Showing papers on "Finite difference published in 2011"

Finite difference solution of mhd radiative boundary layer flow of a nanofluid past a stretching sheet

Abstract: Unsteady heat and mass flow of a nanofluid past a stretching sheet with thermal radiation in the presence of magnetic field is studied. To obtain non-similar equation, continuity, momentum, energy and concentration equations have been non-dimensionalised by usual transformation. The non-similar solutions are presented here which depends on the Magnetic parameter M respectively . The obtained equations have been solved by explicit finite difference method. The temperature and concentration profiles are discussed for the different values of the above parameters with different time steps.

Using singular values to build a subgrid-scale model for large eddy simulations

Abstract: An eddy-viscosity based, subgrid-scale model for large eddy simulations is derived from the analysis of the singular values of the resolved velocity gradient tensor. The proposed σ-model has, by construction, the property to automatically vanish as soon as the resolved field is either two-dimensional or two-component, including the pure shear and solid rotation cases. In addition, the model generates no subgrid-scale viscosity when the resolved scales are in pure axisymmetric or isotropic contraction/expansion. At last, it is shown analytically that it has the appropriate cubic behavior in the vicinity of solid boundaries without requiring any ad-hoc treatment. Results for two classical test cases (decaying isotropic turbulence and periodic channel flow) obtained from three different solvers with a variety of numerics (finite elements, finite differences, or spectral methods) are presented to illustrate the potential of this model. The results obtained with the proposed model are systematically equivalent or slightly better than the results from the Dynamic Smagorinsky model. Still, the σ-model has a low computational cost, is easy to implement, and does not require any homogeneous direction in space or time. It is thus anticipated that it has a high potential for the computation of non-homogeneous, wall-bounded flows.

On Finite-Difference Approximations and Entropy Conditions for Shocks

Abstract: Weak solutions of hyperbolic conservation laws are not uniquely determined by their initial values; an entropy condition is needed to pick out the physically relevant solution. The question arises whether finite-difference approximations converge to this particular solution. It is shown in this paper that, in the case of a single conservation law, monotone schemes, when convergent, always converge to the physically relevant solution. Numerical examples show that this is not always the case with nonmonotone schemes, such as the Lax--Wendroff scheme. 4 figures, 2 tables. (auth)

Introduction to the Finite-Difference Time-Domain (Fdtd) Method for Electromagnetics

Abstract: Introduction to the Finite-Difference Time-Domain (FDTD) Method for Electromagnetics provides a comprehensive tutorial of the most widely used method for solving Maxwell's equations -- the Finite Difference Time-Domain Method. This book is an essential guide for students, researchers, and professional engineers who want to gain a fundamental knowledge of the FDTD method. It can accompany an undergraduate or entry-level graduate course or be used for self-study. The book provides all the background required to either research or apply the FDTD method for the solution of Maxwell's equations to practical problems in engineering and science. Introduction to the Finite-Difference Time-Domain (FDTD) Method for Electromagnetics guides the reader through the foundational theory of the FDTD method starting with the one-dimensional transmission-line problem and then progressing to the solution of Maxwell's equations in three dimensions. It also provides step by step guides to modeling physical sources, lumped-circuit components, absorbing boundary conditions, perfectly matched layer absorbers, and sub-cell structures. Post processing methods such as network parameter extraction and far-field transformations are also detailed. Efficient implementations of the FDTD method in a high level language are also provided. Table of Contents: Introduction / 1D FDTD Modeling of the Transmission Line Equations / Yee Algorithm for Maxwell's Equations / Source Excitations / Absorbing Boundary Conditions / The Perfectly Matched Layer (PML) Absorbing Medium / Subcell Modeling / Post Processing

The Grünwald-Letnikov method for fractional differential equations

Abstract: This paper is devoted to the numerical treatment of fractional differential equations. Based on the Grunwald-Letnikov definition of fractional derivatives, finite difference schemes for the approximation of the solution are discussed. The main properties of these explicit and implicit methods concerning the stability, the convergence and the error behavior are studied related to linear test equations. The asymptotic stability and the absolute stability of these methods are proved. Error representations and estimates for the truncation, propagation and global error are derived. Numerical experiments are given.

