Showing papers on "Finite difference published in 2010"

ADI finite difference schemes for option pricing in the Heston model with correlation

Abstract: This paper deals with the numerical solution of the Heston partial differential equation (PDE) that plays an important role in financial option pricing theory, Heston (1993). A feature of this time-dependent, twodimensional convection-diffusion-reaction equation is the presence of a mixed spatial-derivative term, which stems from the correlation between the two underlying stochastic processes for the asset price and its variance. Semi-discretization of the Heston PDE, using finite difference schemes on non-uniform grids, gives rise to large systems of stiff ordinary differential equations. For the effective numerical solution of these systems, standard implicit time-stepping methods are often not suitable anymore, and tailored timediscretization methods are required. In the present paper, we investigate four splitting schemes of the Alternating Direction Implicit (ADI) type: the Douglas scheme, the Craig–Sneyd scheme, the Modified Craig–Sneyd scheme, and the Hundsdorfer–Verwer scheme, each of which contains a free parameter. ADI schemes were not originally developed to deal with mixed spatialderivative terms. Accordingly, we first discuss the adaptation of the above four ADI schemes to the Heston PDE. Subsequently, we present various numerical examples with realistic data sets from the literature, where we consider European call options as well as down-and-out barrier options. Combined with ample theoretical stability results for ADI schemes that have recently been obtained in In ’t Hout & Welfert (2007, 2009) we arrive at three ADI schemes that all prove to be very effective in the numerical solution of the Heston PDE with a mixed derivative term. It is expected that these schemes will be useful also for general two-dimensional convection-diffusion-reaction equations with mixed derivative terms.

Layer-Adapted Meshes for Reaction-Convection-Diffusion Problems

Abstract: One dimensional problems.- The Analytical Behaviour of Solutions.- Finite Difference Schemes for Convection-Diffusion Problems.- Finite Element and Finite Volume Methods.- Discretisations of Reaction-Convection-Diffusion Problems.- Layer-Adapted Meshes.- Two dimensional problems.- The Analytical Behaviour of Solutions.- Reaction-Diffusion Problems.- Convection-Diffusion Problems.

Depth-integrated, non-hydrostatic model with grid nesting for tsunami generation, propagation, and runup

Abstract: This dissertation describes the formulation, verification, and validation of a dispersive wave model with a shock-capturing scheme, and its implementation for basin-wide evolution and coastal runup of tsunamis using two-way nested computational grids. The depth-integrated formulation builds on the nonlinear shallow-water equations and utilizes a non-hydrostatic pressure term to describe weakly dispersive waves. The semi-implicit, finite difference solution captures flow discontinuities associated with bores or hydraulic jumps through a momentum conservation scheme, which also accounts for energy dissipation in the wave breaking process without the use of an empirical model. An upwind scheme extrapolates the free surface elevation instead of the flow depth to provide the flux in the momentum and continuity equations. This eliminates depth extrapolation errors and greatly improves the model stability, which is essential for computation of energetic breaking waves and runup. The vertical velocity term associated with non-hydrostatic pressure also describes tsunami generation and transfer of kinetic energy due to dynamic seafloor deformation. A depth-dependent Gaussian function smooths bathymetric features smaller than the water depth to improve convergence of the implicit, non-hydrostatic solution. A two-way grid-nesting scheme utilizes the Dirichlet condition of the non-hydrostatic pressure and both the velocity and surface elevation at the grid interface to ensure propagation of dispersive waves and discontinuities through computational grids of different resolution. The inter-grid boundary can adapt to topographic features to model wave transformation processes at optimal resolution and computational efficiency. The computed results show very good agreement with data from previous laboratory experiments for wave propagation, transformation, breaking, and runup over a wide range of conditions. The present model is applied to the 2009 Samoa Tsunami for demonstration and validation. These case studies confirm the validity and effectiveness of the present modeling approach for tsunami research and impact assessment. Since the numerical scheme to the momentum and continuity equations remains explicit, the implicit non-hydrostatic solution is directly applicable to existing nonlinear shallow-water models.

