Laplace-Transform Finite-Difference and Quasistationary Solution Method for Water-Injection/Falloff Tests

SPE Journal ◽  
2013 ◽  
Vol 19 (03) ◽  
pp. 398-409 ◽  
Author(s):  
Azeb D. Habte ◽  
Mustafa Onur
Keyword(s):  
Finite Difference ◽  
Laplace Transform ◽  
Water Injection ◽  
Time Discretization ◽  
Finite Difference Approximation ◽  
Stability And Convergence ◽  
Difference Approximation ◽  
Phase Flow ◽  
Slightly Compressible ◽  
Oil Water

Summary In this work, we present a method for efficiently and accurately simulating the pressure-transient behavior of oil/water flow associated with water-injection/falloff tests. The method uses the Laplace-transform finite-difference (LTFD) method coupled with the well-known Buckley-Leverett frontal-advance formula to solve the radial diffusivity equation describing slightly compressible oil/water two-phase flow. The method is semianalytical in time, and as a result, the issue of time discretization in the finite-difference approximation method is eliminated. Thus, stability and convergence problems caused by time discretization are avoided. Two approaches are presented and compared in terms of accuracy for simulating the tests with multiple-rate injection and falloff periods: One is based on solving the initial-boundary-value (IVB) problem with the initial condition attained from the end of the previous flow period, and the other is based on the conventional superposition on the basis of the single-phase flow of a slightly compressible fluid. The former is shown to always provide a more accurate and efficient solution. The method is quite general in that it allows one to incorporate the effect of wellbore storage and thick-skin and finite outer-boundary conditions. The accuracy of the method was evaluated by considering various synthetic test cases with favorable and unfavorable mobility ratios and by comparing the pressure and pressure-derivative signatures with a commercial black-oil simulator, and an excellent agreement was seen.

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.

Solving Fully Three-Dimensional Microscale Dual Phase Lag Problem Using Mixed-Collocation, Finite Difference Discretization

2012 ◽  
Vol 134 (9) ◽  
Author(s):  
Alaeddin Malek ◽  
Zahra Kalateh Bojdi ◽  
Parisa Nuri Niled Golbarg
Keyword(s):  
Finite Difference ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Finite Difference Approximation ◽  
Stability And Convergence ◽  
Difference Approximation ◽  
Phase Lag ◽  
Dual Phase ◽  
Linear System Of Equations ◽  
Finite Difference Discretization

In the present work, we investigate laser heating of nanoscale thin-films irradiated in three dimensions using the dual phase lag (DPL) model. A numerical solution based on mixed-collocation, finite difference method has been employed to solve the DPL heat conduction equation. Direct substitution in the model transforms the differential equation into a linear system of equations in which related system is solved directly without preconditioning. Consistency, stability, and convergence of the proposed method based on a mixed-collocation, finite difference approximation are proved, and numerical results are presented. The general form of matrices and their corresponding eigenvalues are presented.

Spectrally accurate option pricing under the time fractional Black-Scholes model

2021 ◽  
Vol 63 ◽  
pp. 228-248
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

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.   doi:10.1017/S1446181121000286  

scholarly journals Stable and Efficient Modeling of Anelastic Attenuation in Seismic Wave Propagation

2012 ◽  
Vol 12 (1) ◽  
pp. 193-225 ◽  
Author(s):  
N. Anders Petersson ◽  
Björn Sjögreen
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Half Space ◽  
Material Model ◽  
Second Order ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Space Problem ◽  
Standard Linear Solid ◽  
Standard Linear

AbstractWe develop a stable finite difference approximation of the three-dimensional viscoelastic wave equation. The material model is a super-imposition of N standard linear solid mechanisms, which commonly is used in seismology to model a material with constant quality factor Q. The proposed scheme discretizes the governing equations in second order displacement formulation using 3N memory variables, making it significantly more memory efficient than the commonly used first order velocity-stress formulation. The new scheme is a generalization of our energy conserving finite difference scheme for the elastic wave equation in second order formulation [SIAM J. Numer. Anal., 45 (2007), pp. 1902-1936]. Our main result is a proof that the proposed discretization is energy stable, even in the case of variable material properties. The proof relies on the summation-by-parts property of the discretization. The new scheme is implemented with grid refinement with hanging nodes on the interface. Numerical experiments verify the accuracy and stability of the new scheme. Semi-analytical solutions for a half-space problem and the LOH.3 layer over half-space problem are used to demonstrate how the number of viscoelastic mechanisms and the grid resolution influence the accuracy. We find that three standard linear solid mechanisms usually are sufficient to make the modeling error smaller than the discretization error.

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