Efficient numerical treatment of nonlinearities in the advection–diffusion–reaction equations

Erdoğan, Utku; Sarı, Murat; Koçak, Hüseyin

doi:10.1108/hff-05-2017-0198

Cited by 9 publications

(8 citation statements)

References 24 publications

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“…With the use of the definition of the Fréchet derivative, 45

ϕ^{'} false(u_{k} false) false(θ_{k} false) = \frac{\partial}{\partial ϵ} ϕ false(u_{k} + ϵ θ_{k} false) {false|}_{ϵ = 0},

…”

Section: The Numerical Methodsmentioning

confidence: 99%

“…Note that the linearization technique helps to avoid considering the convergence issues of the Newton iteration applied to the nonlinear algebraic system containing many unknowns at each time step if an implicit method is used in time discretization. 45 Thus, the technique allows us to use large time step size when using implicit time discretization scheme. Notice also that the Fréchet derivative supports to enhance the convergence order of the proposed iterative scheme.…”

Section: 𝜌𝜆(S)svmentioning

confidence: 99%

“…Therefore, the Fréchet derivative has been applied to the illiquid European call option pricing model mentioned above and then Newton iteration technique has been performed to linearize the model. Note that the linearization technique helps to avoid considering the convergence issues of the Newton iteration applied to the nonlinear algebraic system containing many unknowns at each time step if an implicit method is used in time discretization 45 . Thus, the technique allows us to use large time step size when using implicit time discretization scheme.…”

Section: Introductionmentioning

confidence: 99%

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A Fréchet derivative‐based novel approach to option pricing models in illiquid markets

Gülen

Sarı

2021

Math Methods in App Sciences

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Nonlinear option pricing models have been increasingly concerning in financial industries since they build more accurate values by regarding more realistic assumptions such as transaction cost, market liquidity, or uncertain volatility. This study defines a nonclassical numerical method to effectively capture the behavior of the nonlinear option pricing model in illiquid markets where the implementation of a dynamic hedging strategy affects the price of the underlying asset. Unlike the conventional numerical approaches, this study describes a numerical scheme based on the Newton iteration technique and the Fréchet derivative for linearization of the model. The linearized time‐dependent PDE is then discretized by a sixth‐order finite difference scheme in space and a second‐order trapezoidal rule in time. The computations revealed that the current approach appears to be somewhat more effective to some extent and at the same time economical for illustrative examples compared to the existing competitors. In addition, this method helps to prevent considering the convergence issues of the Newton approach applied to the nonlinear algebraic system.

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“…With the use of the definition of the Fréchet derivative, 45

ϕ^{'} false(u_{k} false) false(θ_{k} false) = \frac{\partial}{\partial ϵ} ϕ false(u_{k} + ϵ θ_{k} false) {false|}_{ϵ = 0},

…”

Section: The Numerical Methodsmentioning

confidence: 99%

Section: 𝜌𝜆(S)svmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A Fréchet derivative‐based novel approach to option pricing models in illiquid markets

Gülen

Sarı

2021

Math Methods in App Sciences

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“…In 2016, Celik (2016) obtained numerical solution for the GBHE using Chebyshev wavelet collocation method. Recently, Erdogan et al (2019) and Sari et al (2019) have discussed the behavior of the advection-diffusion-reaction equation. Finding out analytical solitary wave solution of Burgers-Huxley equation is right now an open problem of study.…”

Section: Ec 378mentioning

confidence: 99%

A high-resolution method based on off-step non-polynomial spline approximations for the solution of Burgers-Fisher and coupled nonlinear Burgers’ equations