Unconditionally Stable Finite Difference, Nonlinear Multigrid Simulation of the Cahn-Hilliard-Hele-Shaw System of Equations

Abstract: We present an unconditionally energy stable and solvable finite difference scheme for the Cahn-Hilliard-Hele-Shaw (CHHS) equations, which arise in models for spinodal decomposition of a binary fluid in a Hele-Shaw cell, tumor growth and cell sorting, and two phase flows in porous media. We show that the CHHS system is a specialized conserved gradient-flow with respect to the usual Cahn-Hilliard (CH) energy, and thus techniques for bistable gradient equations are applicable. In particular, the scheme is based on a convex splitting of the discrete CH energy and is semi-implicit. The equations at the implicit time level are nonlinear, but we prove that they represent the gradient of a strictly convex functional and are therefore uniquely solvable, regardless of time step-size. Owing to energy stability, we show that the scheme is stable in the $L_{s}^{\infty}(0,T;H_{h}^{1})$ norm, and, assuming two spatial dimensions, we show in an appendix that the scheme is also stable in the $L_{s}^{2}(0,T;H_{h}^{2})$ norm. We demonstrate an efficient, practical nonlinear multigrid method for solving the equations. In particular, we provide evidence that the solver has nearly optimal complexity. We also include a convergence test that suggests that the global error is of first order in time and of second order in space.

Essential Computational Fluid Dynamics

Abstract: PREFACE. 1 What Is CFD? 1.1. Introduction. 1.2. Brief History of CFD. 1.3. Outline of the Book. References and Suggested Reading. I Fundamentals. 2 Governing Equations of Fluid Dynamics and Heat Transfer. 2.1. Preliminary Concepts. 2.2. Mass Conservation. 2.3. Conservation of Chemical Species. 2.4. Conservation of Momentum. 2.5. Conservation of Energy. 2.6. Equation of State. 2.7. Equations in Integral Form. 2.8. Equations in Conservation Form. 2.9. Equations in Vector Form. 2.10. Boundary Conditions. References and Suggested Reading. Problems. 3 Partial Differential Equations. 3.1. Model Equations Formulation of a PDE Problem. 3.2. Mathematical Classification of PDE of Second Order. 3.3. Numerical Discretization: Different Kinds of CFD. References and Suggested Reading. Problems. 4 Basics of Finite Difference Approximation. 4.1. Computational Grid. 4.2. Finite Differences and Interpolation. 4.3. Approximation of Partial Differential Equations. 4.4. Development of Finite Difference Schemes. References and Suggested Reading. Problems. 5 Finite Volume Method. 5.1. Introduction and Integral Formulation. 5.2. Approximation of Integrals. 5.3. Methods of Interpolation. 5.4. Boundary Conditions. References and Suggested Reading. Problems. 6 Stability of Transient Solutions. 6.1. Introduction and Definition of Stability. 6.2. Stability Analysis. 6.3. Implicit versus Explicit Schemes-Stability and Efficiency Considerations. References and Suggested Reading. Problems. 7 Application to Model Equations. 7.1. Linear Convection Equation. 7.2. One-Dimensional Heat Equation. 7.3. Burgers and Generic Transport Equations. 7.4. Method of Lines Approach. 7.5. Implicit Schemes: Solution of Tridiagonal Systems by Thomas Algorithm. References and Suggested Reading. Problems. II Methods. 8 Steady-State Problems. 8.1. Problems Reducible to Matrix Equations. 8.2. Direct Methods. 8.3. Iterative Methods. 8.4. Systems of Nonlinear Equations. References and Suggested Reading. Problems. 9 Unsteady Problems of Fluid Flows and Heat Transfer. 9.1. Introduction. 9.2. Compressible Flows. 9.3. Unsteady Conduction Heat Transfer. References and Suggested Reading. Problems. 10 Incompressible Flows. 10.1. General Considerations. 10.2. Discretization Approach. 10.3. Projection Method for Unsteady Flows. 10.4. Projection Methods for Steady-State Flows. 10.5. Other Methods. References and Suggested Reading. Problems. III Art of CFD. 11 Turbulence. 11.1. Introduction. 11.2. Direct Numerical Simulation (DNS). 11.3. Reynolds-Averaged Navier-Stokes (RANS) Models. 11.4. Large-Eddy Simulation (LES). References and Suggested Reading. Problems. 12 Computational Grids. 12.1. Introduction: Need for Irregular and Unstructured Grids. 12.2. Irregular Structured Grids. 12.3. Unstructured Grids. References and Suggested Reading. Problems. 13 Conducting CFD Analysis. 13.1. Overview: Setting and Solving a CFD Problem. 13.2. Errors and Uncertainty. 13.3. Adaptive Grids. References and Suggested Reading. INDEX.

