Order Finite Difference Scheme Research Articles

A New Approach for the Black–Scholes Model with Linear and Nonlinear Volatilities

Since financial engineering problems are of great importance in the academic community, effective methods are still needed to analyze these models. Therefore, this article focuses mainly on capturing the discrete behavior of linear and nonlinear Black–Scholes European option pricing models. To achieve this, this article presents a combined method; a sixth order finite difference (FD6) scheme in space and a third–order strong stability preserving Runge–Kutta (SSPRK3) over time. The computed results are compared with available literature and the exact solution. The computed results revealed that the current method seems to be quite strong both quantitatively and qualitatively with minimal computational effort. Therefore, this method appears to be a very reliable alternative and flexible to implement in solving the problem while preserving the physical properties of such realistic processes.

Temporal stability analysis for multiple similarity solutions of viscous incompressible flows in porous channels with moving walls

• We consider the temporal stability of multiple solutions for flows in a porous channel with moving walls for the first time. • We construct second order finite difference schemes for eigenproblems associated with the linear stability analysis . • We verify the linear stability results for the first time by solving the perturbed nonlinear time dependent equation. In this paper, a viscous, incompressible laminar fluid flow along a uniformly porous channel with expanding or contracting walls is considered. We present multiple symmetric steady-state solutions of this flow problem at several different expanding ratios, and use the linear stability theory to analyse the temporal stability for these solutions under symmetric, antisymmetric and general perturbations. We construct second order finite difference schemes for the eigenvalue problems with boundary conditions associated with those perturbations, and observe that most of these solutions which are stable under symmetric perturbations are unstable under antisymmetric perturbations. Furthermore, we verify the linear stability analysis results by directly solving the original perturbed nonlinear time dependent problem, and find that both stability results are consistent.

Numerical Solution of Heston-Hull-White Three-Dimensional PDE with a High Order FD Scheme

A new numerical method for tackling the three-dimensional Heston–Hull–White partial differential equation (PDE) is proposed. This PDE has an application in pricing options when not only the asset price and the volatility but also the risk-free rate of interest are coming from stochastic nature. To solve this time-dependent three-dimensional PDE as efficiently as possible, high order adaptive finite difference (FD) methods are applied for the application of method of lines. It is derived that the new estimates have fourth order of convergence on non-uniform grids. In addition, it is proved that the overall procedure is conditionally time-stable. The results are upheld via several numerical tests.

A new high order ADI numerical difference formula for time-fractional convection-diffusion equation

Based on exponential transformation, quadratic interpolation polynomial and Padé approximation, a new high order finite difference scheme is proposed for solving the two-dimensional (2D) time-fractional convection-dominated diffusion equation (of order α ∈ (0, 1)). The resulting scheme is of (3−α)-order accuracy in time and fourth-order accuracy in space. In order to reduce the amount of computation, the alternating direction implicit (ADI) scheme is further developed. Numerical experiments are given to demonstrate the high accuracy and robustness of our new scheme.

Tensor-Train Format Solution with Preconditioned Iterative Method for High Dimensional Time-Dependent Space-Fractional Diffusion Equations with Error Analysis

In this paper, a first order implicit finite difference scheme with Krylov subspace linear system solver is employed to solving time-dependent space-fractional diffusion equations in high dimensions where the initial condition and source term are in tensor-train (TT) format with low TT-ranks. In the time-marching process, TT-format of the solution is maintained and the increment of TT-ranks due to addition is moderated by rounding. The error introduced by rounding is shown to be consistent with the first order finite difference scheme. On the other hand, the linear systems involved in the solution process are shown to possess Toeplitz-like structure so that the complexity and required memory for Krylov subspace solver can be optimized. Further reduction in complexity is made by utilizing a circulant preconditioner which accelerates the convergence rate of Krylov subspace method dramatically. Numerical examples for problems up to 20 dimensions are presented.

