scholarly journals Exponential Time Integration and Second-Order Difference Scheme for a Generalized Black-Scholes Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Zhongdi Cen ◽  
Anbo Le ◽  
Aimin Xu
Keyword(s):  
Difference Scheme ◽  
Time Integration ◽  
Second Order ◽  
Integration Scheme ◽  
Uniform Mesh ◽  
Central Difference ◽  
Exponential Time ◽  
Black Scholes ◽  
Time Integration Scheme ◽  
Exponential Time Integration

We apply an exponential time integration scheme combined with a central difference scheme on a piecewise uniform mesh with respect to the spatial variable to evaluate a generalized Black-Scholes equation. We show that the scheme is second-order convergent for both time and spatial variables. It is proved that the scheme is unconditionally stable. Numerical results support the theoretical results.

Design for a Hybrid Absorbing Element in the Time Domain Using PML and Infinite Element Concepts

2014 ◽  
Author(s):  
Jeffrey L. Cipolla
Keyword(s):  
Time Domain ◽  
Time Integration ◽  
Low Frequency ◽  
Second Order ◽  
Integration Scheme ◽  
Lumped Mass ◽  
Central Difference ◽  
Infinite Element ◽  
Time Integration Scheme ◽  
The Time Domain

We introduce an approach blending the Perfectly Matched Layer (PML) and infinite element paradigms, to achieve better performance and wider applicability than either approach alone. In this paper, we address the specific challenges of unbounded problems when using time-domain explicit finite elements: 1. The algorithm must be spatially local, to minimize storage and communication cost, 2. It must contain second-order time derivatives for compatibility with the explicit central-difference time integration scheme, 3. Its coefficient for the second-order derivatives must be diagonal (“lumped mass”), 4. It must be time-stable when used with central-differences, 5. It must converge to the correct low-frequency (Laplacian) limit, 6. It should exhibit high accuracy across typically encountered dynamic frequencies, i.e. at short to long wavelengths, 7. Its user interface should be as simple as possible. Here, we will describe the derivation of a time-domain implementation of the hybrid PML/infinite element, and discuss its advantages for implementation.

Studies on the Accuracy Stability and Efficiency of a New Time Integration Scheme for Ballast Modeling Using the Discrete Element Method

2014 ◽  
Author(s):  
G. F. Mathews ◽  
R. L. Mullen ◽  
D. C. Rizos
Keyword(s):  
Particle Interaction ◽  
Time Integration ◽  
Discrete Element ◽  
Critical Discussion ◽  
Implicit Time ◽  
Integration Scheme ◽  
Computationally Efficient ◽  
Time Step ◽  
Central Difference ◽  
Time Integration Scheme

This paper presents the development of a semi-implicit time integration scheme, originally developed for structural dynamics in the 1970’s, and its implementation for use in Discrete Element Methods (DEM) for rigid particle interaction, and interaction of elastic bodies that are modeled as a cluster of rigid interconnected particles. The method is developed in view of ballast modeling that accounts for the flexibility of aggregates and the arbitrary shape and size of granules. The proposed scheme does not require any matrix inversions and is expressed in an incremental form making it appropriate for non-linear problems. The proposed method focuses on improving the efficiency, stability and accuracy of the solutions, as compared to current practice. A critical discussion of the findings of the studies is presented. Extended verification and assessment studies demonstrate that the proposed algorithm is unconditionally stable and accurate even for large time step sizes. It is demonstrated that the proposed method is at least as computationally efficient as the Central Difference Method. Guidelines for the implementation of the method to ballast modeling are discussed.

Fast Exponential Time Integration for Pricing Options in Stochastic Volatility Jump Diffusion Models

2014 ◽  
Vol 4 (1) ◽  
pp. 52-68 ◽  
Author(s):  
Hong-Kui Pang ◽  
Hai-Wei Sun
Keyword(s):  
Stochastic Volatility ◽  
Temporal Integration ◽  
Time Integration ◽  
Computational Cost ◽  
Jump Diffusion ◽  
Integration Scheme ◽  
Arnoldi Method ◽  
Exponential Time ◽  
Jump Diffusion Model ◽  
Exponential Time Integration

AbstractThe stochastic volatility jump diffusion model with jumps in both return and volatility leads to a two-dimensional partial integro-differential equation (PIDE). We exploit a fast exponential time integration scheme to solve this PIDE. After spatial discretization and temporal integration, the solution of the PIDE can be formulated as the action of an exponential of a block Toeplitz matrix on a vector. The shift-invert Arnoldi method is employed to approximate this product. To reduce the computational cost, matrix splitting is combined with the multigrid method to deal with the shift-invert matrix-vector product in each inner iteration. Numerical results show that our proposed scheme is more robust and efficient than the existing high accurate implicit-explicit Euler-based extrapolation scheme.

UNSTEADY FLOW COMPUTATIONS OVER MOVING BODY USING DYNAMIC MESHES

2004 ◽  
Vol 01 (03) ◽  
pp. 507-518
Author(s):  
J. C. MANDAL ◽  
J. BALLMANN
Keyword(s):  
Compressible Flows ◽  
Time Integration ◽  
Time Discretization ◽  
Second Order ◽  
Integration Scheme ◽  
Arbitrary Lagrangian Eulerian ◽  
Time Step ◽  
Naca0012 Airfoil ◽  
Moving Body ◽  
Time Integration Scheme

An efficient implicit unstructured grid algorithm for solving unsteady inviscid compressible flows over moving body employing an Arbitrary Lagrangian Eulerian formulation is presented. In the present formulation, the time discretization is performed using a second-order accurate 3-point time integration scheme and the upwind-biased space discretization using second-order accurate finite volume formulation with Venkatakrishnan limiter. The face-velocities of the control volumes are computed using Geometric Conservation Laws. The nonlinear system arising from the implicit formulation is solved using an ILU preconditioned Newton–Krylov iteration at every time step. The computed results for two test cases involving harmonically oscillating NACA0012 airfoil are presented in order to demonstrate the efficacy of the present solver.

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