A probabilistic numerical method for fully nonlinear parabolic PDEs

Abstract: We consider the probabilistic numerical scheme for fully nonlinear PDEs suggested in \cite{cstv}, and show that it can be introduced naturally as a combination of Monte Carlo and finite differences scheme without appealing to the theory of backward stochastic differential equations. Our first main result provides the convergence of the discrete-time approximation and derives a bound on the discretization error in terms of the time step. An explicit implementable scheme requires to approximate the conditional expectation operators involved in the discretization. This induces a further Monte Carlo error. Our second main result is to prove the convergence of the latter approximation scheme, and to derive an upper bound on the approximation error. Numerical experiments are performed for the approximation of the solution of the mean curvature flow equation in dimensions two and three, and for two and five-dimensional (plus time) fully-nonlinear Hamilton-Jacobi-Bellman equations a! rising in the theory of portfolio optimization in financial mathematics.

Stable P-wave modeling for reverse-time migration in tilted TI media

Abstract: We present an approach for P-wave modeling in inhomogeneous transversely isotropic media with tilted symmetry axis (TTI media), suitable for anisotropic reverse-time migration. The proposed approach is based on wave equations derived from first principles — the equations of motion and Hooke's law — under the acoustic TI approximation. Consequently, no assumptions are made about the spatial variation of medium parameters. A rotation of the stress and strain tensors to a local coordinate system, aligned with the TI-symmetry axis, makes it possible to benefit from the simple and sparse form of the TI-elastic tensor in that system. The resulting wave equations can be formulated either as a set of five first-order or as a set of two second-order partial differential equations. For the constant-density case, the second-order TTI wave equations involve mixed and nonmixed second-order spatial derivatives with respect to global, nonrotated coordinates. We propose a numerical implementation of these equations using high-order centered finite differences. To minimize modeling artifacts related to the use of centered first-derivative operators, we use discrete second-derivative operators for the nonmixed second-order spatial derivatives and repeated discrete first-derivative operators for the mixed derivatives. Such a combination of finite-difference operators leads to a stable wave propagator, provided that the operators are designed properly. In practice, stability is achieved by slightly weighting down terms that contain mixed derivatives. This has a minor, practically negligible, effect on the kinematics of wave propagation. The stability of the presented scheme in inhomogeneous TTI models with rapidly varying anisotropic symmetry axis direction is demonstrated with numerical examples.

Comparing Numerical Methods for Isothermal Magnetized Supersonic Turbulence

Abstract: Many astrophysical applications involve magnetized turbulent flows with shock waves. Ab initio star formation simulations require a robust representation of supersonic turbulence in molecular clouds on a wide range of scales imposing stringent demands on the quality of numerical algorithms. We employ simulations of supersonic super-Alfvenic turbulence decay as a benchmark test problem to assess and compare the performance of nine popular astrophysical MHD methods actively used to model star formation. The set of nine codes includes: ENZO, FLASH, KT-MHD, LL-MHD, PLUTO, PPML, RAMSES, STAGGER, and ZEUS. These applications employ a variety of numerical approaches, including both split and unsplit, finite difference and finite volume, divergence preserving and divergence cleaning, a variety of Riemann solvers, and a range of spatial reconstruction and time integration techniques. We present a comprehensive set of statistical measures designed to quantify the effects of numerical dissipation in these MHD solvers. We compare power spectra for basic fields to determine the effective spectral bandwidth of the methods and rank them based on their relative effective Reynolds numbers. We also compare numerical dissipation for solenoidal and dilatational velocity components to check for possible impacts of the numerics on small-scale density statistics. Finally, we discuss the convergence of various characteristics for the turbulence decay test and the impact of various components of numerical schemes on the accuracy of solutions. The nine codes gave qualitatively the same results, implying that they are all performing reasonably well and are useful for scientific applications. We show that the best performing codes employ a consistently high order of accuracy for spatial reconstruction of the evolved fields, transverse gradient interpolation, conservation law update step, and Lorentz force computation. The best results are achieved with divergence-free evolution of the magnetic field using the constrained transport method and using little to no explicit artificial viscosity. Codes that fall short in one or more of these areas are still useful, but they must compensate for higher numerical dissipation with higher numerical resolution. This paper is the largest, most comprehensive MHD code comparison on an application-like test problem to date. We hope this work will help developers improve their numerical algorithms while helping users to make informed choices about choosing optimal applications for their specific astrophysical problems.

