Accurate Finite Difference Method Research Articles

RBF–based IMEX finite difference schemes for pricing option under liquidity switching

In this work, we introduce two accurate and efficient finite difference methods based on radial basis functions (RBF–FD) for pricing European and American options under liquidity shocks. The problem is formulated as the semi-linear complementarity and PDE systems for American and European options, respectively. In the context of temporal semi-discretization, we provide two backward difference formulas of order one and two (BDF1 & BDF2). Furthermore, we discuss the stability and convergence properties of the proposed methods. For the American option, to solve the semi-linear system of complementarity problems (LCPs), we combine the RBF-FD approaches with an operator splitting (OS) method. To illustrate the efficiency and accuracy of the suggested methods, we provide numerical examples for both European and American call options and verify them with the existing work in the literature. In numerical discussion, we show the Greeks (Delta & Gamma) plots for the American options.

A Time-Domain Wavenumber Integration Model for Underwater Acoustics Based on the High-Order Finite Difference Method

Simulating the acoustic field excited by pulse sound sources holds significant practical value in computational ocean acoustics. Two primary methods exist for modeling underwater acoustic propagation in the time domain: the Fourier synthesis technique based on frequency decomposition and the time-domain underwater acoustic propagation model (TD-UAPM). TD-UAPMs solve the wave equation in the time domain without requiring frequency decomposition, providing a more intuitive explanation of the physical process of sound energy propagation over time. However, time-stepping numerical methods can accumulate numerical errors, making it crucial to improve the algorithm’s accuracy for TD-UAPMs. Herein, the time-domain wavenumber integration model SPARC was improved by replacing the second-order finite element method (FEM) with the high-order accuracy finite difference method (FDM). Furthermore, the matched interface and boundary (MIB) method was used to process the seabed more accurately. The improved model was validated using three classic underwater acoustic benchmarks. By comparing the acoustic solutions obtained using the FDM and the FEM, it is evident that the improved model requires fewer grid points while maintaining the same level of accuracy, leading to lower computational costs and faster processing compared to the original model.

A Finite Difference Technique for Numerical Solution of the Boundary Value Problem in ODEs of Order Three

In the article, we study the approximate numerical solution to the boundary value problem in ordinary differential equations. In the present article, a third-order two-point boundary value problem is considered for discussion. We developed a second order accurate finite difference method for the approximate numerical solution of the considered problem. We took a special boundary condition; we did not find this boundary condition in the literature. We have discussed the standard convergence analysis of the proposed method. Numerical experiments on linear, nonlinear, and obstacle problems approve the order of accuracy and efficiency of the method.

Vortex dynamics of accelerated flow past a mounted wedge

This study is concerned with the simulation of a complex fluid flow problem involving flow past a wedge mounted on a wall for channel Reynolds numbers Rec = 1560, 6621, and 6873 in uniform and accelerated flow medium. The transient Navier–Stokes (N–S) equations governing the flow have been discretized using a recently developed second order spatially and temporally accurate compact finite difference method on a nonuniform Cartesian grid by the authors. Almost all the flow characteristics of a well-known laboratory experiment of Pullin and Perry [“Some flow visualization experiments on the starting vortex,” J. Fluid Mech. 97(2), 239–255 (1980)] have been captured by our numerical simulation, and we provide a qualitative and quantitative assessment of the same. Furthermore, the influence of the parameter m, controlling the intensity of acceleration, has been discussed in detail along with the intriguing consequence of non-dimensionalization of the N–S equations pertaining to such flows. The simulation of the flow across a time span significantly greater than the aforesaid lab experiment is the current study's major achievement. Meanwhile, a grid independence study performed in the process confirmed that our simulation is devoid of any under-resolution or numerical artifact. For the accelerated flow, the onset of shear layer instability leading to a more complicated flow toward transition to turbulence has also been aptly resolved. The quality of our simulation is validated by the close similarity of our simulation to the high Reynolds number experimental results of Lian and Huang [“Starting flow and structures of the starting vortex behind bluff bodies with sharp edges,” Exp. Fluids 8(1–2), 95–103 (1989)] for the accelerated flow across a typical flat plate. All three steps of vortex shedding, including the exceedingly intricate threefold structure, have been captured quite efficiently.

