Showing papers on "Finite difference method published in 2011"

Multiphase Flow and Transport Processes in the Subsurface: A Contribution to the Modeling of Hydrosystems

Abstract: 1. Introduction.- 1.1 Problem classification.- 1.2 Problem formulation and exact definition of the subject.- 1.2.1 Application of the different models.- 1.2.2 Remarks on the term model.- 1.2.3 Objective and structure of this book.- 2. Fundamental principles of conceptual modeling.- 2.1 Preliminary remarks.- 2.1.1 General remarks.- 2.1.2 Definitions and fundamental terms.- 2.2 System properties.- 2.2.1 Mass and mole fractions.- 2.2.2 Density.- 2.2.3 Viscosity.- 2.2.4 Specific enthalpy, specific internal energy.- 2.2.5 Surface tension.- 2.2.6 Specific heat capacity.- 2.3 Phase state, phase transition, phase change.- 2.3.1 Phase state.- 2.3.2 Phase transition, phase change.- 2.4 Capillarity.- 2.4.1 Microscopic capillarity.- 2.4.2 Macroscopic capillarity.- 2.4.3 Capillarity in fractures.- 2.5 Hysteresis.- 2.6 Definition of different saturations.- 2.7 Relative permeability.- 2.7.1 Permeability.- 2.7.2 Relative permeability at the micro scale.- 2.7.3 Relative permeability at the macro scale.- 2.7.4 Relative permeability-saturation relation in fractures.- 2.7.5 Fracture-matrix interaction.- 2.8 Pressure and temperature dependence of porosity.- Mathematical modeling.- 3.1 General balance equation.- 3.1.1 Preconditions and assumptions.- 3.1.2 The Reynolds transport theorem in integral form.- 3.1.3 Derivation of the general balance equation.- 3.1.4 Initial and boundary conditions.- 3.1.5 Choice of the primary variables.- 3.2 Continuity equation per phase.- 3.2.1 Time derivative.- 3.3 Momentum equation and Darcy's law.- 3.3.1 General remarks.- 3.3.2 Darcy's law of single-phase flow.- 3.3.3 Generalization of Darcy's law for multiphase flow.- 3.4 General form of the multiphase flow equation.- 3.4.1 Pressure formulation.- 3.4.2 Pressure-saturation formulation.- 3.4.3 Saturation formulation.- 3.4.4 Mathematical modeling for three-phase infiltration and remobilization processes.- 3.5 Transport equation.- 3.5.1 Basic transport equation.- 3.5.2 Transport in a multiphase system.- 3.5.3 Description of the mass transfer between phases.- 3.5.4 Multicomponent transport processes in the gas phase.- 3.6 Energy equation.- 3.7 Multiphase/multicomponent system.- 4. Numerical modeling.- 4.1 Classification.- 4.1.1 Problem and special solution methods.- 4.1.2 Fundamentals of discretization.- 4.1.3 Conservative discretization.- 4.1.4 Weighted residual method.- 4.2 Finite element and finite volume methods.- 4.2.1 Spatial discretization.- 4.2.2 Choice of element types.- 4.2.3' Galerkin finite element method.- 4.2.4 Sub domain collocation - finite volume method.- 4.2.5 Time discretization.- 4.3 Linearization of the multiphase problem.- 4.3.1 Weak nonlinearities.- 4.3.2 Strong nonlinearities.- 4.3.3 Handling of the nonlinearities.- 4.3.4 Example: Linearized two-phase equation.- 4.4 Discussion of the instationary hyperbolic (convective) transport equation.- 4.4.1 Classification of hyperbolic differential equations.- 4.4.2 A linear hyperbolic transport equation.- 4.4.3 A quasilinear hyperbolic transport equation - Buckley-Levereit equation.- 4.4.4 Analytical solutions for the Buckley-Lev ereit problem.- 4.5 Special discretization methods.- 4.5.1 Motivation.- 4.5.2 Upwind method - finite difference method.- 4.5.3 Explicit upwind method of first order - Fully Upwind.- 4.5.4 Multidimensional upwind method of first order.- 4.5.5 Explicit upwind method of higher order - TVD techniques.- 4.5.6 Implicit upwind method of first order - Fully Upwind.- 4.5.7 Petrov-Galerkin finite element method.- 4.5.8 Additional remarks on conservative discretization.- 4.5.9 Flux-corrected method.- 4.5.10 Mixed-hybrid finite element methods.- 5. Comparison of the different discretization methods.- 5.1 Discretization.- 5.1.1 Finite element Galerkin method.- 5.1.2 Sub domain collocation finite volume method (box method).- 5.2 Boundedness principle - discussion of a monotonic solution.- 5.3 Comparative study of the different methods in homogeneous porous media.- 5.3.1 Multiphase flow without capillary pressure effects - Buckley-Lev ereit problem.- 5.3.2 Multiphase flow with capillary pressure effects - McWhorter problem.- 5.4 Heterogeneity effects.- 5.5 Comparative study of the methods for flow in heterogeneous porous media.- 5.6 Five-spot waterflood problem.- 6. Test problems - applications.- 6.1 DNAPL-Infiltration.- 6.2 LNAPL-Infiltration.- 6.3 Non-isothermal multiphase/multicomponent flow.- 6.3.1 Heat pipe.- 6.3.2 Study of bench-scale experiments.- 7. Final remarks.

