A method for exponential propagation of large systems of stiff nonlinear differential equations

Friesner, Richard A.; Tuckerman, Laurette S.; Dornblaser, Bright; Russo, Thomas V.

doi:10.1007/bf01060992

Cited by 107 publications

(89 citation statements)

References 11 publications

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“…The development of numerical procedures for approximating the action of f (A) to a vector has received considerable attention in the last two decades. This is possibly related to the significant increase of methods for the numerical solution of partial differential equations that either directly approximate the exact solution (see, e.g., [20], [47], [24], [23]), or employ matrix functional integrators; see, e.g., [30], [28], [29]. In addition, large-scale advanced scientific applications often require function evaluations of matrices; see, e.g., [5], [43], [55], [17].…”

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confidence: 99%

A new investigation of the extended Krylov subspace method for matrix function evaluations

Knizhnerman

Simoncini

2010

Numerical Linear Algebra App

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Abstract. For large square matrices A and functions f , the numerical approximation of the action of f (A) to a vector v has received considerable attention in the last two decades. In this paper we investigate the Extended Krylov subspace method, a technique that was recently proposed to approximate f (A)v for A symmetric. We provide a new theoretical analysis of the method, which improves the original result for A symmetric, and gives a new estimate for A nonsymmetric. Numerical experiments confirm that the new error estimates correctly capture the linear asymptotic convergence rate of the approximation. By using recent algorithmic improvements, we also show that the method is computationally competitive with respect to other enhancement techniques.

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confidence: 99%

A new investigation of the extended Krylov subspace method for matrix function evaluations

Knizhnerman

Simoncini

2010

Numerical Linear Algebra App

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“…This bound is independent of the spatial resolution and diverges less rapidly in the limit ǫ → 0 + than the physical time scale, which is proportional to ǫ −1 . It would be interesting to explore whether a more sophisticated explicit time-stepping technique such as a matrix exponential method [40,41] or a fully implicit method [42,43] …”

Section: Discussionmentioning

confidence: 99%

Efficient algorithm on a nonstaggered mesh for simulating Rayleigh-Bénard convection in a box

Chiam

Lai²,

Greenside³

2003

Phys. Rev. E

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“…To apply the integration factor technique to the compact discretization form (10), we multiply (10) by exponential matrix e −At from the left, and e −Bt from the right to obtain (11) Integration of (11) over one time step from t n to t n+1 ≡ t n + Δt, where Δt is the time step, leads to (12) To construct a scheme of rth order truncation error, we approximate the integrand in (12), (13) using a (r − 1)th order Lagrange polynomial at a set of interpolation points t n+1 , t n ,…, t n+2−r : (14) where (15) The specific form of the polynomial (15) at low orders is listed in Table 1. In terms of (τ), (12) takes the form, (16) So the new r-th order implicit schemes are (17) where α 1 , α 0 , α −1 ,···, α −r+2 are coefficients calculated from the integrals of the polynomial in (τ), (18) In Table 2, the value of coefficients, α −j , for schemes of order up to four are listed.…”

Section: Two-dimensionsmentioning

confidence: 99%

“…If matrices A, C (p) , and B do not commute, one may need to consider the eigenspace of matrices A and B in order to evaluate (22) explicitly [11,12]. Assuming an eigenvalue decomposition (24) is (26) where can be evaluated recursively through integration by parts.…”

Section: Two-dimensionsmentioning

confidence: 99%

Compact integration factor methods in high spatial dimensions

Nie

Wan

Zhang

et al. 2008

Journal of Computational Physics

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The dominant cost for integration factor (IF) or exponential time differencing (ETD) methods is the repeated vector-matrix multiplications involving exponentials of discretization matrices of differential operators. Although the discretization matrices usually are sparse, their exponentials are not, unless the discretization matrices are diagonal. For example, a two-dimensional system of N × N spatial points, the exponential matrix is of a size of N 2 × N 2 based on direct representations. The vector-matrix multiplication is of O(N 4 ), and the storage of such matrix is usually prohibitive even for a moderate size N. In this paper, we introduce a compact representation of the discretized differential operators for the IF and ETD methods in both two-and three-dimensions. In this approach, the storage and CPU cost are significantly reduced for both IF and ETD methods such that the use of this type of methods becomes possible and attractive for two-or three-dimensional systems. For the case of two-dimensional systems, the required storage and CPU cost are reduced to O(N 2 ) and O(N 3 ), respectively. The improvement on three-dimensional systems is even more significant. We analyze and apply this technique to a class of semi-implicit integration factor method recently developed for stiff reaction-diffusion equations. Direct simulations on test equations along with applications to a morphogen system in two-dimensions and an intra-cellular signaling system in three-dimensions demonstrate an excellent efficiency of the new approach.

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A method for exponential propagation of large systems of stiff nonlinear differential equations

Cited by 107 publications

References 11 publications

A new investigation of the extended Krylov subspace method for matrix function evaluations

A new investigation of the extended Krylov subspace method for matrix function evaluations

Efficient algorithm on a nonstaggered mesh for simulating Rayleigh-Bénard convection in a box

Compact integration factor methods in high spatial dimensions

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