An Alternative Formulation of Finite Difference Weighted ENO Schemes with Lax--Wendroff Time Discretization for Conservation Laws

2013 ◽  
Vol 35 (2) ◽  
pp. A1137-A1160 ◽  
Author(s):  
Yan Jiang ◽  
Chi-Wang Shu ◽  
Mengping Zhang
Keyword(s):  
Finite Difference ◽  
Conservation Laws ◽  
Time Discretization ◽  
Alternative Formulation ◽  
Eno Schemes ◽  
Weighted Eno

scholarly journals Implicit Weighted ENO Schemes for the Three-Dimensional Incompressible Navier–Stokes Equations

1998 ◽  
Vol 146 (1) ◽  
pp. 464-487 ◽  
Author(s):  
Jaw-Yen Yang ◽  
Shih-Chang Yang ◽  
Yih-Nan Chen ◽  
Chiang-An Hsu
Keyword(s):  
Stokes Equations ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Eno Schemes ◽  
Weighted Eno

SPECTRALLY ACCURATE OPTION PRICING UNDER THE TIME-FRACTIONAL BLACK–SCHOLES MODEL

2021 ◽  
pp. 1-21
Author(s):  
GERALDINE TOUR ◽  
NAWDHA THAKOOR ◽  
DÉSIRÉ YANNICK TANGMAN
Keyword(s):  
Finite Difference ◽  
Option Pricing ◽  
Time Discretization ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Spectral Approximation ◽  
Barrier Option ◽  
Order Accuracy ◽  
Black Scholes ◽  
Richardson Extrapolation Method

Abstract We propose a Legendre–Laguerre spectral approximation to price the European and double barrier options in the time-fractional framework. By choosing an appropriate basis function, the spectral discretization is used for the approximation of the spatial derivatives of the time-fractional Black–Scholes equation. For the time discretization, we consider the popular $L1$ finite difference approximation, which converges with order $\mathcal {O}((\Delta \tau )^{2-\alpha })$ for functions which are twice continuously differentiable. However, when using the $L1$ scheme for problems with nonsmooth initial data, only the first-order accuracy in time is achieved. This low-order accuracy is also observed when solving the time-fractional Black–Scholes European and barrier option pricing problems for which the payoffs are all nonsmooth. To increase the temporal convergence rate, we therefore consider a Richardson extrapolation method, which when combined with the spectral approximation in space, exhibits higher order convergence such that high accuracies over the whole discretization grid are obtained. Compared with the traditional finite difference scheme, numerical examples clearly indicate that the spectral approximation converges exponentially over a small number of grid points. Also, as demonstrated, such high accuracies can be achieved in much fewer time steps using the extrapolation approach.

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