Operator splitting kernel based numerical method for a generalized Leland’s model

Koleva, Miglena N.; Vulkov, Lubin G.

doi:10.1016/j.cam.2014.07.019

Cited by 6 publications

(2 citation statements)

References 15 publications

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“…Remark 3.5. There have been many existing works of operator splitting numerical approximation to nonlinear PDEs, such as [19,95] for reaction-diffusion systems, [5,66,82,86] for the nonlinear Schrödinger equation, [4] for the incompressible magnetohydrodynamics system, [8] for the delay equation, [29] for the nonlinear evolution equation, [30] for the Vlasov-type equation, [53] for a generalized Leland's mode, [91,92] for the "Good" Boussinesq equation, [56] for the Allen-Cahn equation, [57] for the molecular beamer epitaxy (MBE) equation, [94] for nonlinear solvation problem, etc. On the other hand, different computational stages correspond to different physical energy for most existing works of operator splitting, so that a combined energy stability estimate becomes very challenging at a theoretical level.…”

Section: The Operator Splitting Scheme For the Reaction-diffusion Systemmentioning

confidence: 99%

A structure-preserving, operator splitting scheme for reaction-diffusion equations with detailed balance

Liu,

Wang,

Wang

2020

Preprint

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In this paper, we propose and analyze a positivity-preserving, energy stable numerical scheme for certain type reaction-diffusion systems involving the Law of Mass Action with detailed balance condition. The numerical scheme is constructed based on a recently developed energetic variational formulation, in which the reaction part is reformulated in terms of reaction trajectories, and both the reaction and the diffusion parts dissipate the same free energy. This subtle fact opens a path of an operator splitting scheme for these systems. At the reaction stage, we solve equations of reaction trajectories by treating all the logarithmic terms in the reformulated form implicitly due to their convex nature. The positivity-preserving property and unique solvability can be theoretically proved, based on the singular behavior of the logarithmic function around the limiting value. Moreover, the energy stability of this scheme at the reaction stage can be proved by a careful convexity analysis. Similar techniques are used to establish the positivity-preserving property and energy stability for the standard semi-implicit solver at the diffusion stage. As a result, a combination of these two stages leads to a positivity-preserving and energy stable numerical scheme for the original reaction-diffusion system. To our best knowledge, it is the first time to report an energy-dissipation-lawbased operator splitting scheme to a nonlinear PDE with variational structures. Several numerical examples are presented to demonstrate the robustness of the proposed operator splitting scheme.

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Section: The Operator Splitting Scheme For the Reaction-diffusion Systemmentioning

confidence: 99%

A structure-preserving, operator splitting scheme for reaction-diffusion equations with detailed balance

Liu,

Wang,

Wang

2020

Preprint

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show abstract

“…Remark 3.3. There have been many existing works of operator splitting numerical approximation to nonlinear PDEs, such as [12,13,58] for reaction-diffusion systems, [5,7,38,47,48] for the nonlinear Schrödinger equation, [4] for the incompressible magnetohydrodynamics system, [6] for the delay equation, [20] for the nonlinear evolution equation, [21] for the Vlasov-type equation, [30] for a generalized Leland's mode, [54,55] for the "Good" Boussinesq equation, [32] for the Allen-Cahn equation, [33] for the molecular beamer epitaxy (MBE) equation, [57] for nonlinear solvation problem, etc. A few convergence estimates have also been reported for gradient flow with polynomial energy potential, such as [33,55].…”

mentioning

confidence: 99%

Convergence analysis of the variational operator splitting scheme for a reaction-diffusion system with detailed balance

Li¹,

Wang²,

Wang³

et al. 2021

Preprint

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We present a detailed convergence analysis for an operator splitting scheme proposed in [C. Liu et al., J. Comput. Phys., 436, 110253, 2021] for a reaction-diffusion system with detailed balance. The numerical scheme has been constructed based on a recently developed energetic variational formulation, in which the reaction part is reformulated in terms of the reaction trajectory, and both the reaction and diffusion parts dissipate the same free energy. The scheme is energy stable and positivity-preserving. In this paper, the detailed convergence analysis and error estimate are performed for the operator splitting scheme. The nonlinearity in the reaction trajectory equation, as well as the implicit treatment of nonlinear and singular logarithmic terms, impose challenges in numerical analysis. To overcome these difficulties, we make use of the convex nature of the logarithmic nonlinear terms, which are treated implicitly in the chemical reaction stage. In addition, a combination of a rough error estimate and a refined error estimate leads to a desired bound of the numerical error in the reaction stage, in the discrete maximum norm. Furthermore, a discrete maximum principle yields the evolution bound of the numerical error function at the diffusion stage. As a direct consequence, a combination of the numerical error analysis at different stages and the consistency estimate for the operator splitting results in the convergence estimate of the numerical scheme for the full reaction-diffusion system.

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A second order operator splitting numerical scheme for the “good” Boussinesq equation

Zhang

Wang

Huang

et al. 2017

Applied Numerical Mathematics

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Operator splitting kernel based numerical method for a generalized Leland’s model

Cited by 6 publications

References 15 publications

A structure-preserving, operator splitting scheme for reaction-diffusion equations with detailed balance

A structure-preserving, operator splitting scheme for reaction-diffusion equations with detailed balance

Convergence analysis of the variational operator splitting scheme for a reaction-diffusion system with detailed balance

A second order operator splitting numerical scheme for the “good” Boussinesq equation

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