Standard Finite Difference Schemes Research Articles

High-Order Flux Reconstruction Schemes with Minimal Dispersion and Dissipation

Modal analysis of the flux reconstruction (FR) formulation is performed to obtain the semi-discrete and fully-discrete dispersion relations, using which, the wave properties of physical as well as spurious modes are characterized. The effect of polynomial order, correction function and solution points on the dispersion, dissipation and relative energies of the modes are investigated. Using this framework, a new set of linearly stable high-order FR schemes is proposed that minimizes wave propagation errors for the range of resolvable wavenumbers. These schemes provide considerably reduced error for advection in comparison to the Discontinuous Galerkin scheme and benefit from having an explicit differential update. The corresponding resolving efficiencies compare favorably to those of standard high-order compact finite difference schemes. These theoretical expectations are verified by a comparison of proposed and existing FR schemes in advecting a scalar quantity on uniform as well as non-uniform grids.

A positivity-preserving numerical scheme for option pricing model with transaction costs under jump-diffusion process

Option pricing model and numerical method with transaction costs under jump-diffusion process of Merton is studied in this paper. Partial integro-differential equation satisfied by the option value is derived by delta-hedge method, which is a nonlinear Black–Scholes equation with an infinite integral, and it is difficult to obtain the analytic solution. Based on a nonstandard approximation of the second partial derivative, a double discretization strategy is introduced for the unbounded part of the spatial domain and a positivity-preserving numerical scheme is developed. The scheme is not only unconditionally stable and positive, but also allows us to solve the discrete equation explicitly, and after modifying it becomes consistent. The numerical results for European call option and digital call option are compared to the standard finite difference scheme. It turns out that the proposed scheme is efficient and reliable.

Numerical study of Rayleigh-type acoustic streaming based on a three-dimensional incompressible flow model

Rayleigh-type acoustic streaming induced by a plane standing wave in a rectangular parallelepiped is numerically studied on the assumption that the streaming motion is an incompressible flow and induced by the so-called limiting velocity on the outer edge of the acoustic boundary layer on the wall of the rectangular parallelepiped. Solving the three-dimensional incompressible flow equations with a standard finite difference scheme, we show that the streamline indicates chaotic behaviors even when the streaming velocity field converges to a time-independent state (steady flow) for moderate Reynolds numbers. Based on the result, we can discuss an efficiency of mixing by the time-independent Rayleigh-type acoustic streaming motions in three-dimensional boxes.

Modified Finite Difference Schemes on Uniform Grids for Simulations of the Helmholtz Equation at Any Wave Number

We construct modified forward, backward, and central finite difference schemes, specifically for the Helmholtz equation, by using the Bloch wave property. All of these modified finite difference approximations provide exact solutions at the nodes of the uniform grid for the second derivative present in the Helmholtz equation and the first derivative in the radiation boundary conditions for wave propagation. The most important feature of the modified schemes is that they work for large as well as low wave numbers, without the common requirement of a very fine mesh size. The superiority of the modified finite difference schemes is illustrated with the help of numerical examples by making a comparison with standard finite difference schemes.

Complex step-based low-rank extended Kalman filtering for state-parameter estimation in subsurface transport models

Summary The accuracy of groundwater flow and transport model predictions highly depends on our knowledge of subsurface physical parameters. Assimilation of contaminant concentration data from shallow dug wells could help improving model behavior, eventually resulting in better forecasts. In this paper, we propose a joint state-parameter estimation scheme which efficiently integrates a low-rank extended Kalman filtering technique, namely the Singular Evolutive Extended Kalman (SEEK) filter, with the prominent complex-step method (CSM). The SEEK filter avoids the prohibitive computational burden of the Extended Kalman filter by updating the forecast along the directions of error growth only, called filter correction directions. CSM is used within the SEEK filter to efficiently compute model derivatives with respect to the state and parameters along the filter correction directions. CSM is derived using complex Taylor expansion and is second order accurate. It is proven to guarantee accurate gradient computations with zero numerical round-off errors, but requires complexifying the numerical code. We perform twin-experiments to test the performance of the CSM-based SEEK for estimating the state and parameters of a subsurface contaminant transport model. We compare the efficiency and the accuracy of the proposed scheme with two standard finite difference-based SEEK filters as well as with the ensemble Kalman filter (EnKF). Assimilation results suggest that the use of the CSM in the context of the SEEK filter may provide up to 80% more accurate solutions when compared to standard finite difference schemes and is competitive with the EnKF, even providing more accurate results in certain situations. We analyze the results based on two different observation strategies. We also discuss the complexification of the numerical code and show that this could be efficiently implemented in the context of subsurface flow models.

