Showing papers on "Finite difference method published in 2013"
TL;DR: This paper shows that the discontinuous Galerkin collocation spectral element method with Gauss--Lobatto points (DGSEM-GL) satisfies the discrete summation-by-parts (SBP) property and can thus be classified as an SBP-SAT (simultaneous approximation term) scheme with a diagonal norm operator.
Abstract: This paper shows that the discontinuous Galerkin collocation spectral element method with Gauss--Lobatto points (DGSEM-GL) satisfies the discrete summation-by-parts (SBP) property and can thus be classified as an SBP-SAT (simultaneous approximation term) scheme with a diagonal norm operator. In the same way, SBP-SAT finite difference schemes can be interpreted as discontinuous Galerkin-type methods with a corresponding weak formulation based on an inner-product formulation common in the finite element community. This relation allows the use of matrix-vector notation (common in the SBP-SAT finite difference community) to show discrete conservation for the split operator formulation of scalar nonlinear conservation laws for DGSEM-GL and diagonal norm SBP-SAT. Based on this result, a skew-symmetric energy stable discretely conservative DGSEM-GL formulation (applicable to general diagonal norm SBP-SAT schemes) for the nonlinear Burgers equation is constructed.
324 citations
TL;DR: A comparison technique is used to derive a new Entropy Stable Weighted Essentially Non-Oscillatory (SSWENO) finite difference method, appropriate for simulations of problems with shocks.
286 citations
TL;DR: Some computationally effective numerical methods are proposed for simulating the multi-term time-fractional wave-diffusion equations and can be extended to other kinds of themulti-term fractional time-space models with fractional Laplacian.
Abstract: In this paper, the multi-term time-fractional wave-diffusion equations are considered. The multi-term time fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0,1], [1,2), [0,2), [0,3), [2,3) and [2,4), respectively. Some computationally effective numerical methods are proposed for simulating the multi-term time-fractional wave-diffusion equations. The numerical results demonstrate the effectiveness of theoretical analysis. These methods and techniques can also be extended to other kinds of the multi-term fractional time-space models with fractional Laplacian.
278 citations
TL;DR: In this paper, two finite difference/element approaches for the time-fractional subdiffusion equation with Dirichlet boundary conditions are developed, in which the time direction is approximated by the fractional linear multistep method and the space direction by the finite element method.
Abstract: In this paper, two finite difference/element approaches for the time-fractional subdiffusion equation with Dirichlet boundary conditions are developed, in which the time direction is approximated by the fractional linear multistep method and the space direction is approximated by the finite element method. The two methods are unconditionally stable and convergent of order $O(\tau^q+h^{r+1})$ in the $L^2$ norm, where $q=2-\beta$ or 2 when the analytical solution to the subdiffusion equation is sufficiently smooth, $\beta\,(0<\beta<1)$ is the order of the fractional derivative, $\tau$ and $h$ are the step sizes in time and space, respectively, and $r$ is the degree of the polynomial space. The corresponding schemes for the subdiffusion equation with Neumann boundary conditions are presented as well, where the stability and convergence are shown. Numerical examples are provided to verify the theoretical analysis. Comparisons between the algorithms derived in this paper and the existing algorithms are given, ...
268 citations
Book•
17 Dec 2013
TL;DR: In this paper, the authors present the foundations of mimetic finite difference method, inner products and reconstruction operators, as well as the diffusion problem in mixed form and in primal form.
Abstract: 1 Model elliptic problems.- 2 Foundations of mimetic finite difference method.- 3 Mimetic inner products and reconstruction operators.- 4 Mimetic discretization of bilinear forms.- 5 The diffusion problem in mixed form.- 6 The diffusion problem in primal form.- 7 Maxwells equations. 8. The Stokes problem. 9 Elasticity and plates.- 10 Other linear and nonlinear mimetic schemes.- 11 Analysis of parameters and maximum principles.- 12 Diffusion problem on generalized polyhedral meshes.