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.

An Energy Stable and Convergent Finite-Difference Scheme for the Modified Phase Field Crystal Equation

Abstract: We present an unconditionally energy stable finite difference scheme for the Modified Phase Field Crystal equation, a generalized damped wave equation for which the usual Phase Field Crystal equation is a special degenerate case. The method is based on a convex splitting of a discrete pseudoenergy and is semi-implicit. The equation at the implicit time level is nonlinear but represents the gradient of a strictly convex function and is thus uniquely solvable, regardless of time step-size. We present a local-in-time error estimate that ensures the pointwise convergence of the scheme.

Coordinated scheduling of electricity and natural gas infrastructures with a transient model for natural gas flow

Abstract: This paper focuses on transient characteristics of natural gas flow in the coordinated scheduling of security-constrained electricity and natural gas infrastructures. The paper takes into account the slow transient process in the natural gas transmission systems. Considering their transient characteristics, natural gas transmission systems are modeled as a set of partial differential equations (PDEs) and algebraic equations. An implicit finite difference method is applied to approximate PDEs by difference equations. The coordinated scheduling of electricity and natural gas systems is described as a bi-level programming formulation from the independent system operator’s viewpoint. The objective of the upper-level problem is to minimize the operating cost of electric power systems while the natural gas scheduling optimization problem is nested within the lower-level problem. Numerical examples are presented to verify the effectiveness of the proposed solution and to compare the solutions for steady-state and transient models of natural gas transmission systems. V C 2011 American Institute of Physics. [doi:10.1063/1.3600761]

A stable TTI reverse time migration and its implementation

Abstract: Modeling and reverse time migration based on the tilted transverse isotropic (TTI) acoustic wave equation suffers from instability in media of general inhomogeniety, especially in areas where the tilt abruptly changes. We develop a stable TTI acoustic wave equation implementation based on the original elastic anisotropic wave equation. We, specifically, derive a vertical transversely isotropic wave system of equations that is equivalent to their elastic counterpart and introduce the self-adjoint differential operators in rotated coordinates to stabilize the TTI acoustic wave equations. Compared to the conventional formulations, the new system of equations does not add numerical complexity; a stable solution can be found by either a pseudospectral method or a high-order explicit finite difference scheme. We demonstrate by examples that our method provides stable and high-quality TTI reverse time migration images.

Re-initialization Free Level Set Evolution via Reaction Diffusion

Abstract: This paper presents a novel reaction-diffusion (RD) method for implicit active contours, which is completely free of the costly re-initialization procedure in level set evolution (LSE). A diffusion term is introduced into LSE, resulting in a RD-LSE equation, to which a piecewise constant solution can be derived. In order to have a stable numerical solution of the RD based LSE, we propose a two-step splitting method (TSSM) to iteratively solve the RD-LSE equation: first iterating the LSE equation, and then solving the diffusion equation. The second step regularizes the level set function obtained in the first step to ensure stability, and thus the complex and costly re-initialization procedure is completely eliminated from LSE. By successfully applying diffusion to LSE, the RD-LSE model is stable by means of the simple finite difference method, which is very easy to implement. The proposed RD method can be generalized to solve the LSE for both variational level set method and PDE-based level set method. The RD-LSE method shows very good performance on boundary anti-leakage, and it can be readily extended to high dimensional level set method. The extensive and promising experimental results on synthetic and real images validate the effectiveness of the proposed RD-LSE approach.