Finite difference and spectral collocation methods for the solution of semilinear time fractional convection-reaction-diffusion equations with time delay

In this paper, an efficient numerical method is constructed to solve the nonlinear fractional convection-reaction-diffusion equations with time delay. Firstly, we discretize the time fractional derivative with a second order finite difference scheme. Then, Chebyshev spectral collocation method is utilized to space component and to obtain full discretization of problem. We show that the proposed method is unconditionally stable and convergent. Numerical experiments are carried out to demonstrate the accuracy of the proposed method and to compare the results with analytical solutions and the numerical solutions of other schemes in the literature. The results show that the present method is accurate and efficient. It is illustrated that the numerical results are in good agreement with theoretical ones.

Uniform asymptotic stability of a Logistic model with feedback control of fractional order and nonstandard finite difference schemes

Abstract In this paper, we propose and study a Logistic model with feedback control of fractional order that is extended directly from a system of ordinary differential equations. Uniform asymptotic stability of this model is established based on an appropriate Lyapunov function and an important consequence of this result; we present a simple proof for the global stability of the original system of ordinary differential equations. Besides, to numerically solve and simulate the proposed fractional model, unconditionally positive nonstandard finite difference schemes are constructed and analyzed. Finally, the numerical simulations obtained by the constructed nonstandard finite difference schemes are compared with the Grunwald–Letnikov scheme to reveal that the proposed scheme is convenient for solving the proposed model and confirm the validity of the established results.

A Computational Method Based on the Moving Least-Squares Approach for Pricing Double Barrier Options in a Time-Fractional Black–Scholes Model

The mathematical modeling in trade and finance issues is the key purpose in the computation of the value and considering option during preferences in contract. This paper investigates the pricing of double barrier options when the price change of the underlying is considered as a fractal transmission system. Due to the outstanding memory effect present in the fractional derivatives, approximating financial options with regards to their hereditary characteristics can be well interpreted and stated. Motivated by the reason mentioned, relatively reliable and also efficient numerical approaches have to be found while facing with fractional differential equations. The main objective of the current paper is to obtain the approximation solution of the time fractional Black–Scholes model of order $$0<\alpha \le 1$$ governing European options based on the moving least-squares (MLS) method. In proposed method, firstly, the mentioned equation is discretized in the time sense based on finite difference scheme of order $${\mathcal {O}}(\delta t^{2-\alpha })$$ and then approximated by using MLS approach in the space variable. Furthermore, the stability and convergence of the proposed method are discussed in detail throughout the paper. Numerical evidences and comparisons demonstrate that the proposed method is very accurate and efficient.

Numerical Technique for Resolving the Dual Phase Lag Heat Conduction in Thin Film Metal

In this article, a three-time level finite difference scheme is used to resolve the dual phase lag’s (DPL) heat conduction in a micro scale gold film subjected to spontaneous temperature boundary conditions without knowing the heat flux. Finite difference analog of DPL equation on applying to the intermediate grid points of the computational domain results into a system of linear, algebraic equations which can be solved using Thomas’ algorithm to finally obtain the transient temperature solution distributions in the film. The solution predicted by the DPL model is compared with that obtained by the single-phase Cattaneo–Vernotte’s model. Further, the way in which non-Fourier’s temperature distributions affected by the diffusion due to the increase in Heat Conduction Model numbers agree with the predecessor’s published results. The results by both the models revealed a finite thermal wave speed in the film contrasting the infinite speed of heat propagation as stated by the classical Fourier’s thermal model. Low spatial step and higher order finite difference schemes are recommended for better accurate numerical results of the non-Fourier’s temperature distributions occurring in the very short transient period between the instants of the suddenly applied spatial temperature gradient and the reaching of the steady state conditions.