229 citations
TL;DR: In this paper, a reaction-diffusion (RD) method for implicit active contours is proposed, which is completely free of the costly reinitialization procedure in level set evolution (LSE).
Abstract: This paper presents a novel reaction-diffusion (RD) method for implicit active contours that is completely free of the costly reinitialization procedure in level set evolution (LSE). A diffusion term is introduced into LSE, resulting in an RD-LSE equation, from which a piecewise constant solution can be derived. In order to obtain a stable numerical solution from the RD-based LSE, we propose a two-step splitting method to iteratively solve the RD-LSE equation, where we first iterate the LSE equation, then solve 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 reinitialization 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 partial differential equation-based level set method. The RD-LSE method shows very good performance on boundary antileakage. The extensive and promising experimental results on synthetic and real images validate the effectiveness of the proposed RD-LSE approach.
177 citations
TL;DR: In this article, the stability and convergence of fractional finite difference methods with respect to the generalized discrete Gronwall inequality (GDFI) was analyzed. But the authors did not consider the high order methods based on convolution.
Abstract: Fractional finite difference methods are useful to solve the fractional differential equations. The aim of this article is to prove the stability and convergence of the fractional Euler method, the fractional Adams method and the high order methods based on the convolution formula by using the generalized discrete Gronwall inequality. Numerical experiments are also presented, which verify the theoretical analysis.
152 citations
TL;DR: Numerical tests revealed that if the number of points selected by DEIM algorithm reached 50, the approximation errors due to POD/DEIM and POD reduced systems have the same orders of magnitude, thus supporting the theoretical results existing in the literature.
147 citations
TL;DR: Two finite difference schemes are constructed to solve a class of initial-boundary value time fractional diffusion-wave equations based on its equivalent partial integro-differential equations and it is proved that their two schemes are convergent with first- order accuracy in temporal direction and second-order accuracy in spatial direction.
Abstract: Time fractional diffusion-wave equations are generalizations of classical diffusion and wave equations which are used in modeling practical phenomena of diffusion and wave in fluid flow, oil strata and others. In this paper we construct two finite difference schemes to solve a class of initial-boundary value time fractional diffusion-wave equations based on its equivalent partial integro-differential equations. Under the weak smoothness conditions, we prove that our two schemes are convergent with first-order accuracy in temporal direction and second-order accuracy in spatial direction. Numerical experiments are carried out to demonstrate the theoretical analysis.
125 citations
TL;DR: In this paper, the authors presented a numerical solution for the flow of a Newtonian fluid over an impermeable stretching sheet with a power law surface velocity, slip velocity and variable thickness.
Abstract: This article presents a numerical solution for the flow of a Newtonian fluid over an impermeable stretching sheet with a power law surface velocity, slip velocity and variable thickness. The flow is caused by a nonlinear stretching of a sheet. The governing partial differential equations are transformed into a nonlinear ordinary differential equation which is using appropriate boundary conditions for various physical parameters. The numerical solutions of the resulting nonlinear ODEs are found by using the efficient finite difference method (FDM). The effects of the slip parameter and the wall thickness parameter on the flow profile are presented. Moreover, the local skin friction is presented. Comparison of the obtained numerical results is made with previously published results in some special cases, and excellent agreement is noted. The results attained in this paper confirm the idea that FDM is a powerful mathematical tool and can be applied to a large class of linear and nonlinear problems arising in different fields of science and engineering.
120 citations
TL;DR: A two-dimensional mountainous mass flow dynamic procedure solver (Massflow-2D) using the MacCormack-TVD finite difference scheme is proposed and it is established that the model predictions agree well with both the analytical solution as well as the field observations.
TL;DR: A novel method is presented for the parallelization of electromagnetic pseudo-spectral solvers that requires only local FFTs and exchange of local guard cell data between neighboring regions, by taking advantage of the properties of DFTs, the linearity of Maxwell's equations and the finite speed of light.