An efficient fourth-order low dispersive finite difference scheme for a 2-D acoustic wave equation

In this paper, we propose an efficient fourth-order compact finite difference scheme with low numerical dispersion to solve the two-dimensional acoustic wave equation. Combined with the alternating direction implicit (ADI) technique and Padé approximation, the standard second-order finite difference scheme can be improved to fourth-order and solved as a sequence of one-dimensional problems with high computational efficiency. However such compact higher-order methods suffer from high numerical dispersion. To suppress numerical dispersion, the compact and non-compact stages are interlinked to produce a hybrid scheme, in which the compact stage is based on Padé approximation in both y and temporal dimensions while the non-compact stage is based on Padé approximation in y dimension only. Stability analysis shows that the new scheme is conditionally stable and superior to some existing methods in terms of the Courant–Friedrichs–Lewy (CFL) condition. The dispersion analysis shows that the new scheme has lower numerical dispersion in comparison to the existing compact ADI scheme and the higher-order locally one-dimensional (LOD) scheme. Three numerical examples are solved to demonstrate the accuracy and efficiency of the new method.

Fully implicit ADI schemes for solving the nonlinear Poisson-Boltzmann equation

Abstract The Poisson-Boltzmann (PB) model is an effective approach for the electrostatics analysis of solvated biomolecules. The nonlinearity associated with the PB equation is critical when the underlying electrostatic potential is strong, but is extremely difficult to solve numerically. In this paper, we construct two operator splitting alternating direction implicit (ADI) schemes to efficiently and stably solve the nonlinear PB equation in a pseudo-transient continuation approach. The operator splitting framework enables an analytical integration of the nonlinear term that suppresses the nonlinear instability. A standard finite difference scheme weighted by piecewise dielectric constants varying across the molecular surface is employed to discretize the nonhomogeneous diffusion term of the nonlinear PB equation, and yields tridiagonal matrices in the Douglas and Douglas-Rachford type ADI schemes. The proposed time splitting ADI schemes are different from all existing pseudo-transient continuation approaches for solving the classical nonlinear PB equation in the sense that they are fully implicit. In a numerical benchmark example, the steady state solutions of the fully-implicit ADI schemes based on different initial values all converge to the time invariant analytical solution, while those of the explicit Euler and semi-implicit ADI schemes blow up when the magnitude of the initial solution is large. For the solvation analysis in applications to real biomolecules with various sizes, the time stability of the proposed ADI schemes can be maintained even using very large time increments, demonstrating the efficiency and stability of the present methods for biomolecular simulation.

Conditioning and stability of finite difference schemes on uniform meshes for a singularly perturbed parabolic convection-diffusion equation

For a singularly perturbed parabolic convection-diffusion equation, the conditioning and stability of finite difference schemes on uniform meshes are analyzed. It is shown that a convergent standard monotone finite difference scheme on a uniform mesh is not ɛ-uniformly well conditioned or ɛ-uniformly stable to perturbations of the data of the grid problem (here, ɛ is a perturbation parameter, ɛ ∈ (0, 1]). An alternative finite difference scheme is proposed, namely, a scheme in which the discrete solution is decomposed into regular and singular components that solve grid subproblems considered on uniform meshes. It is shown that this solution decomposition scheme converges ɛ-uniformly in the maximum norm at an O(N−1lnN + N0−1) rate, where N + 1 and N0 + 1 are the numbers of grid nodes in x and t, respectively. This scheme is ɛ-uniformly well conditioned and ɛ-uniformly stable to perturbations of the data of the grid problem. The condition number of the solution decomposition scheme is of order O(δ−2lnδ−1 + δ0−1); i.e., up to a logarithmic factor, it is the same as that of a classical scheme on uniform meshes in the case of a regular problem. Here, δ = N−1lnN and δ0 = N0−1 are the accuracies of the discrete solution in x and t, respectively.