A ghost-point based second order accurate finite difference method on uniform orthogonal grids for electromagnetic scattering around curved perfect electric conductors with corners

We propose a finite difference method to solve Maxwell's equations in time domain in the presence of a perfect electric conductor (PEC) that impedes the propagation of electromagnetic waves. Our method relies on the level set characterization of the interface and the PDE-based extension technique which allows us to construct fictitious ghost values inside the PEC without scrutinizing the local geometries of its boundaries. As opposed to a locally perturbed body fitting grid used in [1], we promote a much simpler uniform orthogonal grid which necessitates a new approach for the extension technique to work. A novelty of our work is the introduction of what we call guest values which are accurately estimated boundary values deliberately and carefully misplaced near the interface. We stipulate a mild requirement on the accuracy of our ghost values that they need only be locally second-order accurate. Nevertheless, the resulting accuracy of our method is second order thanks to the application of back and forth error correction and compensation, which also relaxes CFL conditions. We demonstrate the effectiveness of our approach with some numerical examples.

Local-basis difference potentials method for elliptic PDEs in complex geometry

We develop efficient and high-order accurate finite difference methods for elliptic partial differential equations in complex geometry in the difference potentials framework. The main novelty of the developed schemes is the use of local basis functions defined at near-boundary grid points. The use of local basis functions allows unified numerical treatment of (i) explicitly and implicitly defined geometry; (ii) geometry of more complicated shapes, such as those with corners, multi-connected domain, etc; and (iii) different types of boundary conditions. This geometrically flexible approach is complementary to the classical difference potentials method using global basis functions, especially in the case where a large number of global basis functions are needed to resolve the boundary, or where the optimal global basis functions are difficult to obtain. Fast Poisson solvers based on FFT are employed for standard, centered finite difference stencils regardless of the designed order of accuracy. Proofs of convergence of difference potentials in maximum norm are outlined both theoretically and numerically.

A fast and accurate elastic reverse-time migration method based on decoupled elastic wave equations

The extrapolation and decomposition of different mode waves are very important to efficiently obtain multi-wave depth images for elastic reverse-time migration (ERTM). To achieve this goal successfully, we firstly develop an efficient and accurate staggered-grid finite-difference method (SFDM) based on decoupled elastic wave equations. This developed SFDM uses two sets of difference operator lengths for decoupled elastic wave equations to respectively perform the extrapolation of decoupled P- and S-wavefields and employs the least squares (LS) to minimize the objective functions respectively constructed from the time-space dispersion equations of P- and S-waves to generate their corresponding difference coefficients. This developed method can theoretically be regarded as the optimal SFDM based on least squares (LSSFDM), and it can provide the high-precision separated wavefields of different mode waves simultaneously and reduce the computing time effectively for elastic wave modeling under the given accuracy requirements. The modeling results of different models illustrate that our developed LSSFDM based on decoupled elastic wave equations can generate the decomposed P- and S-wavefields and suppress their numerical dispersions effectively. Secondly, we present a fast and accurate ERTM method based on this developed LSSFDM to successfully achieve the multi-wave imaging results, including PP, PS, SP and SS images. The ERTM examples of different models demonstrate that our ERTM based on this developed LSSFDM can accurately obtain the high-precision depth images of multimode waves and effectively reduce the computing time of wavefield extrapolation.

Numerical study on the scattering of acoustic waves by a compact vortex

A new family of compact vortex models is developed and taken as base vortical flows to numerically study the acoustic scattering by solving the two-dimensional Euler equations in the time domain with high-order accurate finite-difference methods and nonreflecting boundary conditions. The computations of scattered fields with very small amplitude are found to be in excellent agreement with a benchmark provided by previous studies. Simulations for the scattering from a Taylor vortex reveal that the amplitude of the scattered fields is strongly influenced by two dimensionless quantities, the vortex strength Mv based on the maximal velocity of the vortex, and the acoustic length-scale ratio λ/R defined as the acoustic wavelength relative to the vortex core size. To have a deep understanding of the roles played by these two quantities, another significant quantity used for describing quantitatively the total amount of scattering, namely, scattered sound power, is introduced. Thereupon, on the basis of a global analysis of scale effects of these two dimensionless quantities on the scattered sound power, the scattering defined in a physical coordinate system with Mv and λ/R is divided into three domains, long-wave domain, resonance domain, and geometrical-acoustics domain. For each domain, we examine the influence of Mv and λ/R in detail and derive the explicit scaling laws involved in the strength of the scattered field and these two dimensionless quantities separately. Furthermore, the computations for the scattering from a high-order compact vortex are conducted at a wide range of Mv and λ/R and compared with the results from the Taylor vortex in each domain to gain some insights into the acoustic scattering by a compact vortex.

