Error Estimate of Fourth-Order Compact Scheme for Linear Schrödinger Equations

2010 ◽  
Vol 47 (6) ◽  
pp. 4381-4401 ◽  
Author(s):  
Hong-lin Liao ◽  
Zhi-zhong Sun ◽  
Han-sheng Shi
Keyword(s):  
Error Estimate ◽  
Fourth Order ◽  
Compact Scheme ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Fourth Order Compact Scheme

A fourth-order real-space algorithm for solving local Schrödinger equations

2001 ◽  
Vol 115 (15) ◽  
pp. 6841-6846 ◽  
Author(s):  
J. Auer ◽  
E. Krotscheck ◽  
Siu A. Chin
Keyword(s):  
Fourth Order ◽  
Real Space ◽  
Schrodinger Equations ◽  
Schrödinger Equations

FOURTH-ORDER COMPACT SCHEME FOR OPTION PRICING UNDER THE MERTON’S AND KOU’S JUMP-DIFFUSION MODELS

2018 ◽  
Vol 21 (04) ◽  
pp. 1850027 ◽  
Author(s):  
KULDIP SINGH PATEL ◽  
MANI MEHRA
Keyword(s):  
Fourth Order ◽  
Initial Conditions ◽  
Linear Equations ◽  
Jump Diffusion ◽  
Diffusion Models ◽  
Compact Scheme ◽  
Discrete Problem ◽  
Option Prices ◽  
Fully Discrete ◽  
Fourth Order Compact Scheme

In this paper, a compact scheme with three time levels is proposed to solve the partial integro-differential equation that governs the option prices in jump-diffusion models. In the proposed compact scheme, the second derivative approximation of the unknowns is approximated using the value of these unknowns and their first derivative approximations, thereby allowing us to obtain a tridiagonal system of linear equations for a fully discrete problem. Moreover, the consistency and stability of the proposed compact scheme are proved. Owing to the low regularity of typical initial conditions, a smoothing operator is employed to ensure the fourth-order convergence rate. Numerical illustrations concerning the pricing of European options under the Merton’s and Kou’s jump-diffusion models are presented to validate the theoretical results.

The Conservative Splitting High-Order Compact Finite Difference Scheme for Two-Dimensional Schrödinger Equations

2017 ◽  
Vol 15 (01) ◽  
pp. 1750079
Author(s):  
Bo Wang ◽  
Dong Liang ◽  
Tongjun Sun
Keyword(s):  
Difference Scheme ◽  
Fourth Order ◽  
Two Dimensions ◽  
High Order ◽  
Optimal Error Estimate ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Two Dimensional ◽  
Time Step ◽  
Order Accuracy

In this paper, a new conservative and splitting fourth-order compact difference scheme is proposed and analyzed for solving two-dimensional linear Schrödinger equations. The proposed splitting high-order compact scheme in two dimensions has the excellent property that it preserves the conservations of charge and energy. We strictly prove that the scheme satisfies the charge and energy conservations and it is unconditionally stable. We also prove the optimal error estimate of fourth-order accuracy in spatial step and second-order accuracy in time step. The scheme can be easily implemented and extended to higher dimensional problems. Numerical examples are presented to confirm our theoretical results.

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