Stability of a standard finite difference scheme for a singularly perturbed convection-diffusion equation

Stability of a standard finite difference scheme for a singularly perturbed convection-diffusion equation

PARAMETER-UNIFORM FINITE ELEMENT METHOD FOR TWO-PARAMETER SINGULARLY PERTURBED PARABOLIC REACTION-DIFFUSION PROBLEMS

In this paper, parameter-uniform numerical methods for a class of singularly perturbed one-dimensional parabolic reaction-diffusion problems with two small parameters on a rectangular domain are studied. Parameter-explicit theoretical bounds on the derivatives of the solutions are derived. The method comprises a standard implicit finite difference scheme to discretize in temporal direction on a uniform mesh by means of Rothe's method and finite element method in spatial direction on a piecewise uniform mesh of Shishkin type. The method is shown to be unconditionally stable and accurate of order O(N-2( ln N)2 + Δt). Numerical results are given to illustrate the parameter-uniform convergence of the numerical approximations.

Numerical simulation of dynamics of dislocation arrays and long-range stress fields of nonplanar dislocation arrays

We present a simulation method for the dynamics of dislocation arrays. In this numerical method, dislocation arrays are considered as continuous surfaces in three dimensions, and the level set representation is used for these dislocation array surfaces. The level set representation of the surfaces has the advantage of automatically handling the topological changes occurring during the evolution, and simple implementation using standard accurate finite difference schemes on a uniform grid. The driving force of the evolution of the dislocation array surfaces comes from both the long-range interaction of the constituent dislocations and their local curvature effect. The long-range interaction, which is expressed by a complicated integral over the whole dislocation array surface, is calculated efficiently using the fast Fourier transform (FFT) method. Simulations are performed for dislocation arrays bypassing different particles under applied stress and are compared with those of a single dislocation. The long-range nature of the stress fields of nonplanar infinite dislocation arrays is discussed, and is shown to be essentially different from that by a single dislocation. Examples also show that such long-range stress fields may contribute to the formation of inhomogeneities of dislocation distributions.

Spatial Reconstruction of Exchange Field Interactions With a Finite Difference Scheme Based on Unstructured Meshes

A finite difference scheme suitable for unstructured meshes is here applied to the spatial reconstruction of the exchange field in Landau-Lifshitz-Gilbert equation. This method determines second-order derivatives by solving a over-determined set of linear equations through norm minimization of a Taylor series expansion functional. In this way, the local character of the exchange field computation typical of standard finite difference schemes is preserved, but similarly to finite element schemes irregular meshes can be handled. In this paper, the accuracy and computational efficiency of the proposed approach is tested in comparison with finite element and structured mesh based finite difference schemes.

Dielectric waveguides of arbitrary cross sectional shape

Numerical guided mode solutions of arbitrary cross sectional shaped waveguides are obtained using a finite difference (FD) technique. The standard FD scheme is appropriately modified to capture all discontinuities, due to the change of the refractive index, across the waveguides’ interfaces taking into account the shape of each interface at the same time. The method is applied to the vector Helmholtz equation formulated to describe the electric or magnetic fields in the waveguide (one field is retrieved from the other through Maxwell’s equations). Computational cost is kept to a minimum by exploiting sparse matrix algebra. The waveguides under study have arbitrary cross sectional shape and arbitrary refractive index profile.