Effect of anisotropic resistance characteristic on boundary-layer transitional flow

It is observed that the feather surface exhibits anisotropic resistances for the streamwise and spanwise flows. To obtain a qualitative understanding about the effect of this anisotropic resistance feature of surface on the boundary-layer transitional flow over a flat plate, a simple phenomenological model for the anisotropic resistance is established in this paper. By means of the large eddy simulation (LES) with high-order accurate finite difference method, the numerical investigations are conducted. The numerical results show that with the spanwise resistance hindering the formation of vortexes, the transition from laminar flow to turbulent flow can be delayed, and turbulence is weakened when the flow becomes fully turbulent, which leads to significant drag reduction for the plate. On the contrary, the streamwise resistance renders the flow less stable, which leads to the earlier transition and enhances turbulence in the turbulent region, causing a drag increase for the plate. Thus, it is indicated that a surface with large resistance for spanwise flow and small resistance for streamwise flow can achieve significant drag reduction. The present results highlight the anisotropic resistance characteristic near the feather surface for drag reduction, and shed a light on the study of bird’s efficient flight.

Compact Finite Difference Scheme with Hermite Interpolation for Pricing American Put Options Based on Regime Switching Model

American put options with the regime-switching model is a system of coupled free boundary problems. In this study, we present an accurate finite difference method coupled with the Hermite interpolation for solving this system. To this end, we first employ the logarithmic transformation to map the free boundary for each regime to a fixed interval and then eliminate the first-order derivatives in the transformed model by taking derivatives to obtain a system of partial differential equations which we call the asset-delta-gamma-speed equations. We then discretize the system using the fourth-order compact scheme coupled with the Crank–Nicholson method. At the same time, the influence of other asset options and option sensitivities are estimated based on the third-order Hermite interpolation. As such, the overall scheme consists of four tridiagonal linear systems, which can be easily solved using the Thomas algorithm and the Gauss–Seidel iteration. The obtained scheme is then applied for the model with two, four, and sixteen regimes, respectively. Our results show that the scheme provides an accurate solution that is fast in computation as compared with other existing numerical methods.

A numerical method for the solution of the three-dimensional acoustic wave equation in a marine environment considering complex sources

The need to understand how complex acoustic sources propagate noise in a realistic environment is of growing interest. In this work, we propose a numerical model for the simulation of noise generated by sources with directivity and propagating in ocean waveguides. The numerical model solves the wave equation for the acoustic pressure in the physical space using a second-order accurate finite difference method (FDTD). The source is implemented using an improved form of the hard-source method, which implicitly takes into consideration the reflection of acoustic waves associated to the presence of the ocean free surface, through the use of the method of images. This novel method is shown to improve the results with respect to the standard hard source implementation. We first validate the numerical method considering both analytical solutions and benchmark cases for the case of a monopole and then explore the acoustic energy patterns developed in the case of dipole and quadrupole sources. Specifically, the algorithm and the implementation of complex sources are evaluated first in a semi-infinite fluid layer and then considering two classical waveguides: the Ideal one and the Pekeris one. The comparison with analytical results shows the numerical method’s accuracy and that incorporating free surface effects in the hard source implementation improves results. In addition, the study shows that the acoustic response in the near field, of the order of few kilometers from the source, is strongly influenced by the source’s directivity and orientation relative to the free surface. The results of this paper have implications for future research aimed at characterizing and quantifying ship propeller noise in realistic waveguides. Indeed, this preliminary work is necessary to proceed to more complex numerical experiments, such as considering a real propeller signal as a source or considering stratification of the medium or propagation in confined domains such as experimental tanks.

Study on the combination influences of homotype rough element on hypersonic blunt cone flow

Abstract To study the influence of the homotype rough element combination and spacing on the flow around the hypersonic blunt cone, the high-order accuracy finite difference method is used to directly simulate the flow field around the hypersonic compressible blunt cone, and the influence of the homotype of rough element on the boundary layer parameters such as heat flux and friction coefficient is discussed. It is concluded that two K-type rough elements lead to the formation of three compression waves and two expansion waves in the flow field. On the contrary, two D-type rough elements form two compression waves and three expansion waves in the flow field, and a long flow separation will be formed between two K-type and D-type rough elements. When the spacing of rough elements is 1, the drag reduction and heat reduction effect of K+K rough element is better than that of D+D rough element. When the spacing between the two rough elements is greater than a certain length, the two rough elements will not affect each other.

