scholarly journals On the convergence of two point block backward differentiation formula for second order ODEs

2016 ◽  
Author(s):  
Nooraini Zainuddin ◽  
Zarina Bibi Ibrahim ◽  
Noraini Jamaludin
Keyword(s):  
Second Order ◽  
Differentiation Formula ◽  
Backward Differentiation Formula

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.

scholarly journals Numerical solution of second order ODE directly by two point block backward differentiation formula

2015 ◽  
Author(s):  
Nooraini Zainuddin ◽  
Zarina Bibi Ibrahim ◽  
Khairil Iskandar Othman ◽  
Mohamed Suleiman ◽  
Noraini Jamaludin
Keyword(s):  
Numerical Solution ◽  
Second Order ◽  
Differentiation Formula ◽  
Backward Differentiation Formula
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