Analysis on two approaches for high order accuracy finite difference computation

We analyze two approaches for enhancing the accuracy of the standard second order finite difference schemes in solving one dimensional elliptic partial differential equations. These are the fourth order compact difference scheme and the fourth order scheme based on the Richardson extrapolation techniques. We study the truncation errors of these approaches and comment on their regularity requirements and computational costs. We present numerical experiments to demonstrate the validity of our analysis.

New code for equilibriums and quasiequilibrium initial data of compact objects

We present a new code, named COCAL - Compact Object CALculator, for the computation of equilibriums and quasiequilibrium initial data sets of single or binary compact objects of all kinds. In the cocal code, those solutions are calculated on one or multiple spherical coordinate patches covering the initial hypersurface up to the asymptotic region. The numerical method used to solve field equations written in elliptic form is an adaptation of self-consistent field iterations in which Green's integral formula is computed using multipole expansions and standard finite difference schemes. We extended the method so that it can be used on a computational domain with excised regions for a black hole and a binary companion. Green's functions are constructed for various types of boundary conditions imposed at the surface of the excised regions for black holes. The numerical methods used in cocal are chosen to make the code simpler than any other recent initial data codes, accepting the second order accuracy for the finite difference schemes. We perform convergence tests for time symmetric single black hole data on a single coordinate patch, and binary black hole data on multiple patches. Then, we apply the code to obtain spatially conformally flat binary black hole initial data using boundary conditions including the one based on the existence of equilibrium apparent horizons.

New Second‐Order Finite Difference Scheme for the Problem of Contaminant in Groundwater Flow

We develop a new efficient second‐order finite difference scheme for two‐dimensional problem of contaminant in groundwater flow. Theoretical analysis shows that the scheme is second‐order convergence in the L2 norm and is unconditionally stable. Numerical results demonstrate that the error measures of the second‐order scheme are several times smaller than those of the standard implicit finite difference scheme.

A Positivity‐Preserving Numerical Scheme for Nonlinear Option Pricing Models

A positivity‐preserving numerical method for nonlinear Black‐Scholes models is developed in this paper. The numerical method is based on a nonstandard approximation of the second partial derivative. The scheme is not only unconditionally stable and positive, but also allows us to solve the discrete equation explicitly. Monotone properties are studied in order to avoid unwanted oscillations of the numerical solution. The numerical results for European put option and European butterfly spread are compared to the standard finite difference scheme. It turns out that the proposed scheme is efficient and reliable.

Solving eigenvalue problems on curved surfaces using the Closest Point Method

Eigenvalue problems are fundamental to mathematics and science. We present a simple algorithm for determining eigenvalues and eigenfunctions of the Laplace–Beltrami operator on rather general curved surfaces. Our algorithm, which is based on the Closest Point Method, relies on an embedding of the surface in a higher-dimensional space, where standard Cartesian finite difference and interpolation schemes can be easily applied. We show that there is a one-to-one correspondence between a problem defined in the embedding space and the original surface problem. For open surfaces, we present a simple way to impose Dirichlet and Neumann boundary conditions while maintaining second-order accuracy. Convergence studies and a series of examples demonstrate the effectiveness and generality of our approach.

Efficient computational methods for singular free boundary problems using smoothing variable substitutions

We study a class of singular free boundary problems for the degenerate m-Laplacian. Taking into account the behavior of the solution in the neighborhood of the singular points, a variable substitution is introduced, which makes the solution smooth in all the domain. Then, a standard finite difference scheme is used to discretize the problem. The numerical results suggest that in this way the second order convergence of the finite difference scheme is recovered, in spite of the singularities.

A nonstandard finite difference scheme for contaminant transport with kinetic langmuir sorption

This study focuses on a contaminant transport model with Langmuir sorption under nonequilibrium conditions. The numerical instabilities of the standard finite difference schemes including the upwind method are investigated. By using the nonstandard finite difference method, a better finite difference model is constructed. The numerical simulation on a specific system configuration proves the advantages of the new finite difference model. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 767–785, 2011