Accurate and Efficient Finite Difference Method for the Black–Scholes Model with No Far-Field Boundary Conditions

Accurate and Efficient Finite Difference Method for the Black–Scholes Model with No Far-Field Boundary Conditions

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

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-orderαin the time-fractional BS model. Numerical experiments demonstrate the efficiency and fastness of the proposed scheme.

An accurate and efficient multiscale finite-difference frequency-domain method for the scalar Helmholtz equation

Frequency-domain numerical modeling of the seismic wave equation can readily describe frequency-dependent seismic wave behaviors, yet it is computationally challenging to perform in finely discretized or large-scale geologic models. Conventional finite-difference frequency-domain (FDFD) methods for solving the Helmholtz equation usually lead to large linear systems that are difficult to solve with a direct or iterative solver. Parallel strategies and hybrid solvers can partially alleviate the computational burden by improving the performance of the linear system solver. We have developed a novel multiscale FDFD method to eventually construct a dimension-reduced linear system from the scalar Helmholtz equation based on the general framework of heterogeneous multiscale method. The methodology associated with multiscale basis functions in the multiscale finite-element method is applied to the local microscale problems of this multiscale FDFD method. Solved from frequency- and medium-dependent local Helmholtz problems, these multiscale basis functions capture fine-scale medium heterogeneities and are finally incorporated into the dimension-reduced linear system by a coupling of scalar Helmholtz problem solutions at two scales. We use several highly heterogeneous models to verify the performance in terms of the accuracy, efficiency, and memory cost of our multiscale method. The results indicate that our new method can solve the scalar Helmholtz equation in complicated models with high accuracy and quite low time and memory costs compared to the conventional FDFD methods.

Implications of the third-grade nanomaterials lubrication problem in terms of radiative heat flux: A Keller box analysis

Steady flow of non-Newtonian nanofluid impinging on a vertical lubricated surface is numerically investigated. The rheological behaviour of the base fluid is characterized by the constitutive equation of third-grade fluid and the power-law fluid model is used for lubricant. Higher-order chemical reaction and nonlinear radiative heat flux are also taken into account. The governing nonlinear partial differential equations are simplified using the Prandtl boundary layer theory and similarity transformations. An efficient and accurate implicit finite difference method is applied to approximate the resulting nonlinear coupled differential equations with complicated boundary conditions for several values of parameters of interest. The velocity, temperature, nanoparticles concentration profiles Nusselt number, and streamlines are illustrated through graphs for full slip and no–slip case as well as assisting and opposing flow cases. The temperature and concentration profiles are also described for linear and non-linear thermal radiation and for different order chemical reactions. The numerical results are compared with the existing literature and found good agreement. It is found that heat and mass transfer rates reduces over the lubricated surface compared with the rough surface.

Study on the Influence of Isolated Roughness on Hypersonic Flow over Blunt Wedge

To investigate the problem of the influence of the depth of the isolated D-type roughness element on the hypersonic flow around the blunt wedge boundary, the high-order accuracy finite difference method is used to carry out the direct numerical simulation, the influence of the depth of the isolated D-type roughness element on the interaction between the free flow and the wall is analysed, and the influence of the D-type roughness element on the wall pressure, the wall friction resistance and the wall heat flow is discussed. The results show that the strength of compression wave and expansion wave increases with the increase of the depth of D-type roughness element, but the position of compression wave and expansion wave does not change; With the increase of the depth of the D-type element, the change speed of the flow parameters will be accelerated, the peak value will increase and the trough value will decrease, and the position of the peak and trough will gradually move away from the center of the element with the increase of the depth of the element. When the depth of the D-type element is greater than or equal to 0.08, a vortex will form in the D-type element, and the length of the vortex increases with the deepening of the element. The D-type element will not change the flow state of the shear layer near it.

