An unconditionally stable splitting method for the Allen–Cahn equation with logarithmic free energy

We propose an unconditionally stable numerical scheme for a 2D dynamic [Formula: see text]-tensor model of nematic liquid crystals. This dynamic [Formula: see text]-tensor model is an [Formula: see text]-gradient flow generated by the liquid crystal free energy that contains a cubic term, which is physically relevant but makes the free energy unbounded from below, and for this reason, has been avoided in other numerical studies. The unboundedness of the energy brings significant difficulty in analyzing the model and designing numerical schemes. By using a stabilizing technique, we construct an unconditionally stable scheme, and establish its unique solvability and convergence. Our convergence analysis also leads to, as a byproduct, the well-posedness of the original PDE system for the 2D [Formula: see text]-tensor model. Several numerical examples are presented to validate and demonstrate the effectiveness of the scheme.

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An unconditionally stable splitting scheme for a class of nonlinear parabolic equations

IMA Journal of Numerical Analysis ◽

10.1093/imanum/19.4.609 ◽

1999 ◽

Vol 19 (4) ◽

pp. 609-635 ◽

Cited By ~ 32

Author(s):

K. Karlsen

Keyword(s):

Parabolic Equations ◽

Nonlinear Parabolic Equations ◽

Splitting Scheme ◽

Unconditionally Stable ◽

Nonlinear Parabolic ◽

Stable Splitting

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An Unconditionally Stable, Positivity-Preserving Splitting Scheme for Nonlinear Black-Scholes Equation with Transaction Costs

The Scientific World JOURNAL ◽

10.1155/2014/525207 ◽

2014 ◽

Vol 2014 ◽

pp. 1-11 ◽

Cited By ~ 1

Author(s):

Jianqiang Guo ◽

Wansheng Wang

Keyword(s):

Transaction Costs ◽

Splitting Method ◽

Euler Method ◽

Discrete Equation ◽

Call Option ◽

Backward Euler Method ◽

Splitting Scheme ◽

Unconditionally Stable ◽

Backward Euler ◽

Black Scholes

This paper deals with the numerical analysis of nonlinear Black-Scholes equation with transaction costs. An unconditionally stable and monotone splitting method, ensuring positive numerical solution and avoiding unstable oscillations, is proposed. This numerical method is based on the LOD-Backward Euler method which allows us to solve the discrete equation explicitly. The numerical results for vanilla call option and for European butterfly spread are provided. It turns out that the proposed scheme is efficient and reliable.

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Encultured minds, not error reduction minds

Behavioral and Brain Sciences ◽

10.1017/s0140525x19002826 ◽

2020 ◽

Vol 43 ◽

Author(s):

Robert Mirski ◽

Mark H. Bickhard ◽

David Eck ◽

Arkadiusz Gut

Keyword(s):

Free Energy ◽

Error Reduction ◽

Energy Principle ◽

Free Energy Principle ◽

Current Article

Abstract There are serious theoretical problems with the free-energy principle model, which are shown in the current article. We discuss the proposed model's inability to account for culturally emergent normativities, and point out the foundational issues that we claim this inability stems from.

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