Analysis and Applications of the Exponential Time Differencing Schemes and Their Contour Integration Modifications

Du, Qiang; Zhu, Wenchao

doi:10.1007/s10543-005-7141-8

Cited by 110 publications

(73 citation statements)

References 12 publications

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“…A quite convenient way to numerically solve equations of the same kind as (20) is to use Exponential Time Differencing Runge-Kutta (ETDRK) methods [47,48,49,50]. If h denotes the (assumed constant) time step size, the first order and fourth order schemes [49] can respectively be written as…”

Section: Numerical Strategymentioning

confidence: 99%

“…Following [49,48], in order to avoid these errors for small values of |Lh|, the numerical evaluation of (e z − 1)/z makes use of a contour integral in the complex plane around z = 0. Preliminary tests on one-dimensional Michelson-Sivashinsky equation [21] showed it was more convenient to use the ETDRK4 method in terms of stability and CPU effort.…”

Section: Numerical Strategymentioning

confidence: 99%

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Assessment of the Evolution Equation Modelling approach for three-dimensional expanding wrinkled premixed flames

Albin

D'Angelo

2012

Combustion and Flame

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. Assessment of the Evolution Equation Modelling approach for threedimensional expanding wrinkled premixed flames. Combustion and Flame, Elsevier, 2012, 159 (5), pp.1932-1948. <10.1016/j.combustflame.2011 AbstractDirect Numerical Simulations (DNS), Evolution Equation Modelling (EEM) and Experimental results from the literature (EXP) are presented and analyzed for an expanding propane/air flame. DNS results are obtained thanks to the in-house finite-difference code HAllegro. Computed (DNS/EEM) and measured (EXP) equivalent radii R P and R S , mean stretch k and consumption velocity S C , as well as sample front shapes, are compared using to the same post-processing methodology. Small perturbations in the EEM input parameters induce comparatively small shifts in the compared results, showing the robustness of the approach. When slightly adapting only one O(1) parameter for the EEM strategy (the effective turbulent forcing amplitude felt by the flame), DNS and EEM show quite fair agreement one with the other and also with EXP, except for one of the experiments at early times. In the context of expanding flames, this validated EEM methodology can constitute a reliable tool to compute realisticly large sized flames.

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Section: Numerical Strategymentioning

confidence: 99%

Section: Numerical Strategymentioning

confidence: 99%

Assessment of the Evolution Equation Modelling approach for three-dimensional expanding wrinkled premixed flames

Albin

D'Angelo

2012

Combustion and Flame

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“…The stability analysis for cETD schemes is similar to the stability analysis of ETD [2,14,6,7,9], and the stability analysis of cIIF presented above.…”

Section: Stability Analysis Of Ciif Methodsmentioning

confidence: 81%

“…Using the second order central difference discretization on the diffusion, we obtain a system of nonlinear ODEs (6) Next we define three matrices U, A and B by (7) (8) and (9) In terms of these three matrices, the semi-discretized form (6) becomes (10) This formulation is based on a compact representation previously developed for solving a twodimensional Poisson's equation and other related separable equations [10].…”

Section: Two-dimensionsmentioning

confidence: 99%

“…However, the exact treatment of the differential operator requires evaluating exponentials of the approximation matrix for the linear differential operator. For periodic systems, this calculation is cheap both in CPU and storage because the approximation matrix can be diagonalized in the Fourier space [1][2][3][4][5][6][7]. For non-periodic systems, in which the approximation matrices are not diagonal, storage and calculation of exponentials of the matrices are significantly more expensive.…”

Section: Introductionmentioning

confidence: 99%

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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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Pattern formation in superdiffusion predator–prey‐like problems with integer‐ and noninteger‐order derivatives

Owolabi

Karaağaç

Băleanu

2020

Math Methods in App Sciences

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This paper focuses on the modeling and application of fractional derivative to model the interactions between two different species in which the one named predator depends on the other called prey solely for survival. The interaction between predator and prey has been one of the most intriguing and interesting subjects in applied mathematical biology and ecology. In the models, the classical reaction–diffusion equations subject to the Neumann boundary conditions are formulated on a finite but large domain x ∈ [0, L] by replacing the second‐order spatial derivatives with the fractional Laplacian operator of order 1 < α ≤ 2, which is classified as superdiffusion process. We examine the resulting coupled reaction–diffusion models for linear stability analysis and derive conditions under which the spatial patterns is evolved. In a view to understand our theoretical findings, the species spatial interactions is described in one and two dimensions. Through numerical experiments, we observe that a number of patterns can arise, including Turing spots, spiral‐like structures, and seemingly complex spatiotemporal distributions.

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Analysis and Applications of the Exponential Time Differencing Schemes and Their Contour Integration Modifications

Cited by 110 publications

References 12 publications

Assessment of the Evolution Equation Modelling approach for three-dimensional expanding wrinkled premixed flames

Assessment of the Evolution Equation Modelling approach for three-dimensional expanding wrinkled premixed flames

Compact integration factor methods in high spatial dimensions

Pattern formation in superdiffusion predator–prey‐like problems with integer‐ and noninteger‐order derivatives

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