TL;DR: This paper deals with the kinetic theory modeling crowd dynamics with the aim of showing how the dynamics at the microscale is transferred to the dynamics of collective behaviors.
Abstract: This paper deals with the kinetic theory modeling crowd dynamics with the aim of showing how the dynamics at the microscale is transferred to the dynamics of collective behaviors. The derivation of a new model is followed by a qualitative analysis of the initial value problem. Existence of solutions is proved for arbitrary large times, while simulations are developed by computational schemes based on splitting methods, where the transport equations are treated by finite difference methods for hyperbolic equations. Some preliminary reasoning toward the modeling of panic conditions is proposed.
TL;DR: In this paper, convergence theorems for these methods are proved under various assumptions on the coupling operator, and convergence results for the stationary and evolutive versions of the mean field type models are shown.
Abstract: Mean field type models describing the limiting behavior of stochastic differential games as the number of players tends to $+\infty$ have been recently introduced by Lasry and Lions. Numerical methods for the approximation of the stationary and evolutive versions of such models have been proposed by the authors in previous works. Here, convergence theorems for these methods are proved under various assumptions on the coupling operator.
Journal Article•
TL;DR: In this article, the stability and convergence of fractional finite difference methods with respect to the generalized discrete Gronwall inequality (GDFI) was analyzed. But the authors did not consider the high order methods based on convolution.
Abstract: Fractional finite difference methods are useful to solve the fractional differential equations. The aim of this article is to prove the stability and convergence of the fractional Euler method, the fractional Adams method and the high order methods based on the convolution formula by using the generalized discrete Gronwall inequality. Numerical experiments are also presented, which verify the theoretical analysis.
TL;DR: An improved lattice Boltzmann equation (LBE) method is presented to capture the interface between different phases and solve the pressure and velocity fields, which can recover the correct Cahn-Hilliard equation (CHE) and Navier-Stokes equations.
Abstract: A phase-field-based hybrid model that combines the lattice Boltzmann method with the finite difference method is proposed for simulating immiscible thermocapillary flows with variable fluid-property ratios. Using a phase field methodology, an interfacial force formula is analytically derived to model the interfacial tension force and the Marangoni stress. We present an improved lattice Boltzmann equation (LBE) method to capture the interface between different phases and solve the pressure and velocity fields, which can recover the correct Cahn-Hilliard equation (CHE) and Navier-Stokes equations. The LBE method allows not only use of variable mobility in the CHE, but also simulation of multiphase flows with high density ratio because a stable discretization scheme is used for calculating the derivative terms in forcing terms. An additional convection-diffusion equation is solved by the finite difference method for spatial discretization and the Runge-Kutta method for time marching to obtain the temperature field, which is coupled to the interfacial tension through an equation of state. The model is first validated against analytical solutions for the thermocapillary driven convection in two superimposed fluids at negligibly small Reynolds and Marangoni numbers. It is then used to simulate thermocapillary migration of a three-dimensional deformable droplet and bubble at various Marangoni numbers and density ratios, and satisfactory agreement is obtained between numerical results and theoretical predictions.
TL;DR: In this paper, a unified solution framework is presented for one-, two-or three-dimensional complex non-symmetric eigenvalue problems, respectively governing linear modal instability of incompressible fluid flows in rectangular domains having two, one or no homogeneous spatial directions.
TL;DR: In this article, the authors studied laminar, steady state flow in helically coiled tubes at a constant wall temperature, numerically and experimentally, using a finite difference method with projection algorithm using FORTRAN programming language.
TL;DR: A stable and high-order accurate finite difference method for problems in earthquake rupture dynamics in complex geometries with multiple faults using an isotropic elastic solid cut by pre-existing fault interfaces, resulting in a provably stable discretization.
Abstract: We develop a stable and high-order accurate finite difference method for problems in earthquake rupture dynamics in complex geometries with multiple faults. The bulk material is an isotropic elastic solid cut by pre-existing fault interfaces that accommodate relative motion of the material on the two sides. The fields across the interfaces are related through friction laws which depend on the sliding velocity, tractions acting on the interface, and state variables which evolve according to ordinary differential equations involving local fields.
