An ETD Crank‐Nicolson method for reaction‐diffusion systems

Kleefeld, B.; Wade, Bruce A.

doi:10.1002/num.20682

Cited by 41 publications

(31 citation statements)

References 26 publications

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“…However, we shall use A and F based on an abstract formulation for convenience of the development of the numerical scheme and its analysis. Therefore, we reset the initial value problem (2.1) to be posed in a Banach space

scriptX

, as follows, see . We consider A to be a linear, self–adjoint, positive definite, closed operator with a compact inverse, defined on a dense domain

D (A) \subset X

.…”

Section: Partial Integral Differential Equationsmentioning

confidence: 99%

“…Let

0 < k \leq k_{0}

, for some k 0 , be the fixed time step and t n = nk ,

0 \leq n \leq N, k = Δ t

. Using Duhamel's principle, an approach similar to Kleefeld et al and Yousuf et al , we can show that

u_{normalα} (t_{n}), 1 \leq α \leq m_{o}, 0 < n \leq N

satisfy the following recurrent formula:

u_{normalα} (t_{n + 1}) = e^{- k A_{normalα}} u_{normalα} (t_{n}) + \int_{0}^{k} e^{- A_{α} false(k - normalτ false)} F_{normalα} (u_{1} (t_{n} + τ), u_{2} (t_{n} + τ), \dots, u_{m_{o}} (t_{n} + τ), t_{n} + τ) d τ

The integral in (2.6) can be approximated by a class of exponential time differencing (ETD) numerical schemes, see for example and references therein. Denoting the approximation to

u_{normalα} (t_{n})

u_{α, n}

and the approximation to

a_{normalα} (t_{n})

a_{α,}

…”

Section: Partial Integral Differential Equationsmentioning

confidence: 99%

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Numerical solution of systems of partial integral differential equations with application to pricing options

Yousuf

2018

Numerical Methods Partial

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We introduce and analyze a strongly stable numerical method designed to yield good performance under challenging conditions of irregular or mismatched initial data for solving systems of coupled partial integral differential equations (PIDEs). Spatial derivatives are approximated using second order central difference approximations by treating the mixed derivative terms in a special way. The integral operators are approximated using one and two–dimensional trapezoidal rule on an equidistant grid. Computational complexity of the method for solving large systems of PIDEs is discussed. A detailed treatment for the consistency, stability, and convergence of the proposed method is provided. Two asset American option under regime–switching with jump–diffusion model when solved using a penalty term, leads to a system of two dimensional PIDEs with mixed derivatives. This model involves double probability density function which brings more challenges to the numerical solution in already a complicated partial integral differential equation. The complexity of the dense jump probability generator, the nonlinear penalty term and the regime–switching terms are treated efficiently, while maintaining the stability and convergence of the method. The impact of the jump intensity and other parameters is shown in the graphs. Numerical experiments are performed to demonstrated efficiency, accuracy, and reliability of the proposed approach.

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scriptX

, as follows, see . We consider A to be a linear, self–adjoint, positive definite, closed operator with a compact inverse, defined on a dense domain

D (A) \subset X

.…”

Section: Partial Integral Differential Equationsmentioning

confidence: 99%

“…Let

0 < k \leq k_{0}

, for some k 0 , be the fixed time step and t n = nk ,

0 \leq n \leq N, k = Δ t

. Using Duhamel's principle, an approach similar to Kleefeld et al and Yousuf et al , we can show that

u_{normalα} (t_{n}), 1 \leq α \leq m_{o}, 0 < n \leq N

satisfy the following recurrent formula:

u_{normalα} (t_{n + 1}) = e^{- k A_{normalα}} u_{normalα} (t_{n}) + \int_{0}^{k} e^{- A_{α} false(k - normalτ false)} F_{normalα} (u_{1} (t_{n} + τ), u_{2} (t_{n} + τ), \dots, u_{m_{o}} (t_{n} + τ), t_{n} + τ) d τ