Some domain decomposition algorithms for elliptic problems

Abstract: We discuss domain decomposition methods with which the often very large linear systems of algebraic equations, arising when elliptic problems are discretized by finite differences or finite elements, can be solved with the aid of exact or approximate solvers for the same equations restricted to subregions. The interaction between the subregions, to enforce appropriate continuity requirements, is handled by an iterative method, often a preconditioned conjugate gradient method. Much of the work is local and can be carried out in parallel. We first explore how ideas from structural engineering computations naturally lead to certain matrix splittings. In preparation for the detailed design and analysis of related domain decomposition methods, we then consider the Schwarz alternating algorithm, discovered in 1869. That algorithm can conveniently be expressed in terms of certain projections. We develop these ideas further and discuss an interesting additive variant of the Schwarz method. This also leads to the development of a general framework, which already has proven quite useful in the study of a variety of domain decomposition methods and certain related algorithms. We demonstrate this by developing several algorithms and by showing how their rates of convergence can be estimated. One of them is a Schwarz-type method, for which the subregions overlap, while the others are so called iterative substructuring methods, where the subregions do not overlap. Compared to previous studies of iterative substructuring methods, our proof is simpler and in one case it can be completed without using a finite element extension theorem. Such a theorem has, to our knowledge, always been used in the previous analysis in all but the very simplest cases.

Existence Results for Nonlinear Fractional Difference Equation

Abstract: This paper is concerned with the initial value problem to a nonlinear fractional difference equation with the Caputo like difference operator. By means of some fixed point theorems, global and local existence results of solutions are obtained. An example is also provided to illustrate our main result.

A review of the spectral, pseudo-spectral, finite-difference and finite-element modelling techniques for geophysical imaging

Abstract: Modelling methods are nowadays at the heart of any geophysical interpretation approach. These are heavily relied upon by imaging techniques in elastodynamics and electromagnetism, where they are crucial for the extraction of subsurface characteristics from ever larger and denser datasets. While high-frequency or one-way approximations are very powerful and efficient, they reach their limits when complex geological settings and solutions of full equations are required at finite frequencies. A review of three important formulations is carried out here: the spectral method, which is very efficient and accurate but generally restricted to simple earth structures and often layered earth structures; the pseudo-spectral, finite-difference and finite-volume methods based on strong formulation of the partial differential equations, which are easy to implement and currently represent a good compromise between accuracy, efficiency and flexibility and the continuous or discontinuous Galerkin finite-element methods that are based on the weak formulation, which lead to more accurate earth representations and therefore to more accurate solutions, although with higher computational costs and more complex use. The choice between these different approaches is still difficult and depends on the applications. Guidelines are given here through discussion of the requirements for imaging/inversion.

Convergent Finite Difference Solvers for Viscosity Solutions of the Elliptic Monge-Ampère Equation in Dimensions Two and Higher

Abstract: The elliptic Monge-Ampere 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 Monge-Ampere 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 nondifferentiable.

Obtaining Potential Field Solutions with Spherical Harmonics and Finite Differences

Abstract: Potential magnetic field solutions can be obtained based on the synoptic magnetograms of the Sun. Traditionally, a spherical harmonics decomposition of the magnetogram is used to construct the current- and divergence-free magnetic field solution. This method works reasonably well when the order of spherical harmonics is limited to be small relative to the resolution of the magnetogram, although some artifacts, such as ringing, can arise around sharp features. When the number of spherical harmonics is increased, however, using the raw magnetogram data given on a grid that is uniform in the sine of the latitude coordinate can result in inaccurate and unreliable results, especially in the polar regions close to the Sun. We discuss here two approaches that can mitigate or completely avoid these problems: (1) remeshing the magnetogram onto a grid with uniform resolution in latitude and limiting the highest order of the spherical harmonics to the anti-alias limit; (2) using an iterative finite difference algorithm to solve for the potential field. The naive and the improved numerical solutions are compared for actual magnetograms and the differences are found to be rather dramatic. We made our new Finite Difference Iterative Potential-field Solver (FDIPS) a publicly available code so that other researchers can also use it as an alternative to the spherical harmonics approach.

