Second-order Accurate Finite Difference Scheme Research Articles

Unconditionally Stable and Second-Order Accurate Explicit Finite Difference Schemes Using Domain Transformation: Part I One-Factor Equity Problems

We introduce a class of stable and second-order accurate finite difference schemes that resolve a number of problems when approximating the solution of option pricing models in computational finance using the finite difference method (FDM). In particular, we show how to avoid having to apply truncation methods to the domain of integration as well as the resulting ad-hoc experimentation in choosing suitable numerical boundary conditions that we must apply on the boundary of the truncated domain. Instead, we transform the PDE option model (which is originally defiined on a semi-infinite interval) to a PDE that is defined on a bounded interval. We then apply the elegant Fichera theory to help us determine which boundary conditions to apply to the transformed PDE. We show that the new system is well-posed by proving energy inequalities in the space of square-integrable functions. Having done that, we adapt the Alternating Direction Explicit (ADE) (a method that dates from the first half of the last century) method to compute an approximation to the solution of the transformed PDE. We prove that the explicit scheme is unconditionally stable and second-order accurate. The scheme is very easy to model, to program and to parallelise and the generalization to n-factor problems is easily motivated. We examine equity problems and we compare the ADE approach (in terms of accuracy and speedup) with more traditional finite difference schemes and with the Monte Carlo method. Finally, we stress-test the scheme by varying critical parameters over a spectrum of values and the results are noted. The methods in this article - in particular the combination of pure and applied techniques - are relatively new in the computational finance literature in our opinion, in particular the realization that a PDE on a semi-infinite interval can be transformed to one on a bounded interval and that the new PDE can be approximated by schemes other than Crank-Nicolson and its workarounds. Finally, we shall report on n-factor models in future articles.

Numerical study of a nuclear fuel element dissipating fission heat into its surrounding fluid medium

The objective of the present work is twofold – the first to establish the criterion for the boundary layer solution to be accurate enough in the study of conjugate heat transfer problem associated with a rectangular nuclear fuel element washed by upward moving coolant and the second to predict the critical thermal performance characteristics of the fuel element with uniform volumetric energy generation. Accordingly, employing stream function–vorticity formulation, equations governing the steady, two-dimensional flow and thermal fields in the coolant are solved simultaneously with the steady, two-dimensional heat conduction equation for the fuel element using second-order accurate finite difference schemes. Keeping the Prandtl number constant at 0.005 for liquid sodium as coolant, numerical results are presented for wide range of aspect ratio, conduction–convection parameter, energy generation parameter and Reynolds number. It is found that for all value of aspect ratio greater than 15, numerical prediction using the boundary layer approximation based model is quite accurate enough. It is also concluded that other parameters being kept constant, the increase in the maximum fuel element temperature due to increase in aspect ratio beyond 15 is negligible. Further, it is found that a relatively higher value of conduction–convection parameter reduces the coolant pumping power requirement to a large extent.

Signal processing of acoustic signals in the time domain with an active nonlinear nonlocal cochlear model

A two-space dimensional active nonlinear nonlocal cochlear model is formulated in the time domain to capture nonlinear hearing effects such as compression, multi-tone suppression and difference tones. The micromechanics of the basilar membrane (BM) are incorporated to model active cochlear properties. An active gain parameter is constructed in the form of a nonlinear nonlocal functional of BM displacement. The model is discretized with a boundary integral method and numerically solved using an iterative second-order accurate finite difference scheme. A block matrix structure of the discrete system is exploited to simplify the numerics with no loss of accuracy. Model responses to multiple frequency stimuli are shown in agreement with hearing experiments. A nonlinear spectrum is computed from the model, and compared with FFT spectrum for noisy tonal inputs. The discretized model is efficient and accurate, and can serve as a useful auditory signal processing tool.

