scholarly journals A second-order energy stable backward differentiation formula method for the epitaxial thin film equation with slope selection

2018 ◽  
Vol 34 (6) ◽  
pp. 1975-2007 ◽  
Author(s):  
Wenqiang Feng ◽  
Cheng Wang ◽  
Steven M. Wise ◽  
Zhengru Zhang
Keyword(s):  
Thin Film ◽  
Second Order ◽  
Epitaxial Thin Film ◽  
Differentiation Formula ◽  
Backward Differentiation Formula ◽  
Order Energy ◽  
Slope Selection ◽  
Thin Film Equation ◽  
Energy Stable ◽  
Formula Method

A Second Order Energy Stable Linear Scheme for a Thin Film Model Without Slope Selection

2018 ◽  
Vol 76 (3) ◽  
pp. 1905-1937 ◽  
Author(s):  
Weijia Li ◽  
Wenbin Chen ◽  
Cheng Wang ◽  
Yue Yan ◽  
Ruijian He
Keyword(s):  
Thin Film ◽  
Second Order ◽  
Film Model ◽  
Order Energy ◽  
Thin Film Model ◽  
Slope Selection ◽  
Linear Scheme ◽  
Energy Stable ◽  
Stable Linear

scholarly journals Numerical Approximation of the Space Fractional Cahn-Hilliard Equation

2019 ◽  
Vol 2019 ◽  
pp. 1-10
Author(s):  
Zhifeng Weng ◽  
Langyang Huang ◽  
Rong Wu
Keyword(s):  
Numerical Approximation ◽  
Second Order ◽  
Differentiation Formula ◽  
Numerical Examples ◽  
Backward Differentiation Formula ◽  
Hilliard Equation ◽  
Order Case ◽  
Energy Stable ◽  
Cahn Hilliard Equation ◽  
Fourier Spectral Methods

In this paper, a second-order accurate (in time) energy stable Fourier spectral scheme for the fractional-in-space Cahn-Hilliard (CH) equation is considered. The time is discretized by the implicit backward differentiation formula (BDF), along with a linear stabilized term which represents a second-order Douglas-Dupont-type regularization. The semidiscrete schemes are shown to be energy stable and to be mass conservative. Then we further use Fourier-spectral methods to discretize the space. Some numerical examples are included to testify the effectiveness of our proposed method. In addition, it shows that the fractional order controls the thickness and the lifetime of the interface, which is typically diffusive in integer order case.

scholarly journals Backward differentiation formula finite difference schemes for diffusion equations with an obstacle term

2020 ◽  
Author(s):  
Olivier Bokanowski ◽  
Kristian Debrabant
Keyword(s):  
Finite Difference ◽  
Mathematical Finance ◽  
Second Order ◽  
Difference Schemes ◽  
Diffusion Equations ◽  
Finite Difference Schemes ◽  
Differentiation Formula ◽  
Backward Differentiation Formula ◽  
Hamilton Jacobi Bellman ◽  
Nicolson Scheme

Abstract Finite difference schemes, using backward differentiation formula (BDF), are studied for the approximation of one-dimensional diffusion equations with an obstacle term of the form $$\begin{equation*}\min(v_t - a(t,x) v_{xx} + b(t,x) v_x + r(t,x) v, v- \varphi(t,x))= f(t,x).\end{equation*}$$For the scheme building on the second-order BDF formula, we discuss unconditional stability, prove an $L^2$-error estimate and show numerically second-order convergence, in both space and time, unconditionally on the ratio of the mesh steps. In the analysis an equivalence of the obstacle equation with a Hamilton–Jacobi–Bellman equation is mentioned, and a Crank–Nicolson scheme is tested in this context. Two academic problems for parabolic equations with an obstacle term with explicit solutions and the American option problem in mathematical finance are used for numerical tests.

A Linear Energy Stable Scheme for a Thin Film Model Without Slope Selection

2011 ◽  
Vol 52 (3) ◽  
pp. 546-562 ◽  
Author(s):  
Wenbin Chen ◽  
Sidafa Conde ◽  
Cheng Wang ◽  
Xiaoming Wang ◽  
Steven M. Wise
Keyword(s):  
Thin Film ◽  
Film Model ◽  
Thin Film Model ◽  
Slope Selection ◽  
Energy Stable ◽  
Energy Stable Scheme ◽  
Stable Scheme
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