The Temporal Second Order Difference Schemes Based on the Interpolation Approximation for Solving the Time Multi-term and Distributed-Order Fractional Sub-diffusion Equations

2017 ◽  
Vol 73 (1) ◽  
pp. 93-121 ◽  
Author(s):  
Guang-hua Gao ◽  
Anatoly A. Alikhanov ◽  
Zhi-zhong Sun
Keyword(s):  
Second Order ◽  
Difference Schemes ◽  
Diffusion Equations ◽  
Order Difference ◽  
Distributed Order

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.

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