Design of an FIR filter for the displacement reconstruction using measured acceleration in low‐frequency dominant structures

Abstract: This paper presents a new class of displacement reconstruction scheme using only acceleration measured from a structure. For a given set of acceleration data, the reconstruction problem is formulated as a boundary value problem in which the acceleration is approximated by the second-order central finite difference of displacement. The displacement is reconstructed by minimizing the least-squared errors between measured and approximated acceleration within a finite time interval. An overlapping time window is introduced to improve the accuracy of the reconstructed displacement. The displacement reconstruction problem becomes ill-posed because the boundary conditions at both ends of each time window are not known a priori. Furthermore, random noise in measured acceleration causes physically inadmissible errors in the reconstructed displacement. A Tikhonov regularization scheme is adopted to alleviate the ill-posedness. It is shown that the proposed method is equivalent to an FIR filter designed in the time domain. The fundamental characteristics of the proposed method are presented in the frequency domain using the transfer function and the accuracy function. The validity of the proposed method is demonstrated by a numerical example, a laboratory experiment and a field test. Copyright © 2009 John Wiley & Sons, Ltd.

An easy proof of Jensen’s theorem on the uniqueness of infinity harmonic functions

Abstract: We present a new, easy, and elementary proof of Jensen’s Theorem on the uniqueness of infinity harmonic functions. The idea is to pass to a finite difference equation by taking maximums and minimums over small balls.

Convergent finite difference solvers for viscosity solutions of the elliptic Monge-Amp\`ere equation in dimensions two and higher

Abstract: The elliptic Monge-Amp\`ere equation is a fully nonlinear Partial Differential Equation that originated in geometric surface theory and has been applied in dynamic meteorology, elasticity, geometric optics, image processing and image registration. Solutions can be singular, in which case standard numerical approaches fail. Novel solution methods are required for stability and convergence to the weak (viscosity) solution. In this article we build a wide stencil finite difference discretization for the \MA equation. The scheme is monotone, so the Barles-Souganidis theory allows us to prove that the solution of the scheme converges to the unique viscosity solution of the equation. Solutions of the scheme are found using a damped Newton's method. We prove convergence of Newton's method and provide a systematic method to determine a starting point for the Newton iteration. Computational results are presented in two and three dimensions, which demonstrates the speed and accuracy of the method on a number of exact solutions, which range in regularity from smooth to non-differentiable.

Robust and provably second-order explicit-explicit and implicit-explicit staggered time-integrators for highly non-linear compressible fluid-structure interaction problems

Abstract: An explicit–explicit staggered time-integration algorithm and an implicit–explicit counterpart are presented for the solution of non-linear transient fluid–structure interaction problems in the Arbitrary Lagrangian–Eulerian (ALE) setting. In the explicit–explicit case where the usually desirable simultaneous updating of the fluid and structural states is both natural and trivial, staggering is shown to improve numerical stability. Using rigorous ALE extensions of the two-stage explicit Runge–Kutta and three-point backward difference methods for the fluid, and in both cases the explicit central difference scheme for the structure, second-order time-accuracy is achieved for the coupled explicit–explicit and implicit–explicit fluid–structure time-integration methods, respectively, via suitable predictors and careful stagings of the computational steps. The robustness of both methods and their proven second-order time-accuracy are verified for sample application problems. Their potential for the solution of highly non-linear fluid–structure interaction problems is demonstrated and validated with the simulation of the dynamic collapse of a cylindrical shell submerged in water. The obtained numerical results demonstrate that, even for fluid–structure applications with strong added mass effects, a carefully designed staggered and subiteration-free time-integrator can achieve numerical stability and robustness with respect to the slenderness of the structure, as long as the fluid is justifiably modeled as a compressible medium. Copyright © 2010 John Wiley & Sons, Ltd.