Simulating the Aerodynamics of the NASA John H. Glenn Icing Research Tunnel

The objective of this study is to develop and employ a numerical simulation strategy for predicting the airflow from the spray bars to the test section of the NASA John H. Glenn Icing Research Tunnel (IRT). In particular, predictions of the mean velocity and turbulence distributions were desired throughout this flow domain to later investigate droplet dispersion. Computational airflow results were produced using the WIND code (developed by NPARC) with a second-order accurate finite difference scheme and the shear stress transport k‐Ω urbulence model. The inflow conditions for the flow domain were derived from the IRT measurements just upstream of the spray bars, which reflected the contributions to turbulence from the upstream heat exchanger wake. It was found that inclusion of the spray bar wakes and the air jets (of the spray nozzles) were required to describe the wind-tunnel turbulence distribution. Because it was impractical to simultaneously resolve the overall flow domain (60 ft long), along with the detailed flow around the 10 spray bars and the flow within a hundred air jets (issuing from 1/8-in. nozzle diameters), these features were simulated individually and then algebraically combined together to give an approximate solution. The results of the spray bar wake combined with the heat exchanger flow yielded good prediction of test section mean velocity and turbulence for the jets-off condition. Inclusion of all of the individual air jets also yielded reasonable resulting predictions of mean velocity and turbulence in the test section.

Finite-Volume Models for Unidirectional, Nonlinear, Dispersive Waves

Several finite-volume schemes are developed and applied to simulate nonlinear, dispersive, unidirectional waves propagating over a flat bed. These schemes differ mainly in the treatment of advection, while dispersion is treated the same among the different models. Three methods—linear, total variation diminishing, and essentially nonoscillatory—are used to discretize the advection portion of the governing equation. The linear schemes are analyzed with Von Neumann's method to discern stability limits as well as their damping and dispersion characteristics. In addition, predictions from all of the models are compared with analytical solutions for solitary and cnoidal waves as well as experimental data for undular bores. The finite-volume methods are also compared with a second-order accurate finite-difference scheme. The results indicate that the finite-volume schemes yield more accurate solutions than the finite-difference scheme. In addition, the Warming-Beam and Fromm linear finite-volume schemes yielded the most accurate solutions and were among the most computationally efficient schemes tested.

Depth-Averaged Simulation of Supercritical Flow in Channel with Wavy Sidewall

Supercritical flow in a channel with a wavy sidewall is numerically simulated by solving the two-dimensional (2D) depth-averaged equations using two different second-order accurate finite-difference schemes: ADI and MAC. ADI is an implicit model that uses an alternating-direction-implicit (ADI) scheme to solve the governing equations. MAC is an explicit model employing the MacCormack two-step predictor-corrector scheme. To accurately simulate the wavy sidewall, both models solve the governing equations in transformed computational coordinates. Bottom friction is computed using the Manning formula and the effective stresses are modeled with a constant eddy-viscosity turbulence model. As is customary, the stresses due to depth-averaging are neglected. The computed water depth in the channel is compared with experimental data obtained by Mizumura. The effect of bottom friction, effective stresses, artificial viscosity, grid geometry, boundary conditions, and the Courant-Friedrichs-Lewy (CFL) number are investigated. Similarities and differences in the behavior of the models are observed and discussed.

Dispersion analysis of numerical approximations to plane wave motions in an isotropic elastic solid

It is well known that isotropic, nondispersive continuous hyperbolic problems become dispersive and anisotropic upon discretization. The purpose of this paper is to conduct a dispersion analysis of the nondissipative numerical approximations to plane wave motions in isotropic elastic solids. The discrete formulations considered are: an explicit, second-order accurate finite difference scheme, a consistent mass matrix formulation with linear quadrilateral elements and the corresponding lumped mass matrix formulation. Dispersion relation is derived for each of these formulations. In the context of the finite difference scheme, expressions for group velocity for both the shear and longitudinal waves are derived and the effect of using meshes of unequal size in x and y directions is studied. Results from numerical experiments confirming the predictions of analysis are also presented.

A second-order accurate finite difference scheme for a class of nonlocal parabolic equations with natural boundary conditions

A difference scheme is derived for a class of nonlocal parabolic equations with natural boundary conditions by the method of reduction of order. It is shown that the scheme is uniquely solvable and unconditionally convergent with the convergence rate of order O( h 2 + τ 2). A numerical example with some comparisons is presented.