The method is based on summation-by-parts finite difference operators with irregular geometries handled through coordinate transforms and multi-block meshes. Boundary conditions as well as block interface conditions (whether frictional or otherwise) are enforced weakly through the simultaneous approximation term method, resulting in a provably stable discretization.
The theoretical accuracy and stability results are confirmed with the method of manufactured solutions. The practical benefits of the new methodology are illustrated in a simulation of a subduction zone megathrust earthquake, a challenging application problem involving complex free-surface topography, nonplanar faults, and varying material properties.
TL;DR: It is shown how stable fully discrete high order accurate approximations of the Maxwells' equations, the elastic wave equations and the linearized Euler and Navier-Stokes equations can obtained.
TL;DR: A Symmetrical Conservative Metric Method (SCMM) is newly proposed based on the discussions of the metrics and Jacobian in FDM from geometry viewpoint by following the concept of vectorized surface and cell volume in Finite Volume Methods (FVMs).
TL;DR: In this article, the effects of complex geometries on 3D slope stability using an elastoplastic finite difference method (FDM) with a strength reduction technique were analyzed.
Abstract: This paper analyzes the effects of complex geometries on three-dimensional (3D) slope stability using an elastoplastic finite difference method (FDM) with a strength reduction technique. A series o...
TL;DR: In this paper, the authors deal with the put option pricing problems based on the time-fractional Black-Scholes equation, where the fractional derivative is a so-called modified Riemann-Liouville fractional derivatives.
Abstract: This work deals with the put option pricing problems based on the time-fractional Black-Scholes equation, where the fractional derivative is a so-called modified Riemann-Liouville fractional derivative. With the aid of symbolic calculation software, European and American put option pricing models that combine the time-fractional Black-Scholes equation with the conditions satisfied by the standard put options are numerically solved using the implicit
scheme of the finite difference method.
TL;DR: In this paper, a discretization method for the fractional-order form of Chua's system is proposed, which is an approximation for the right-hand side of the system under study.
Abstract: In this paper we are interested in the fractional-order form of Chua’s system. A discretization process will be applied to obtain its discrete version. Fixed points and their asymptotic stability are investigated. Chaotic attractor, bifurcation and chaos for different values of the fractional-order parameter are discussed. We show that the proposed discretization method is different from other discretization methods, such as predictor-corrector and Euler methods, in the sense that our method is an approximation for the right-hand side of the system under study.
01 Jan 2013
TL;DR: In this paper, several aspects of a finite difference method used to approximate the previously mentioned system of PDEs are discussed, including: existence and uniqueness properties, a priori bounds on the solutions of the discrete schemes, convergence, and algorithms for solving the resulting nonlinear systems of equations.
Abstract: Mean field type models describing the limiting behavior of stochastic differential game problems as the number of players tends to + ∞, have been recently introduced by J-M. Lasry and P-L. Lions. They may lead to systems of evolutive partial differential equations coupling a forward Bellman equation and a backward Fokker–Planck equation. The forward-backward structure is an important feature of this system, which makes it necessary to design new strategies for mathematical analysis and numerical approximation. In this survey, several aspects of a finite difference method used to approximate the previously mentioned system of PDEs are discussed, including: existence and uniqueness properties, a priori bounds on the solutions of the discrete schemes, convergence, and algorithms for solving the resulting nonlinear systems of equations. Some numerical experiments are presented. Finally, the optimal planning problem is considered, i.e. the problem in which the positions of a very large number of identical rational agents, with a common value function, evolve from a given initial spatial density to a desired target density at the final horizon time.
Posted Content•
TL;DR: In this article, the authors deal with the kinetic theory modeling of crowd dynamics with the aim of showing how the dynamics at the micro-scale is transferred to the dynamics of collective behaviors.
