Comparison of Split Step Solvers for Multidimensional Schrödinger Problems

2013 ◽  
Vol 13 (2) ◽  
pp. 237-250 ◽  
Author(s):  
Raimondas Čiegis ◽  
Aleksas Mirinavičius ◽  
Mindaugas Radziunas
Keyword(s):  
Finite Difference ◽  
Numerical Experiments ◽  
Operator Splitting ◽  
Three Dimensional ◽  
Finite Difference Schemes ◽  
Alternating Direction Implicit ◽  
One Dimensional ◽  
Alternating Direction ◽  
Discrete Diffraction ◽  
Reaction Potential

Abstract. This paper presents the analysis of the split step solvers for multidimensional Schrödinger problems. The second-order symmetrical splitting techniques are applied. The standard operator splitting is used to split the linear diffraction and reaction/potential processes. The dimension splitting exploits the commuting property of one-dimensional discrete diffraction operators. Alternating Direction Implicit (ADI) and Locally One-Dimensional (LOD) algorithms are constructed and stability is investigated for two- and three-dimensional problems. Compact high-order approximations are applied to discretize diffraction operators. Results of numerical experiments are presented and convergence of finite difference schemes is investigated.

scholarly journals From Time-Collocated to Leapfrog Fundamental Schemes for ADI and CDI FDTD Methods

Axioms ◽  
2022 ◽  
Vol 11 (1) ◽  
pp. 23
Author(s):  
Eng Leong Tan
Keyword(s):  
Finite Difference ◽  
Time Domain ◽  
Finite Difference Time Domain ◽  
Fdtd Method ◽  
Unconditionally Stable ◽  
Alternating Direction Implicit ◽  
One Dimensional ◽  
Alternating Direction ◽  
Fdtd Methods ◽  
Difference Time

The leapfrog schemes have been developed for unconditionally stable alternating-direction implicit (ADI) finite-difference time-domain (FDTD) method, and recently the complying-divergence implicit (CDI) FDTD method. In this paper, the formulations from time-collocated to leapfrog fundamental schemes are presented for ADI and CDI FDTD methods. For the ADI FDTD method, the time-collocated fundamental schemes are implemented using implicit E-E and E-H update procedures, which comprise simple and concise right-hand sides (RHS) in their update equations. From the fundamental implicit E-H scheme, the leapfrog ADI FDTD method is formulated in conventional form, whose RHS are simplified into the leapfrog fundamental scheme with reduced operations and improved efficiency. For the CDI FDTD method, the time-collocated fundamental scheme is presented based on locally one-dimensional (LOD) FDTD method with complying divergence. The formulations from time-collocated to leapfrog schemes are provided, which result in the leapfrog fundamental scheme for CDI FDTD method. Based on their fundamental forms, further insights are given into the relations of leapfrog fundamental schemes for ADI and CDI FDTD methods. The time-collocated fundamental schemes require considerably fewer operations than all conventional ADI, LOD and leapfrog ADI FDTD methods, while the leapfrog fundamental schemes for ADI and CDI FDTD methods constitute the most efficient implicit FDTD schemes to date.

scholarly journals Comparison of numerical methods to price zero coupon bonds in a two-factor CIR model

2021 ◽  
Vol 20 (1) ◽  
pp. 109-147
Author(s):  
S. Emslie ◽  
S. Mataramvura
Keyword(s):  
Numerical Methods ◽  
Finite Difference ◽  
Factor Model ◽  
The Other ◽  
Finite Difference Schemes ◽  
Bond Price ◽  
Cir Model ◽  
Alternating Direction Implicit ◽  
Alternating Direction ◽  
Runge Kutta

