE Discretizations of Advection-Diffusion Equations

Bergamaschi

Caliari

Vianello

2004

Lecture Notes in Computer Science

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Abstract. We implement an exponential integrator for large and sparse systems of ODEs, generated by FE (Finite Element) discretization with mass-lumping of advection-diffusion equations. The relevant exponentiallike matrix function is approximated by polynomial interpolation, at a sequence of real Leja points related to the spectrum of the FE matrix (ReLPM, Real Leja Points Method). Application to 2D and 3D advection-dispersion models shows speed-ups of one order of magnitude with respect to a classical variable step-size Crank-Nicolson solver. The Advection-Diffusion ModelWe consider the classical evolutionary advection-diffusion problemwith mixed Dirichlet and Neumann boundary conditions on. Equation (1) represents, e.g., a simplified model for solute transport in groundwater flow (advection-dispersion), where c is the solute concentration, D the hydrodynamic dispersion tensor, v the average linear velocity of groundwater flow andWork supported by the research project CPDA028291 "Efficient approximation methods for nonlocal discrete transforms" of the University of Padova, and by the subproject "Approximation of matrix functions in the numerical solution of differential equations" (co-ordinator M. Vianello, University of Padova) of the MIUR PRIN 2003 project "Dynamical systems on matrix manifolds: numerical methods and applications" (co-ordinator L. Lopez, University of Bari). Thanks also to the numerical analysis group at the Dept. of Math. Methods and Models for Appl. Sciences of the University of Padova, for having provided FE matrices for our numerical tests, and for the use of their computing resources.

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“…[13]). In the case of the relevant ODEs system (2) (with stationary b and q), its variable step-size version writes as…”

Section: Crank-nicolson (Cn) Methodsmentioning

confidence: 99%

Section: An Exponential Integrator Via Mass-lumpingmentioning

confidence: 97%

See 1 more Smart Citation

The ReLPM Exponential Integrator for FE Discretizations of Advection-Diffusion Equations

Bergamaschi

Caliari

Vianello

2004

Lecture Notes in Computer Science

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“…Although CN might not be considered the best choice for time integration of ADR problems, it is an efficient and robust method still widely used on stiff large-scale problems in engineering applications, and a sound baseline benchmark for an advection-diffusion solver (cf. [19]). In the case of the relevant ODEs system (2) it writes as…”

Section: Numerical Examplesmentioning

confidence: 99%

The LEM exponential integrator for advection–diffusion–reaction equations

Caliari

Vianello

Bergamaschi

2007

Journal of Computational and Applied Mathematics

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We implement a second-order exponential integrator for semidiscretized advection-diffusion-reaction equations, obtained by coupling exponential-like Euler and Midpoint integrators, and computing the relevant matrix exponentials by polynomial interpolation at Leja points. Numerical tests on 2D models discretized in space by finite differences or finite elements, show that the Leja-Euler-Midpoint (LEM) exponential integrator can be up to 5 times faster than a classical second-order implicit solver.

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“…7 work includes that of [48,75,76] in which systems of ordinary differential equations are solved by projecting onto a Krylov subspace and integrated in time with polynomial methods based on Chebyshev expansion. Recently, Pini and Gambolati [63] introduced a method in which a system of ordinary differential equations is projected onto a Krylov subspace and integrated by the Crank Nicolson method.…”

Section: Notationmentioning

confidence: 99%

Exponential integrators for the incompressible Navier-Stokes equations.

Newman¹

2004

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We provide an algorithm and analysis of a high order projection scheme for time integration of the incompressible Navier-Stokes equations (NSE). The method is based on a projection onto the subspace of divergence-free (incompressible) functions interleaved with a Krylov-based exponential time integration (KBEI). These time integration methods provide a high order accurate, stable approach with many of the advantages of explicit methods, and can reduce the computational resources over conventional methods. The method is scalable in the sense that the computational costs grow linearly with problem size.Exponential integrators, used typically to solve systems of ODEs, utilize matrix vector products of the exponential of the Jacobian on a vector. For large systems, this product can be approximated efficiently by Krylov subspace methods. However, in contrast to explicit methods, KBEIs are not restricted by the time step. While implicit methods require a solution of a linear system with the Jacobian, KBEIs only require matrix vector products of the Jacobian. Furthermore, these methods are based on linearization, so there is no non-linear system solve at each time step.Differential-algebraic equations (DAEs) are ordinary differential equations (ODEs) subject to algebraic constraints. The discretized NSE constitute a system of DAEs, where the incompressibility condition is the algebraic constraint. Exponential integrators can be extended to DAEs with linear constraints imposed via a projection onto the constraint manifold. This results in a projected ODE that is integrated by a KBEI. In this approach, the Krylov subspace satisfies the constraint, hence the solution at the advanced time step automatically satisfies the constraint as well. For the NSE, the projection onto the constraint is typically achieved by a projection induced by the L 2 inner product. We examine this L 2 projection and an H 1 projection induced by the H 1 semi-inner product. The H 1 projection has an advantage over the L 2 projection in that it retains tangential Dirichlet boundary conditions for the flow. Both the H 1 and L 2 projections are solutions to saddle point problems that are efficiently solved by a preconditioned Uzawa algorithm.

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Arnoldi and Crank-Nicolson methods for integration in time of the transport equation

Cited by 10 publications

References 20 publications

The ReLPM Exponential Integrator for FE Discretizations of Advection-Diffusion Equations

The ReLPM Exponential Integrator for FE Discretizations of Advection-Diffusion Equations

The LEM exponential integrator for advection–diffusion–reaction equations

Exponential integrators for the incompressible Navier-Stokes equations.

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