New two-derivative implicit-explicit Runge-Kutta methods for stiff reaction-diffusion systems

Singh, Ankit; Maurya, Vikas; Rajpoot, Manoj K.

doi:10.1016/j.jcp.2022.111610

Cited by 4 publications

(3 citation statements)

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“…This scaling behavior is known as a large-deviation principle, and it describes an exponential decay in time with a 'rate function' I (a). The rate function vanishes at its global minimum which is at a = 0, and its asymptotic behavior at |a| ≪ 1 is I (a) ≃ a 2 /4p, thus providing a smooth matching with the typical-fluctuations Gaussian regime (12). I (a) is closely related to the higher cumulants of the distribution: It is given by the Legendre-Fenchel transform…”

Section: Diffusion Equation In D =mentioning

confidence: 91%

“…We are now interested in analyzing the error in the approximate result (12). This can be done by considering the higher cumulants of the distribution of X j : For an (exactly) Gaussian distribution, all cumulants higher than the second cumulant vanish.…”

Section: Diffusion Equation In D =mentioning

confidence: 99%

“…In these equations, u (x, t) is the concentration of some substance, D > 0 is the diffusion coefficient and in equation (2), F is the velocity due to advection. In simple cases these equations can be solved analytically, but in general they can be difficult to solve, and as a result many numerical methods have been developed [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. One of the simplest methods involves a discretization of time and space, followed by an approximation of the temporal and spatial derivatives by using the method of finite differences.…”

Section: Introductionmentioning

confidence: 99%

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Optimal finite-differences discretization for the diffusion equation from the perspective of large-deviation theory

Smith

2024

J. Stat. Mech.

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When applying the finite-differences method to numerically solve the one-dimensional diffusion equation, one must choose discretization steps Δx, Δt in space and time, respectively. By applying large-deviation theory on the discretized dynamics, we analyze the numerical errors due to the discretization, and find that the (relative) errors are especially large in regions of space where the concentration of particles is very small. We find that the choice Δ t = Δ x 2 / ( 6 D ) , where D is the diffusion coefficient, gives optimal accuracy compared to any other choice (including, in particular, the limit Δ t → 0 ), thus reproducing the known result that may be obtained using truncation error analysis. In addition, we give quantitative estimates for the dynamical lengthscale that describes the size of the spatial region in which the numerical solution is accurate, and study its dependence on the discretization parameters. We then turn to study the advection–diffusion equation, and obtain explicit expressions for the optimal Δt and other parameters of the finite-differences scheme, in terms of Δx, D and the advection velocity. We apply these results to study large deviations of the area swept by a diffusing particle in one dimension, trapped by an external potential ∼ | x | . We extend our analysis to higher dimensions by combining our results from the one dimensional case with the locally one-dimension method.

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Section: Diffusion Equation In D =mentioning

confidence: 91%

Section: Diffusion Equation In D =mentioning

confidence: 99%