In this paper we price a zero coupon bond under a Cox–Ingersoll–Ross (CIR) two-factor model using various numerical schemes. To the best of our knowledge, a closed-form or explicit price functional is not trivial and has been less studied. The use and comparison of several numerical methods to determine the bond price is one contribution of this paper. Ordinary differential equations (ODEs) , finite difference schemes and simulation are the three classes of numerical methods considered. These are compared on the basis of computational efficiency and accuracy, with the second aim of this paper being to identify the most efficient numerical method. The numerical ODE methods used to solve the system of ODEs arising as a result of the affine structure of the CIR model are more accurate and efficient than the other classes of methods considered, with the Runge–Kutta ODE method being the most efficient. The Alternating Direction Implicit (ADI) method is the most efficient of the finite difference scheme methods considered, while the simulation methods are shown to be inefficient. Our choice of considering these methods instead of the other known and apparently new numerical methods (eg Fast Fourier Transform (FFT) method, Cosine (COS) method, etc.) is motivated by their popularity in handling interest rate instruments. Keywords: Cox–Ingersoll–Ross model; numerical methods; Runge–Kutta method; zero-coupon bonds; Alternating Direction Implicit method

scholarly journals New schemes for a two-dimensional inverse problem with temperature overspecification

2001 ◽  
Vol 7 (3) ◽  
pp. 283-297 ◽  
Author(s):  
Mehdi Dehghan
Keyword(s):  
Inverse Problem ◽  
Finite Difference ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Central Processor ◽  
Two Dimensional ◽  
Alternating Direction Implicit ◽  
Fully Implicit ◽  
Implicit Schemes ◽  
Alternating Direction

Two different finite difference schemes for solving the two-dimensional parabolic inverse problem with temperature overspecification are considered. These schemes are developed for indentifying the control parameter which produces, at any given time, a desired temperature distribution at a given point in the spatial domain. The numerical methods discussed, are based on the (3,3) alternating direction implicit (ADI) finite difference scheme and the (3,9) alternating direction implicit formula. These schemes are unconditionally stable. The basis of analysis of the finite difference equation considered here is the modified equivalent partial differential equation approach, developed from the 1974 work of Warming and Hyett [17]. This allows direct and simple comparison of the errors associated with the equations as well as providing a means to develop more accurate finite difference schemes. These schemes use less central processor times than the fully implicit schemes for two-dimensional diffusion with temperature overspecification. The alternating direction implicit schemes developed in this report use more CPU times than the fully explicit finite difference schemes, but their unconditional stability is significant. The results of numerical experiments are presented, and accuracy and the Central Processor (CPU) times needed for each of the methods are discussed. We also give error estimates in the maximum norm for each of these methods.

scholarly journals Nonuniform Finite Difference Scheme for the Three-Dimensional Time-Fractional Black–Scholes Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Sangkwon Kim ◽  
Chaeyoung Lee ◽  
Wonjin Lee ◽  
Soobin Kwak ◽  
Darae Jeong ◽  
...  
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Numerical Experiments ◽  
Operator Splitting ◽  
Three Dimensional ◽  
Nonuniform Grid ◽  
Call Option ◽  
Splitting Scheme ◽  
European Call Option ◽  
Black Scholes

In this study, we present an accurate and efficient nonuniform finite difference method for the three-dimensional (3D) time-fractional Black–Scholes (BS) equation. The operator splitting scheme is used to efficiently solve the 3D time-fractional BS equation. We use a nonuniform grid for pricing 3D options. We compute the three-asset cash-or-nothing European call option and investigate the effects of the fractional-order α in the time-fractional BS model. Numerical experiments demonstrate the efficiency and fastness of the proposed scheme.

Numerical studies of high-dimensional nonlinear viscous and nonviscous wave equations by using finite difference methods

2020 ◽  
Vol 11 (02) ◽  
pp. 2050008 ◽  
Author(s):  
Ding-Wen Deng ◽  
Zhu-An Wang
Keyword(s):  
Finite Difference ◽  
Wave Equations ◽  
Numerical Solutions ◽  
Finite Difference Methods ◽  
Three Dimensional ◽  
Discrete Energy ◽  
Alternating Direction Implicit ◽  
Numerical Studies ◽  
Alternating Direction ◽  
Difference Methods

The numerical solutions of two-dimensional (2D) and three-dimensional (3D) nonlinear viscous and nonviscous wave equations via the unified alternating direction implicit (ADI) finite difference methods (FDMs) are obtained in this paper. By making use of the discrete energy method, it is proven that their numerical solutions converge to exact solutions with an order of two in both time and space with respect to [Formula: see text]-norm. Numerical results confirm that they are relatively accurate and high-resolution, and more successfully simulate the conservation of the energy for nonviscous equations, and the dissipation of the energy for viscous equation.

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