Radial integration BEM for transient heat conduction problems

Abstract: In this paper, a new boundary element analysis approach is presented for solving transient heat conduction problems based on the radial integration method. The normalized temperature is introduced to formulate integral equations, which makes the representation very simple and having no temperature gradients involved. The Green's function for the Laplace equation is adopted in deriving basic integral equations for time-dependent problems with varying heat conductivities and, as a result, domain integrals are involved in the derived integral equations. The radial integration method is employed to convert the domain integrals into equivalent boundary integrals. Based on the central finite difference technique, an implicit time marching solution scheme is developed for solving the time-dependent system of equations. Numerical examples are given to demonstrate the correctness of the presented approach.

Solvable rational extensions of the isotonic oscillator

Abstract: Combining recent results on rational solutions of the Riccati-Schrodinger equations for shape invariant potentials to the finite difference Backlund algorithm and specific symmetries of the isotonic potential, we show that it is possible to generate the three infinite sets (L1, L2 and L3 families) of regular rational solvable extensions of this potential in a very direct and transparent way.

On analogies between nonlinear difference and differential equations

Abstract: In this paper, we point out some similarities between results on the existence and uniqueness of finite order entire solutions of the nonlinear differential equations and differential-difference equations of the form $$f^n+L(z,f)=h.$$ Here n is an integer $\geq 2$, h is a given non-vanishing meromorphic function of finite order, and L(z,f) is a linear differential-difference polynomial, with small meromorphic functions as the coefficients.

Corruption of accuracy and efficiency of Markov chain Monte Carlo simulation by inaccurate numerical implementation of conceptual hydrologic models

Abstract: Conceptual rainfall?runoff models have traditionally been applied without paying much attention to numerical errors induced by temporal integration of water balance dynamics. Reliance on first?order, explicit, fixed?step integration methods leads to computationally cheap simulation models that are easy to implement. Computational speed is especially desirable for estimating parameter and predictive uncertainty using Markov chain Monte Carlo (MCMC) methods. Confirming earlier work of Kavetski et al. (2003), we show here that the computational speed of first?order, explicit, fixed?step integration methods comes at a cost: for a case study with a spatially lumped conceptual rainfall?runoff model, it introduces artificial bimodality in the marginal posterior parameter distributions, which is not present in numerically accurate implementations of the same model. The resulting effects on MCMC simulation include (1) inconsistent estimates of posterior parameter and predictive distributions, (2) poor performance and slow convergence of the MCMC algorithm, and (3) unreliable convergence diagnosis using the Gelman?Rubin statistic. We studied several alternative numerical implementations to remedy these problems, including various adaptive?step finite difference schemes and an operator splitting method. Our results show that adaptive?step, second?order methods, based on either explicit finite differencing or operator splitting with analytical integration, provide the best alternative for accurate and efficient MCMC simulation. Fixed?step or adaptive?step implicit methods may also be used for increased accuracy, but they cannot match the efficiency of adaptive?step explicit finite differencing or operator splitting. Of the latter two, explicit finite differencing is more generally applicable and is preferred if the individual hydrologic flux laws cannot be integrated analytically, as the splitting method then loses its advantage.

High-order finite-difference simulations of marine CSEM surveys using a correspondence principle for wave and diffusion fields

Abstract: The computer time required to solve a typical 3D marine controlled-source electromagnetic surveying (CSEM) simulation can be reduced by more than one order of magnitude by transforming low-frequency Maxwell equations in the quasi-static or diffusive limit to a hyperbolic set of partial differential equations that give a representation of electromagnetic fields in a fictitious wave domain. The dispersion and stability analysis can be made equivalent to that of other types of wave simulation problems such as seismic acoustic and elastic modeling. Second-order to eighth-order spatial derivative operators are implemented for flexibility. Fourth-order and sixth-order methods are the most numerically efficient implementations for this particular scheme. An implementation with high-order operators requires that both electric and magnetic fields are extrapolated simultaneously into the air layer. The stability condition given for high-order staggered-derivative operators here should be equally valid for seismic-wave simulation. The bandwidth of recovered fields in the diffusive domain is independent of the bandwidth of the fields in the fictitious wave domain. The fields in the fictitious wave domain do not represent observable fields. Propagation paths and interaction/reflection amplitudes are not altered by the transform from the fictitious wave domain to the diffusive frequency domain; however, the transform contains an exponential decay factor that damps down late arrivals in the fictitious wave domain. The propagation paths that contribute most to the diffusive domain fields are airwave (shallow water) plus typically postcritical events such as refracted and guided waves. The transform from the diffusive frequency domain to the fictitious wave domain is an ill-posed problem. The transform is nonunique. This gives a large degree of freedom in postulating temporal waveforms for boundary conditions in the fictitious wave domain that reproduce correct diffusive frequency-domain fields.