Fourth order computational method for two parameters singularly perturbed boundary value problem using non-polynomial cubic spline

In this paper, we proposed a fourth order finite difference scheme using non-polynomial cubic spline for the solution of two parameters singularly perturbed two-point boundary value problem having dual boundary layer on a uniform mesh. In this method, the first order derivatives in the non-polynomial cubic spline finite difference scheme are replaced by the higher order finite differences to get the discretisation equation for the problem. The discretisation equation is solved by the tridiagonal solver discrete invariant imbedding. The proposed method is analysed for convergence and a fourth order rate of convergence is proved. The numerical results are compared with exact solutions and the outcomes of other existing numerical methods.

On the second order finite-difference scheme for the solution of the system of one-dimensional equationsof hemodynamics

The paper is devoted to the analysis and optimization of the second order Lax - Wendroff finite-difference scheme with artificial viscosity for the system of 1D equations of hemodynamics. The scheme is constructed for the conservative form of this system. For the elimination of numerical dispersion the artificial viscosity is used. The value of the viscosity is obtained by the approach, based on simultaneous optimization of the dispersive and dissipative characteristics.

Influence of slip velocity on the flow of viscous fluid through a porous medium in a permeable tube with a variable bulk flow rate

Abstract A mathematical model is presented for an axisymmetric flow of a Newtonian fluid through a permeable tube filled with porous medium and slip at the wall. The bulk flow rate is prescribed as a decreasing function of axial distance. The governing coupled partial differential equations are solved analytically using Adomian decomposition method and numerically using a second order finite difference scheme. Numerical method is validated by already published work and a good agreement is observed between the two solutions. Trusting this validity, effects of pertinent parameters on the flow variables such as velocity components, wall shear stress and pressure drop are discussed graphically. This study reveals that the slip parameter (ϕ), reabsorption parameter (α) and permeability parameter (k) have significant influences on flow variables involved in the problem. Creeping flow (Re = 0), flow without porous medium ( k → ∞ ) and flow with no slip ( ϕ = 0 ) at the wall are the limiting cases of this study.

Simulation of plasma turbulence in the periphery of diverted tokamak by using the GBS code

We report on the implementation of diverted magnetic equilibria in GBS and on first simulations in this geometry. GBS is a simulation code used to evolve plasma turbulence in the tokamak periphery by solving the drift-reduced Braginskii's equations. The model equations are written in toroidal coordinates, abandoning flux coordinate systems that are not defined at the X-point. A fourth order finite difference scheme is used for the implementation of the spatial operators on poloidally and toroidally staggered grids. The GBS numerical implementation is verified through the method of manufactured solutions. The code convergence properties are tested on a relatively simple analytical X-point configuration. Finally, the diverted equilibrium from a TCV tokamak discharge is implemented in the new version of GBS. The analysis of the simulation results is focused on blob formation, radial transport, and plasma poloidal rotation mechanisms.

High order locally one-dimensional methods for solving two-dimensional parabolic equations

Based on the locally one-dimensional strategy, we propose two high order finite difference schemes for solving two-dimensional linear parabolic equations. In the first method, fourth order approximation in space and (2,2) Padé formula in time are considered. These lead to a fourth order finite difference scheme in both space and time. For the second method, we employ sixth order approximation in space and (3,3) Padé formula in time. This yields a novel sixth order scheme in both space and time. The methods are proved to be unconditionally stable, and the Sheng–Suzuki barrier is successfully avoided. Numerical experiments are given to illustrate our conclusions as well as computational effectiveness.

A study on a second order finite difference scheme for fractional advection–diffusion equations

Second order finite difference schemes for fractional advection–diffusion equations are considered in this paper. We note that, when studying these schemes, advection terms with coefficients having the same sign as those of diffusion terms need additional estimates. In this paper, by comparing generating functions of the corresponding discretization matrices, we find that sufficiently strong diffusion can dominate the effects of advection. As a result, convergence and stability of schemes are obtained in this situation.