Conservative Properties of Finite Difference Schemes for Incompressible Flow. 2nd Report. Schemes in Staggered and Collocated Grid Systems.

In this report, finite difference schemes in staggered and collocated grid systems are considered in light of the analytical requirements that were discussed in the first report. The standard second-order accurate finite difference scheme with divergence form in a staggered grid system is proper. Proper second-order accurate advective and skew-symmetric forms have been proposed in the staggered grid system. Existing fourth-order accurate convective schemes in a staggered grid system are not proper. Proper higher-order accurate finite difference schemes in a staggered grid system are proposed. This is novel because higher-order staggered grid schemes that conserve momentum and kinetic energy simultaneously do not previously appear in the literature. The pressure term is not conservative in the kinetic energy equation in the collocated grid system, although we can construct proper-second and fourth-order accurate convective schemes in this system.

Numerical and experimental analysis of unsteady heat transfer with periodic variation of inlet temperature in circular ducts

This work focusses on a numerical and experimental analysis of unsteady forced convection in hydrodynamically developed and thermally developing laminar air flow in a circular duct, subjected to a periodic variation of the inlet temperature. The experiments were conducted over a wide range of Reynolds number (281.2 ≤ Re ≤ 1024.3) and inlet frequency (0.01 ≤ β ≤ 0.20 Hz) of the periodic heat input. In the numerical study, the non-uniform inlet temperature amplitude profile derived from the experiments, was included in the numerical model. A fully explicit, second-order accurate finite difference scheme was developed and used for the solution of the unsteady energy equation. Numerical results are obtained with the fully developed parabolic velocity profile under the boundary condition of the first kind, which was verified by the experiments. Temperature variations along the centerline of the circular duct are observed to be thermal oscillations with the same frequency as the inlet periodic heat input and amplitudes that decayed exponentially with distance along the duct. Thermal response along the wall exhibits negligible amplitude variation with changes in Reynolds number and inlet frequency. The variation in the periods and amplitudes of the thermal oscillations are observed to be a function of spacial system variables only. Satisfactory agreement between the numerical and experimental results are obtained.

Computing the Flow around a Submerged Body Using Composite Grids

The subject of this paper is an accurate numerical method for solving the linear two-dimensional steady potential flow around a body which moves in a liquid of finite constant depth at constant speed and distance below the free surface. The differential equation is discretized by a second-order accurate finite difference scheme on a composite grid. The composite grid consists of two overlapping component grids; one curvilinear grid close to the body and one Cartesian grid which covers the surrounding liquid. To solve the problem numerically, the infinite domain is truncated to finite length. The inflow and outflow boundary conditions are formed by making an eigenfunction expansion of the solution ahead of and behind the body. Each eigenfunction is required to be bounded and satisfy the upstream condition at infinity. This is imposed by functional relations between the solution and its normal derivative at the inflow and outflow boundaries. The method is carefully validated and the computed solutions are found to be in very good agreement with existing results.

A second-order finite-difference scheme for computing three-dimensional supersonic flows of an ideal gas

An explicit second-order accurate finite-difference scheme for computing the steady three-dimensional supersonic flows of an inviscid gas is proposed. The features of the scheme are described. The results of systematical computations of conical flows and submerged jets are in good agreement with data obtained using existing second-order accurate schemes. Some samples computations for compound three-dimensional supersonic flows are presented.

TRANSIENT HEAT TRANSFER IN CHANNEL FLOW WITH STEP CHANGE IN INLET TEMPERATURE

A second-order accurate finite-difference scheme based on a modification of the upwind method for the classical wave equation is proposed for transient forced convection in laminar channel flow. Stability analysis is performed, and the modified equation is established. The problem involving a step change in inlet temperature is studied for both slug and parabolic flow inside a parallel-plate duct. The accuracy of the numerical scheme is examined by comparing the results to the exact solutions available for the case of slug flow. Numerical results are presented for the average flow temperature and local Nusselt number as a function of position along the channel at different times, and the thermal wave propagation phenomena are physically interpreted.