Permeability prediction for the meso–macro coupling in the simulation of the impregnation stage of Resin Transfer Moulding

Abstract: The impregnation stage of the Resin Transfer Moulding process can be simulated by solving the Darcy equations on a mould model, with a ‘macro-scale’ finite element method. For every element, a local ‘meso-scale’ permeability must be determined, taking into account the local deformation of the textile reinforcement. This paper demonstrates that the meso-scale permeability can be computed efficiently and accurately by using meso-scale simulation tools. We discuss the speed and accuracy requirements dictated by the macro-scale simulations. We show that these requirements can be achieved for two meso-scale simulators, coupled with a geometrical textile reinforcement modeller. The first solver is based on a finite difference discretisation of the Stokes equations, the second uses an approximate model, based on a 2D simulation of the flow.

Application of the homotopy perturbation method for the Fokker–Planck equation

Abstract: In this paper, we will discuss the solution of an initial value problem of parabolic type. The main objective is to propose an alternative method of solution, one not based on finite difference or finite element or spectral methods. The aim of the present paper is to investigate the application of the Homotopy perturbation method (HPM) for solving the Fokker–Planck equation and some similar equations. This method is a powerful tool for solving various kinds of problems. Employing this technique, it is possible to find the exact solution or an approximate solution of the problem. The results show applicability, accuracy and efficiency of HPM in solving nonlinear differential equations. It is predicted that HPM can be widely applied in science and engineering problems. Copyright © 2008 John Wiley & Sons, Ltd.

On the L∞ convergence of a difference scheme for coupled nonlinear Schrödinger equations

Abstract: In this article, a finite difference scheme for coupled nonlinear Schrodinger equations is studied. The existence of the difference solution is proved by Brouwer fixed point theorem. With the aid of the fact that the difference solution satisfies two conservation laws, the finite difference solution is proved to be bounded in the discrete L"~ norm. Then, the difference solution is shown to be unique and second order convergent in the discrete L"~ norm. Finally, a convergent iterative algorithm is presented.

The complex step approximation to the Fréchet derivative of a matrix function

Abstract: We show that the Frechet derivative of a matrix function f at A in the direction E, where A and E are real matrices, can be approximated by Im f(A + ihE)/h for some suitably small h. This approximation, requiring a single function evaluation at a complex argument, generalizes the complex step approximation known in the scalar case. The approximation is proved to be of second order in h for analytic functions f and also for the matrix sign function. It is shown that it does not suffer the inherent cancellation that limits the accuracy of finite difference approximations in floating point arithmetic. However, cancellation does nevertheless vitiate the approximation when the underlying method for evaluating f employs complex arithmetic. The ease of implementation of the approximation, and its superiority over finite differences, make it attractive when specialized methods for evaluating the Frechet derivative are not available, and in particular for condition number estimation when used in conjunction with a block 1-norm estimation algorithm.

A Multipoint Flux Mixed Finite Element Method on Hexahedra

Abstract: We develop a mixed finite element method for elliptic problems on hexahedral grids that reduces to cell-centered finite differences. The paper is an extension of our earlier paper for quadrilateral and simplicial grids [M. F. Wheeler and I. Yotov, SIAM J. Numer. Anal., 44 (2006), pp. 2082-2106]. The construction is motivated by the multipoint flux approximation method, and it is based on an enhancement of the lowest order Brezzi-Douglas-Duran-Fortin (BDDF) mixed finite element spaces on hexahedra. In particular, there are four fluxes per face, one associated with each vertex. A special quadrature rule is employed that allows for local velocity elimination and leads to a symmetric and positive definite cell-centered system for the pressures. Theoretical and numerical results indicate first-order convergence for pressures and subface fluxes on sufficiently regular grids, as well as second-order convergence for pressures at the cell centers. Second-order convergence for face fluxes is also observed computationally.