Entropy generation of dissipative flow of carbon nanotubes in rotating frame with Darcy-Forchheimer porous medium: A numerical study

This research article explains the influence of magnetic field and volume fraction of carbon nanotubes (CNTs) on the flow over a stretching sheet in rotating frame. We analysed the behaviour of base fluid such as water with both kind of CNTs i.e. multi-wall carbon nanotubes (MWCNTs) and single-wall carbon nanotubes (SWCNTs). The nanofluid transport mechanism is obtained following Xue model. The mathematical model is developed by order of magnitude analysis. Governing partial differential equations are first converted into a system of ordinary differential equations making the use of suitable similarity transformation and then solved by fourth order finite difference scheme using Newton's linearization technique, which helps us to deal with the non-linear terms in a very smooth way. Numerical simulation for the fluid velocity and fluid temperature along with Nusselt number and skin friction coefficient are carried out for a set of values of different regulatory flow parameters that characterize the physics of the flow graphically. An outstanding correlation is perceived in tabular form of mesh density solutions of different length of domain obtained by fourth order finite difference scheme. The Biot number, Brinkmann number and applied magnetic field have vital effects on the thermal energy irreversibility in terms of Bejan number.

The Finite-Difference Scheme of Higher Order of Accuracy for the Two-Dimensional Poisson Equation in a Rectangle with Regard for the Effect of the Dirichlet Boundary Condition

We investigate the finite-difference scheme of higher order of accuracy on a nine-point template for Poisson’s equation in a rectangle with the Dirichlet boundary condition. We substantiate the error estimate taking into account the influence of the boundary condition. We prove that the accuracy order is higher near the sides of the rectangle than at the inner nodes of the grid set and increase in the approximation order has no impact on the boundary effect.

An improved consistent, conservative, non-oscillatory and high order finite difference scheme for variable density low Mach number turbulent flow simulation

A consistent, conservative and truly high order finite difference scheme for low Mach number turbulent variable density flow simulation is presented. In the present scheme, the scalar transport equation is discretized with the explicit weighted compact nonlinear scheme (WCNS) while the continuity and momentum equations are discretized with a conservative centered high order finite difference scheme on a staggered grid. Bounded hybrid high order upwind interpolation is proposed for the evaluation of density on the cell faces of the staggered grid. The use of explicit WCNS along with the bounded density interpolation resolves the issue of degraded order of accuracy in the existing high order finite difference schemes where the centered finite difference scheme for mass and momentum conservation is combined with a total variation stable scheme for scalar transport. With the same difference operator used for the continuity and scalar transport equations, the present scheme is consistent naturally, and resolves the inconsistency problem reported in Trisjono et al. (2016) [6] without employing any modification to a scheme. Such improvements are demonstrated for several test cases.

Total variation bounded flux limiters for high order finite difference schemes solving one-dimensional scalar conservation laws

Total variation bounded flux limiters for high order finite difference schemes solving one-dimensional scalar conservation laws

An energy stable fourth order finite difference scheme for the Cahn–Hilliard equation

In this paper we propose and analyze an energy stable numerical scheme for the Cahn–Hilliard equation, with second order accuracy in time and the fourth order finite difference approximation in space. In particular, the truncation error for the long stencil fourth order finite difference approximation, over a uniform numerical grid with a periodic boundary condition, is analyzed, via the help of discrete Fourier analysis instead of the standard Taylor expansion. This in turn results in a reduced regularity requirement for the test function. In the temporal approximation, we apply a second order BDF stencil, combined with a second order extrapolation formula applied to the concave diffusion term, as well as a second order artificial Douglas–Dupont regularization term, for the sake of energy stability. As a result, the unique solvability, energy stability are established for the proposed numerical scheme, and an optimal rate convergence analysis is derived in the ℓ∞(0,T;ℓ2)∩ℓ2(0,T;Hh2) norm. A few numerical experiments are presented, which confirm the robustness and accuracy of the proposed scheme.