Abstract: This paper deals with the kinetic theory modeling of crowd dynamics with the aim of showing how the dynamics at the micro-scale is transferred to the dynamics of collective behaviors. The derivation of a new model is followed by a qualitative analysis of the initial value problem. Existence of solutions is proved for arbitrary large times, while simulations are developed by computational schemes based on splitting methods, where the transport equations treated by finite difference methods for hyperbolic equations. Some preliminary reasonings toward the modeling of panic conditions are proposed.
TL;DR: A new time-space domain dispersion-relation-based FD stencil can reach the same arbitrary even-order accuracy along all directions, and is more accurate and more stable than the conventional one for the same M.
Book•
07 Feb 2013
TL;DR: In this paper, the authors present a mathematical interpretation of pollution transport in the context of water resources management and water quality measurement in rivers and streams, and the most frequent pollutants in a river.
Abstract: Preface.- 1. Water Quality in the context of Water Resources Management.- 2. Basic notions.- 3. Mathematical interpretation of pollution transport.- 4. Fundamental expressions.- 5. Dispersion in rivers and streams.- 6. The biochemical pollution.- 7. The most frequent pollutants in a river.- 8. Temperature dependence.- 9. Application of the general differential equations.- 10. The steady-state case.- 11. Interpretation in finite terms.- 12. Progress in numerical modelling: the Finite Difference Method.- 13. The finite element method.- 14. The finite volume method.- 15. Multi-dimensional approach.- 16. Thermal Pollution.- 17. Optimisation models.- 18. Model calibration and validation.- 19. Water Quality Measurements and Uncertainty.- 20. Model reliability.- 21. Final thoughts and future trends.- Appendix.- Index.
TL;DR: In this article, an unconditionally energy stable and uniquely solvable finite difference scheme for the Cahn-Hilliard-Brinkman (CHB) system is presented, which is comprised of a CahnHilliard type diffusion equation and a generalized Brinkman equation mod- eling fluid flow.
Abstract: We present an unconditionally energy stable and uniquely solvable finite difference scheme for the Cahn-Hilliard-Brinkman (CHB) system, which is comprised of a Cahn-Hilliard-type diffusion equation and a generalized Brinkman equation mod- eling fluid flow. The CHB system is a generalizationof the Cahn-Hilliard-Stokesmodel and describes two phase very viscous flows in porous media. 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 time and space discrete l ¥ (0,T;H 1 h ) and l 2 (0,T;H 2 ) norms. We also present an efficient, practical non- linear multigrid method - comprised of a standard FAS method for the Cahn-Hilliard part, and a method based on the Vanka smoothing strategy for the Brinkman part - for solving these equations. In particular, we provide evidence that the solver has nearly optimal complexity in typical situations. The solver is applied to simulate spinodal decomposition of a viscous fluid in a porous medium, as well as to the more general problems of buoyancy- and boundary-driven flows. AMS subject classifications: 65M06, 65M12, 65M55, 76T99
TL;DR: In this paper, a partitioned approach by the coupling finite difference method (FDM) and the finite element method(FEM) is developed for simulating the interaction between free surface flow and a thin elastic plate.
Abstract: A partitioned approach by the coupling finite difference method (FDM) and the finite element method (FEM) is developed for simulating the interaction between free surface flow and a thin elastic plate. The FDM, in which the constraint interpolation profile method is applied, is used for solving the flow field in a regular fixed Cartesian grid, and the tangent of the hyperbola for interface capturing with the slope weighting scheme is used for capturing free surface. The FEM is used for solving structural deformation of the thin plate. A conservative momentum-exchange method, based on the immersed boundary method, is adopted to couple the FDM and the FEM. Background grid resolution of the thin plate in a regular fixed Cartesian grid is important to the computational accuracy by using this method. A virtual structure method is proposed to improve the background grid resolution of the thin plate. Both of the flow solver and the structural solver are carefully tested and extensive validations of the coupled FDM–FEM method are carried out on a benchmark experiment, a rolling tank sloshing with a thin elastic plate.