An analysis of conservative finite difference schemes for differential equations with discontinuous coefficients

2007 ◽  
Vol 191 (1) ◽  
pp. 183-192
Author(s):  
Ebru Ozbilge
Keyword(s):  
Finite Difference ◽  
Differential Equations ◽  
Discontinuous Coefficients ◽  
Difference Schemes ◽  
Finite Difference Schemes

scholarly journals Exact Finite-Difference Schemes ford-Dimensional Linear Stochastic Systems with Constant Coefficients

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Peng Jiang ◽  
Xiaofeng Ju ◽  
Dan Liu ◽  
Shaoqun Fan
Keyword(s):  
Finite Difference ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Symplectic Structure ◽  
Stochastic Systems ◽  
First Integral ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Linear Stochastic Systems ◽  
Constant Coefficients

The authors attempt to construct the exact finite-difference schemes for linear stochastic differential equations with constant coefficients. The explicit solutions to Itô and Stratonovich linear stochastic differential equations with constant coefficients are adopted with the view of providing exact finite-difference schemes to solve them. In particular, the authors utilize the exact finite-difference schemes of Stratonovich type linear stochastic differential equations to solve the Kubo oscillator that is widely used in physics. Further, the authors prove that the exact finite-difference schemes can preserve the symplectic structure and first integral of the Kubo oscillator. The authors also use numerical examples to prove the validity of the numerical methods proposed in this paper.

On Finite Difference Schemes for Elliptic Equations with Discontinuous Coefficients

2013 ◽  
Vol 13 (3) ◽  
pp. 281-289
Author(s):  
Manfred Dobrowolski
Keyword(s):  
Finite Element ◽  
Finite Difference ◽  
Elliptic Equations ◽  
Approximation Error ◽  
Discontinuous Coefficients ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
The Finite Element Method ◽  
Order Four ◽  
Element Theory

Abstract. We study the convergence of finite difference schemes for approximating elliptic equations of second order with discontinuous coefficients. Two of these finite difference schemes arise from the discretization by the finite element method using bilinear shape functions. We prove an convergence for the gradient, if the solution is locally in H3. Thus, in contrast to the first order convergence for the gradient obtained by the finite element theory we show that the gradient is superclose. From the Bramble–Hilbert Lemma we derive a higher order compact (HOC) difference scheme that gives an approximation error of order four for the gradient. A numerical example is given.

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