Wavelet-optimized compact finite difference method for convection–diffusion equations

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Mani Mehra ◽  
Kuldip Singh Patel ◽  
Ankita Shukla
Keyword(s):  
Finite Difference ◽  
Two Dimensions ◽  
Finite Difference Approximation ◽  
Diffusion Equations ◽  
Difference Approximation ◽  
Uniform Grid ◽  
Adaptive Grids ◽  
Smooth Functions ◽  
Compact Finite Difference ◽  
Convection Diffusion

AbstractIn this article, compact finite difference approximations for first and second derivatives on the non-uniform grid are discussed. The construction of diffusion wavelets using compact finite difference approximation is presented. Adaptive grids are obtained for non-smooth functions in one and two dimensions using diffusion wavelets. High-order accurate wavelet-optimized compact finite difference (WOCFD) method is developed to solve convection–diffusion equations in one and two dimensions on an adaptive grid. As an application in option pricing, the solution of Black–Scholes partial differential equation (PDE) for pricing barrier options is obtained using the proposed WOCFD method. Numerical illustrations are presented to explain the nature of adaptive grids for each case.

scholarly journals An Efficient Compact Difference Method for Temporal Fractional Subdiffusion Equations

2019 ◽  
Vol 2019 ◽  
pp. 1-9 ◽  
Author(s):  
Lei Ren ◽  
Lei Liu
Keyword(s):  
Finite Difference ◽  
Fourth Order ◽  
Difference Method ◽  
Second Order ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Approximation Formula ◽  
Discrete Energy ◽  
Compact Finite Difference ◽  
Second Order In Time

In this paper, a high-order compact finite difference method is proposed for a class of temporal fractional subdiffusion equation. A numerical scheme for the equation has been derived to obtain 2-α in time and fourth-order in space. We improve the results by constructing a compact scheme of second-order in time while keeping fourth-order in space. Based on the L2-1σ approximation formula and a fourth-order compact finite difference approximation, the stability of the constructed scheme and its convergence of second-order in time and fourth-order in space are rigorously proved using a discrete energy analysis method. Applications using two model problems demonstrate the theoretical results.

Digital simulation of photoacoustic impulse responses

1986 ◽  
Vol 64 (9) ◽  
pp. 1049-1052 ◽  
Author(s):  
Richard M. Miller
Keyword(s):  
Finite Difference ◽  
Impulse Response ◽  
Depth Distribution ◽  
Digital Simulation ◽  
Heat Diffusion ◽  
Finite Difference Approximation ◽  
Diffusion Equations ◽  
Difference Approximation ◽  
Impulse Responses ◽  
Solid Samples

Impulse-response photoacoustic spectroscopy provides information on the depth distribution of chromophores in solid samples. To gain an understanding of the way in which sample properties affect the impulse response, a digital model has been generated. This model is based on discretization of time and space coupled with a finite-difference approximation of the governing heat-diffusion equations. The simulations are compared with experimental results.

scholarly journals A derivation of Griffith functionals from discrete finite-difference models

2020 ◽  
Vol 59 (6) ◽  
Author(s):  
Vito Crismale ◽  
Giovanni Scilla ◽  
Francesco Solombrino
Keyword(s):  
Finite Difference ◽  
Continuum Limit ◽  
Dirichlet Boundary ◽  
Two Dimensions ◽  
Dirichlet Boundary Conditions ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Discretization Step ◽  
Fidelity Term ◽  
Ellipticity Parameter

AbstractWe analyze a finite-difference approximation of a functional of Ambrosio–Tortorelli type in brittle fracture, in the discrete-to-continuum limit. In a suitable regime between the competing scales, namely if the discretization step $$\delta $$ δ is smaller than the ellipticity parameter $$\varepsilon $$ ε , we show the $$\varGamma $$ Γ -convergence of the model to the Griffith functional, containing only a term enforcing Dirichlet boundary conditions and no $$L^p$$ L p fidelity term. Restricting to two dimensions, we also address the case in which a (linearized) constraint of non-interpenetration of matter is added in the limit functional, in the spirit of a recent work by Chambolle, Conti and Francfort.

Sign in / Sign up

Export Citation Format

Share Document