Structure-preserving finite difference schemes for a semilinear thermoelastic system with second order time derivative

2018 ◽  
Vol 35 (3) ◽  
pp. 1213-1244
Author(s):  
Keisuke Yano ◽  
Shuji Yoshikawa
Keyword(s):  
Finite Difference ◽  
Time Derivative ◽  
Second Order ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Structure Preserving

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 CABARET FINITE‐DIFFERENCE SCHEMES FOR THE ONE‐DIMENSIONAL EULER EQUATIONS

2001 ◽  
Vol 6 (2) ◽  
pp. 210-220 ◽  
Author(s):  
V. M. Goloviznin ◽  
T. P. Hynes ◽  
S. A. Karabasov
Keyword(s):  
Time Derivative ◽  
Second Order ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Gas Flows ◽  
Time Layer ◽  
One Dimensional ◽  
Upwind Schemes ◽  
The One ◽  
Solution Accuracy

In the present paper we consider second order compact upwind schemes with a space split time derivative (CABARET) applied to one‐dimensional compressible gas flows. As opposed to the conventional approach associated with incorporating adjacent space cells we use information from adjacent time layer to improve the solution accuracy. Taking the first order Roe scheme as the basis we develop a few higher (i.e. second within regions of smooth solutions) order accurate difference schemes. One of them (CABARET3) is formulated in a two‐time‐layer form, which makes it most simple and robust. Supersonic and subsonic shock‐tube tests are used to compare the new schemes with several well‐known second‐order TVD schemes. In particular, it is shown that CABARET3 is notably more accurate than the standard second‐order Roe scheme with MUSCL flux splitting.

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