Mohanty

Sharma

2020

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Purpose This paper aims to develop a new high accuracy numerical method based on off-step non-polynomial spline in tension approximations for the solution of Burgers-Fisher and coupled nonlinear Burgers’ equations on a graded mesh. The spline method reported here is third order accurate in space and second order accurate in time. The proposed spline method involves only two off-step points and a central point on a graded mesh. The method is two-level implicit in nature and directly derived from the continuity condition of the first order space derivative of the non-polynomial tension spline function. The linear stability analysis of the proposed method has been examined and it is shown that the proposed two-level method is unconditionally stable for a linear model problem. The method is directly applicable to problems in polar systems. To demonstrate the strength and utility of the proposed method, the authors have solved the generalized Burgers-Huxley equation, generalized Burgers-Fisher equation, coupled Burgers-equations and parabolic equation in polar coordinates. The authors show that the proposed method enables us to obtain the high accurate solution for high Reynolds number. Design/methodology/approach In this method, the authors use only two-level in time-direction, and at each time-level, the authors use three grid points for the unknown function u(x,t) and two off-step points for the known variable x in spatial direction. The methodology followed in this paper is the construction of a non-polynomial spline function and using its continuity properties to obtain consistency condition, which is third order accurate on a graded mesh and fourth order accurate on a uniform mesh. From this consistency condition, the authors derive the proposed numerical method. The proposed method, when applied to a linear equation is shown to be unconditionally stable. To assess the validity and accuracy, the method is applied to solve several benchmark problems, and numerical results are provided to demonstrate the usefulness of the proposed method. Findings The paper provides a third order numerical scheme on a graded mesh and fourth order spline method on a uniform mesh obtained directly from the consistency condition. In earlier methods, consistency conditions were only second order accurate. This brings an edge over other past methods. Also, the method is directly applicable to physical problems involving singular coefficients. So no modification in the method is required at singular points. This saves CPU time and computational costs. Research limitations/implications There are no limitations. Obtaining a high accuracy spline method directly from the consistency condition is a new work. Also being an implicit method, this method is unconditionally stable. Practical implications Physical problems with singular and non-singular coefficients are directly solved by this method. Originality/value The paper develops a new method based on non-polynomial spline approximations of order two in time and three (four) in space, which is original and has lot of value because many benchmark problems of physical significance are solved in this method.

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“…In case of time-fractional neutron diffusion models with delayed neutrons [19,29,30], non-local effects are established and sub-diffusive phenomena caught, coming from the heterogeneity of nuclear reactors. Anyway, analytical solutions for this problem are generally not available and only a few efficient numerical techniques have been developed in the literature to approximate the solution of even a 2D fractional [10,31] or non-fractional [32] diffusion models, even in the presence of a reaction term [26,33]. Alternatively, non-local problems accounting for long-range interactions can be treated via non-local integral models [34] and even combined time-fractional and space-nonlocal strategies [35].…”

Section: Introductionmentioning

confidence: 99%

Numerical Solutions of Space-Fractional Advection–Diffusion–Reaction Equations

Salomoni

Marchi

2021

Fractal Fract

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Background: solute transport in highly heterogeneous media and even neutron diffusion in nuclear environments are among the numerous applications of fractional differential equations (FDEs), being demonstrated by field experiments that solute concentration profiles exhibit anomalous non-Fickian growth rates and so-called “heavy tails”. Methods: a nonlinear-coupled 3D fractional hydro-mechanical model accounting for anomalous diffusion (FD) and advection–dispersion (FAD) for solute flux is described, accounting for a Riesz derivative treated through the Grünwald–Letnikow definition. Results: a long-tailed solute contaminant distribution is displayed due to the variation of flow velocity in both time and distance. Conclusions: a finite difference approximation is proposed to solve the problem in 1D domains, and subsequently, two scenarios are considered for numerical computations.

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Efficient numerical treatment of nonlinearities in the advection–diffusion–reaction equations

Cited by 9 publications

References 24 publications

A Fréchet derivative‐based novel approach to option pricing models in illiquid markets

A Fréchet derivative‐based novel approach to option pricing models in illiquid markets

A high-resolution method based on off-step non-polynomial spline approximations for the solution of Burgers-Fisher and coupled nonlinear Burgers’ equations

Numerical Solutions of Space-Fractional Advection–Diffusion–Reaction Equations

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