A weighted finite difference method for the fractional diffusion equation based on the Riemann-Liouville derivative

Abstract: A one dimensional fractional diffusion model with the Riemann-Liouville fractional derivative is studied. First, a second order discretization for this derivative is presented and then an unconditionally stable weighted average finite difference method is derived. The stability of this scheme is established by von Neumann analysis. Some numerical results are shown, which demonstrate the efficiency and convergence of the method. Additionally, some physical properties of this fractional diffusion system are simulated, which further confirm the effectiveness of our method.

A High-Order Unifying Discontinuous Formulation for the Navier-Stokes Equations on 3D Mixed Grids

Abstract: The newly developed unifying discontinuous formulation named the correction pro- cedure via reconstruction (CPR) for conservation laws is extended to solve the Navier-Stokes equations for 3D mixed grids. In the current development, tetrahedrons and triangular prisms are considered. The CPR method can unify several popular high order methods including the dis- continuous Galerkin and the spectral volume methods into a more efficient differential form. By selecting the solution points to coincide with the flux points, solution reconstruction can be com- pletely avoided. Accuracy studies confirmed that the optimal order of accuracy can be achieved with the method. Several benchmark test cases are computed by solving the Euler and compress- ible Navier-Stokes equations to demonstrate its performance.

Recent advances in Serre-Green Naghdi modelling for wave transformation, breaking and runup processes

Abstract: To describe the strongly nonlinear dynamics of waves propagating in the final stages of shoaling and in the surf and swash zones, fully nonlinear models are required. The ability of the Serre or Green Naghdi (S-GN) equations to reproduce this nonlinear processes is reviewed. Two high-order methods for solving S-GN equations, based on Finite Volume approaches, are presented. The first one is based on a quasi-conservative form of the S-GN equations, and the second on a hybrid Finite Volume/Finite Difference method. We show the ability of these two approaches to accurately simulate nonlinear shoaling, breaking and runup processes.

Time-dependent fractional advection-diffusion equations by an implicit MLS meshless method

Abstract: Recently, many new applications in engineering and science are governed by a series of fractional partial differential equations (FPDEs). Unlike the normal partial differential equations (PDEs), the differential order in a FPDE is with a fractional order, which will lead to new challenges for numerical simulation, because most existing numerical simulation techniques are developed for the PDE with an integer differential order. The current dominant numerical method for FPDEs is Finite Difference Method (FDM), which is usually difficult to handle a complex problem domain, and also hard to use irregular nodal distribution. This paper aims to develop an implicit meshless approach based on the moving least squares (MLS) approximation for numerical simulation of fractional advection-diffusion equations (FADE), which is a typical FPDE. The discrete system of equations is obtained by using the MLS meshless shape functions and the meshless strong-forms. The stability and convergence related to the time discretization of this approach are then discussed and theoretically proven. Several numerical examples with different problem domains and different nodal distributions are used to validate and investigate accuracy and efficiency of the newly developed meshless formulation. It is concluded that the present meshless formulation is very effective for the modeling and simulation of the FADE.

Accuracy of transfer matrix approaches for solving the effective mass Schr\"{o}dinger equation

Abstract: The accuracy of different transfer matrix approaches, widely used to solve the stationary effective mass Schrodinger equation for arbitrary one-dimensional potentials, is investigated analytically and numerically. Both the case of a constant and a position dependent effective mass are considered. Comparisons with a finite difference method are also performed. Based on analytical model potentials as well as self-consistent Schrodinger-Poisson simulations of a heterostructure device, it is shown that a symmetrized transfer matrix approach yields a similar accuracy as the Airy function method at a significantly reduced numerical cost, moreover avoiding the numerical problems associated with Airy functions.

An implicit finite-difference time-stepping method for a sub-diffusion equation, with spatial discretization by finite elements

Abstract: The numerical solution for a class of sub-diffusion equations involving a parameter in the range −1 < α < 0 is studied. For the time discretization, we use an implicit finite-difference Crank–Nicolson method and show that the error is of order k2+α , where k denotes the maximum time step. A nonuniform time step is employed to compensate for the singular behaviour of the exact solution at t = 0. We also consider a fully discrete scheme obtained by applying linear finite elements in space to the proposed time-stepping scheme. We prove that the additional error is of order h2 max(1, log k−1), where h is the parameter for the space mesh. Numerical experiments on some sample problems demonstrate our theoretical result.

A Random Choice Finite-difference Scheme for Hyperbolic Conservation Laws

Abstract: In this paper, we show how to modify the random choice method of Glimm by replacing the exact solution of the Riemann problem with an appropriate finite difference approximation. Our modification resolves discontinuities as well as Glimm’s scheme, but is computationally more efficient and is easier to extend to more general situations.