A mixed pseudospectral/finite difference method for the axisymmetric flow in a heated, rotating spherical shell

A numerical solution technique for the axisymmetric flow in a differentially heated, rotating spherical annulus is developed. This method, based on the incompressible Navier-Stokes equations, simulates the Atmospheric General Circulation Experiment (AGCE) proposed for a future Shuttle mission. In the method a pseudospectral technique is used in the latitudinal direction, and a second-order accurate finite difference scheme discretizes time and radial derivatives. This pseudospectral/finite difference (PS/FD) method is applied to a hierarchy of cases of varying difficulty. The difficult cases are characterized by thinner Ekman layers resulting from higher rotation rates, larger difference in boundary temperatures, and stronger body forces. Comparison of the results establishes the higher accuracy and efficiency of the PS/FD method over the pure second-order accurate finite difference (FD) method. This paper discusses the development and performance of a mixed PS/FD model for the AGCE which has been modelled in the past only by pure FD formulations.

Stability and Convergence of Finite Difference Methods for Systems of Nonlinear Reaction-Diffusion Equations

We consider the initial-boundary value problem for the general reaction-diffusion system (*) $U_t = D\nabla ^2 U + \Sigma _j M_j U_{xj} + F$, where U is a vector, $D,M_j $, and F depend on $(x,t,U)$, and D and $M_j$ are diagonal matrices with $D \geqq 0$. It is known that if there is a box S in U-space such that F does not point out of S, then S is invariant for (*). This invariance gives rise to an a priori estimate which in turn yields the existence of a smooth solution.In this paper we construct a family of locally second-order accurate finite difference schemes for (*). We prove that, under certain conditions on the mesh, S is also invariant for the difference equations. This enables us to derive an error bound which is $O(h)$ for t fixed, and which tends to diam S as $t \to \infty $. Next, we obtain a time-independent error bound by imposing a monotonicity condition on F, and we examine the implications of this condition for the solution of (*). Finally we derive a time-independent error bound for a ...

Numerical method for boundary layers with blowing - The exponential box scheme

The paper describes a new numerical scheme based on exponential difference operator concepts combined with Keller's (1968) box scheme approach to produce a stable second-order accurate finite-difference scheme for convection-diffusion problems arising in boundary layer flows in the presence of massive injection through a porous surface. The technique is demonstrated by application to the self-similar boundary layer equations with massive blowing at the surface.

Numerical solution of slender channel laminar flows

The slender channel approximate equations are used to describe flows in curved channels of varying height. The governing equations are transformed in order that similar solutions can be obtained and used as initial profiles. A second-order accurate Crank-Nicolson finite difference scheme is developed with the pressure treated directly as one of the dependent variables. The rechnique is applied to several channel geometries, and solutions are obtained at various Reynolds numbers.

Efficient Computation of "Stiff" Chemically Reacting Flow in Turbulent Free Jets

The chemically reacting flow in axisymmetric, turbulent free jets is solved numerically by a second-order accurate finite-difference scheme that is unconditionally stable with respect to the chemical rates. The transport properties are obtained from the turbulent kinetic energy model of turbulence. The momentum, energy, and turbulence model equations are solved by an explicit differencing scheme, whereas a hybrid expolcit/implicit scheme is used to solve the species conservation equations. High computational efficiency is attained by solving the equations in a natural coordinate system, and by employing the chemical element conservation principles to minimize the order of the matrix inverted to obtain the species concentrations. The scheme computes reacting jet flows in one-fifth the computer time required by any previously reported scheme.

Investigation of some finite-difference techniques for solving the boundary layer equations

The Levy-Lees form of the laminar boundary layer equations is solved with several second-order accurate finite-difference schemes. For incompressible flow the methods investigated include three forms of the Crank-Nicolson scheme, four variations of the Keller box scheme and a modified box scheme. The number of iterations required at each step along the surface to obtain a second-order accurate scheme is studied. The accuracy of the schemes with various step-sizes is determined for the boundary layer flow with a linearly retarded edge velocity. In addition, one form of the Crank-Nicolson scheme is extended to compressible flows and its accuracy and behavior are also examined for the linearly retarded flow case. The results of this investigation show that the coupled continuity-momentum form of the Crank-Nicolson scheme is second-order with one iteration at each step and requires less time than the Keller box scheme.