Discretization errors and free surface stabilization in the finite difference and marker-in-cell method for applied geodynamics: A numerical study

Abstract: The finite difference–marker-in-cell (FD-MIC) method is a popular method in thermomechanical modeling in geodynamics. Although no systematic study has investigated the numerical properties of the method, numerous applications have shown its robustness and flexibility for the study of large viscous deformations. The model setups used in geodynamics often involve large smooth variations of viscosity (e.g., temperature-dependent viscosity) as well large discontinuous variations in material properties (e.g., material interfaces). Establishing the numerical properties of the FD-MIC and showing that the scheme is convergent adds relevance to the applications studies that employ this method. In this study, we numerically investigate the discretization errors and order of accuracy of the velocity and pressure solution obtained from the FD-MIC scheme using two-dimensional analytic solutions. We show that, depending on which type of boundary condition is used, the FD-MIC scheme is a second-order accurate in space as long as the viscosity field is constant or smooth (i.e., continuous). With the introduction of a discontinuous viscosity field characterized by a viscosity jump (η*) within the control volume, the scheme becomes first-order accurate. We observed that the transition from second-order to first-order accuracy will occur with only a small increase in the viscosity contrast (η* ≈ 5). We have employed two methods for projecting the material properties from the Lagrangian markers onto the Eulerian nodes. The methods are based on the size of the interpolation volume (4-cell, 1-cell). The use of a more local interpolation scheme (1-cell) decreases the absolute velocity and pressure discretization errors. We also introduce a stabilization algorithm that damps the potential oscillations that may arise from quasi free surface calculations in numerical codes that employ the strong form of the Stokes equations. This correction term is of particular interest for topographic modeling, since the surface of the Earth is generally represented by a free surface. Including the stabilization enables physically meaningful solutions to be obtained from our simulations, even in cases where the time step value exceeds the isostatic relaxation time. We show that including the stabilization algorithm in our FD stencil does not affect the convergence properties of our scheme. In order to verify our approach, we performed time-dependent simulations of free surface Rayleigh-Taylor instability.

Scalar Wave Equation Modeling with Time-Space Domain Dispersion-Relation-Based Staggered-Grid Finite-Difference Schemes

Abstract: The staggered-grid finite-difference (SFD) method is widely used in numerical modeling of wave equations. Conventional SFD stencils for spatial derivatives are usually designed in the space domain. However, when they are used to solve wave equations, it becomes difficult to satisfy the dispersion relations exactly. Liu and Sen (2009c) proposed a new SFD scheme for one-dimensional (1D) scalar wave equation based on the time–space domain dispersion relation and plane wave theory, which is made to satisfy the exact dispersion relation. This new SFD scheme has greater accuracy and better stability than a conventional scheme under the same discretizations. In this paper, we develop this new SFD scheme further for numerical solution of 2D and 3D scalar wave equations. We demonstrate that the modeling accuracy is second order when the conventional 2 M -th-order space-domain SFD and the second order time-domain finite-difference stencils are directly used to solve the scalar wave equation. However, under the same discretization, our 1D scheme can reach 2 M -th-order accuracy and is always stable; 2D and 3D schemes can reach 2 M -th-order accuracy along 8 and 48 directions, respectively, and have better stability. The advantages of the new schemes are also demonstrated with dispersion analysis, stability analysis, and numerical modeling.

Obtaining Potential Field Solution with Spherical Harmonics and Finite Differences

Abstract: Potential magnetic field solutions can be obtained based on the synoptic magnetograms of the Sun. Traditionally, a spherical harmonics decomposition of the magnetogram is used to construct the current and divergence free magnetic field solution. This method works reasonably well when the order of spherical harmonics is limited to be small relative to the resolution of the magnetogram, although some artifacts, such as ringing, can arise around sharp features. When the number of spherical harmonics is increased, however, using the raw magnetogram data given on a grid that is uniform in the sine of the latitude coordinate can result in inaccurate and unreliable results, especially in the polar regions close to the Sun. We discuss here two approaches that can mitigate or completely avoid these problems: i) Remeshing the magnetogram onto a grid with uniform resolution in latitude, and limiting the highest order of the spherical harmonics to the anti-alias limit; ii) Using an iterative finite difference algorithm to solve for the potential field. The naive and the improved numerical solutions are compared for actual magnetograms, and the differences are found to be rather dramatic. We made our new Finite Difference Iterative Potential-field Solver (FDIPS) a publically available code, so that other researchers can also use it as an alternative to the spherical harmonics approach.