A High Order Accurate Finite Difference Method for the Drinfel\u2019d\u2013Sokolov\u2013Wilson Equation

We analyse numerically the periodic problem and the initial boundary value problem of the Korteweg-de Vries equation and the Drindfeld–Sokolov–Wilson equation using the summation-by-parts simultaneous-approximation-term method. Two sets of boundary conditions are derived for each equation of which stability is shown using the energy method. Numerical analysis is done when the solution interacts with the boundaries. Results show the benefit of higher order SBP operators.

Smooth, cusped and sharp shock waves in a one-dimensional model of a microfluidic drop ensemble

PurposeThe purpose of this paper is to determine both analytically and numerically the existence of smooth, cusped and sharp shock wave solutions to a one-dimensional model of microfluidic droplet ensembles, water flow in unsaturated flows, infiltration, etc., as functions of the powers of the convection and diffusion fluxes and upstream boundary condition; to study numerically the evolution of the wave for two different initial conditions; and to assess the accuracy of several finite difference methods for the solution of the degenerate, nonlinear, advection--diffusion equation that governs the model.Design/methodology/approachThe theory of ordinary differential equations and several explicit, finite difference methods that use first- and second-order, accurate upwind, central and compact discretizations for the convection terms are used to determine the analytical solution for steadily propagating waves and the evolution of the wave fronts from hyperbolic tangent and piecewise linear initial conditions to steadily propagating waves, respectively. The amplitude and phase errors of the semi-discrete schemes are determined analytically and the accuracy of the discrete methods is assessed.FindingsFor non-zero upstream boundary conditions, it has been found both analytically and numerically that the shock wave is smooth and its steepness increases as the power of the diffusion term is increased and as the upstream boundary value is decreased. For zero upstream boundary conditions, smooth, cusped and sharp shock waves may be encountered depending on the powers of the convection and diffusion terms. For a linear diffusion flux, the shock wave is smooth, whereas, for a quadratic diffusion flux, the wave exhibits a cusped front whose left spatial derivative decreases as the power of the convection term is increased. For higher nonlinear diffusion fluxes, a sharp shock wave is observed. The wave speed decreases as the powers of both the convection and the diffusion terms are increased. The evolution of the solution from hyperbolic tangent and piecewise linear initial conditions shows that the wave back adapts rapidly to its final steady value, whereas the wave front takes much longer, especially for piecewise linear initial conditions, but the steady wave profile and speed are independent of the initial conditions. It is also shown that discretization of the nonlinear diffusion flux plays a more important role in the accuracy of first- and second-order upwind discretizations of the convection term than either a conservative or a non-conservative discretization of the latter. Second-order upwind and compact discretizations of the convection terms are shown to exhibit oscillations at the foot of the wave’s front where the solution is nil but its left spatial derivative is largest. The results obtained with a conservative, centered second--order accurate finite difference method are found to be in good agreement with those of the second-order accurate, central-upwind Kurganov--Tadmor method which is a non-oscillatory high-resolution shock-capturing procedure, but differ greatly from those obtained with a non-conservative, centered, second-order accurate scheme, where the gradients are largest.Originality/valueA new, one-dimensional model for microfluidic droplet transport, water flow in unsaturated flows, infiltration, etc., that includes high-order convection fluxes and degenerate diffusion, is proposed and studied both analytically and numerically. Its smooth, cusped and sharp shock wave solutions have been determined analytically as functions of the powers of the nonlinear convection and diffusion fluxes and the boundary conditions. These solutions are used to assess the accuracy of several finite difference methods that use different orders of accuracy in space, and different discretizations of the convection and diffusion fluxes, and can be used to assess the accuracy of other numerical procedures for one-dimensional, degenerate, convection--diffusion equations.

A residual-based artificial viscosity finite difference method for scalar conservation laws

In this paper, we present an accurate, stable and robust shock-capturing finite difference method for solving scalar non-linear conservation laws. The spatial discretization uses high-order accurate upwind summation-by-parts finite difference operators combined with weakly imposed boundary conditions via simultaneous-approximation-terms. The method is an extension of the residual-based artificial viscosity methods developed in the finite- and spectral element communities to the finite difference setting. The three main ingredients of the proposed method are: (i) shock detection provided by a residual-based error estimator; (ii) first-order viscosity applied in regions with strong discontinuities; (iii) additional dampening of spurious oscillations provided by high-order dissipation from the upwind finite difference operators. The method is shown to be stable for skew-symmetric discretizations of the advective flux. Accuracy and robustness are shown by solving several benchmark problems in 2D for convex and non-convex fluxes.