The integral in (2.6) can be approximated by a class of exponential time differencing (ETD) numerical schemes, see for example and references therein. Denoting the approximation to

u_{normalα} (t_{n})

u_{α, n}

and the approximation to

a_{normalα} (t_{n})

a_{α,}

…”

Section: Partial Integral Differential Equationsmentioning

confidence: 99%

Numerical solution of systems of partial integral differential equations with application to pricing options

Yousuf

2018

Numerical Methods Partial

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“…Nevertheless, a comparison between two maps shows that in general, the instability is favored in the NLS-MB case, which means that the rogue wave solution and MI develop effectively over shorter length scales than in the MB case. Besides, we study numerically the dynamics and stability of these rogue waves, using the exponential time differencing Crank-Nicolson (ETDCN) scheme with Padé approximation [46], which is proved to be stable and second-order convergent for such kind of stiff problems. To evaluate the dynamics and stability, three numerical games are played, for given system parameters s = 1, a = 1.5, σ = 1, φ = Fig.…”

Section: And Numerical Simulationsmentioning

confidence: 99%

Optical Peregrine rogue waves of self-induced transparency in a resonant erbium-doped fiber

Chen

Baronio

et al. 2017

Opt. Express

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Abstract:The resonant interaction of an optical field with two-level doping ions in a cryogenic optical fiber is investigated within the framework of nonlinear Schrödinger and Maxwell-Bloch equations. We present explicit fundamental rational rogue wave solutions in the context of self-induced transparency for the coupled optical and matter waves. It is exhibited that the optical wave component always features a typical Peregrine-like structure, while the matter waves involve more complicated yet spatiotemporally balanced amplitude distribution. The existence and stability of these rogue waves is then confirmed by numerical simulations, and they are shown to be excited amid the onset of modulation instability. These solutions can also be extended, using the same analytical framework, to include higher-order dispersive and nonlinear effects, highlighting their universality.

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“…For temporal integration, the integration factor (IF) and exponential time differencing (ETD) methods are effective ways to deal with the temporal stability constraints arising from high-order spatial derivatives on uniform meshes [20, 21, 22]. The IF and ETD methods usually treat linear operators of the highest-order derivatives exactly, and hence, they provide good temporal stability by allowing larger sizes of time step in temporal updates [23, 24, 20].…”

Section: Introductionmentioning

confidence: 99%

Semi-implicit integration factor methods on sparse grids for high-dimensional systems

Wang

Chen

Nie

2015

Journal of Computational Physics

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Numerical methods for partial differential equations in high-dimensional spaces are often limited by the curse of dimensionality. Though the sparse grid technique, based on a one-dimensional hierarchical basis through tensor products, is popular for handling challenges such as those associated with spatial discretization, the stability conditions on time step size due to temporal discretization, such as those associated with high-order derivatives in space and stiff reactions, remain. Here, we incorporate the sparse grids with the implicit integration factor method (IIF) that is advantageous in terms of stability conditions for systems containing stiff reactions and diffusions. We combine IIF, in which the reaction is treated implicitly and the diffusion is treated explicitly and exactly, with various sparse grid techniques based on the finite element and finite difference methods and a multi-level combination approach. The overall method is found to be efficient in terms of both storage and computational time for solving a wide range of PDEs in high dimensions. In particular, the IIF with the sparse grid combination technique is flexible and effective in solving systems that may include cross-derivatives and non-constant diffusion coefficients. Extensive numerical simulations in both linear and nonlinear systems in high dimensions, along with applications of diffusive logistic equations and Fokker-Planck equations, demonstrate the accuracy, efficiency, and robustness of the new methods, indicating potential broad applications of the sparse grid-based integration factor method.

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An ETD Crank‐Nicolson method for reaction‐diffusion systems

Cited by 41 publications

References 26 publications

Numerical solution of systems of partial integral differential equations with application to pricing options

Numerical solution of systems of partial integral differential equations with application to pricing options

Optical Peregrine rogue waves of self-induced transparency in a resonant erbium-doped fiber

Semi-implicit integration factor methods on sparse grids for high-dimensional systems

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