Abstract: Introduction to Fractional Calculus Fractional Integrals and Derivatives Some Other Properties of Fractional Derivatives Some Other Fractional Derivatives and Extensions Physical Meanings Fractional Initial and Boundary Problems Numerical Methods for Fractional Integral and Derivatives Approximations to Fractional Integrals Approximations to Riemann-Liouville Derivatives Approximations to Caputo Derivatives Approximation to Riesz Derivatives Matrix Approach Short Memory Principle Other Approaches Numerical Methods for Fractional Ordinary Differential Equations Introduction Direct Methods Integration Methods Fractional Linear Multistep Methods Finite Difference Methods for Fractional Partial Differential Equations Introduction One-Dimensional Time-Fractional Equations One-Dimensional Space-Fractional Differential Equations One-Dimensional Time-Space Fractional Differential Equations Fractional Differential Equations in Two Space Dimensions Galerkin Finite Element Methods for Fractional Partial Differential Equations Mathematical Preliminaries Galerkin FEM for Stationary Fractional Advection Dispersion Equation Galerkin FEM for Space-Fractional Diffusion Equation Galerkin FEM for Time-Fractional Differential Equations Galerkin FEM for Time-Space Fractional Differential Equations Bibliography Index

Abstract: Free convection in a square differentially heated porous cavity filled with a nanofluid is numerically investigated. The mathematical model has been formulated in dimensionless stream function and temperature taking into account the Darcy–Boussinesq approximation. The Tiwari and Das’ nanofluid model with new more realistic empirical correlations for the physical properties of the nanofluids has been used for numerical analysis. The governing equations have been solved numerically on the basis of a second-order accurate finite difference method. The developed algorithm has been validated by direct comparisons with previously published papers and the results have been found to be in good agreement. The results have been presented in terms of the streamlines, isotherms, local, and average Nusselt numbers at left vertical wall at a wide range of key parameters.

Abstract: In this article, we develop a two-grid algorithm based on the mixed finite element (MFE) method for a nonlinear fourth-order reaction-diffusion equation with the time-fractional derivative of Caputo-type. We formulate the problem as a nonlinear fully discrete MFE system, where the time integer and fractional derivatives are approximated by finite difference methods and the spatial derivatives are approximated by the MFE method. To solve the nonlinear MFE system more efficiently, we propose a two-grid algorithm, which is composed of two steps: we first solve a nonlinear MFE system on a coarse grid by nonlinear iterations, then solve the linearized MFE system on the fine grid by Newton iteration. Numerical stability and optimal error estimate O ( k Δ 2 - α + h r + 1 + H 2 r + 2 ) in L 2 -norm are proved for our two-grid scheme, where k Δ , h and H are the time step size, coarse grid mesh size, and fine grid mesh size, respectively. We implement the two-grid algorithm, and present the numerical results justifying our theoretical error estimate. The numerical tests also show that the two-grid method is much more efficient than solving the nonlinear MFE system directly.

Abstract: We present an error analysis for an unconditionally energy stable, fully discrete finite difference scheme for the Cahn-Hilliard-Hele-Shaw equa- tion, a modified Cahn-Hilliard equation coupled with the Darcy flow law. The scheme, proposed by S. M. Wise, is based on the idea of convex splitting. In this paper, we rigorously prove first order convergence in time and second or- der convergence in space. Instead of the (discrete)L ∞ (0,T;L 2 )∩L 2 (0,T;H 2 h ) error estimate, which would represent the typical approach, we provide a dis- creteL ∞ (0,T;H 1 )∩L 2 (0,T;H 3 h )e rror estimate for the phase variable, which allows us to treat the nonlinear convection term in a straightforward way. Our convergence is unconditionalin the sense that the time stepsis in no way constrained by the mesh spacingh .T his is accomplished with the help of anL 2 (0,T;H 3 h ) bound of the numerical approximation of the phase variable. To facilitate both the stability and convergence analyses, we establish a finite difference analog of a Gagliardo-Nirenberg type inequality.

Abstract: Current study contains adaption of Haar wavelet discretization method (HWDM) for FG beams and its accuracy estimates. The convergence analysis is performed for differential equations covering a wide class of composite and nanostructures. Corresponding error bound has been derived. It has been shown that the order of convergence of the HWDM can be increased from two to four by applying Richardson extrapolation method. The theoretical estimates are validated by numerical samples considering FGM beam as a model problem. The results obtained by applying HWDM are compared with the results of finite difference method (FDM).

Abstract: We introduce numerical methods for simulating the diffusive motion of rigid bodies of arbitrary shape immersed in a viscous fluid. We parameterize the orientation of the bodies using normalized quaternions, which are numerically robust, space efficient, and easy to accumulate. We construct a system of overdamped Langevin equations in the quaternion representation that accounts for hydrodynamic effects, preserves the unit-norm constraint on the quaternion, and is time reversible with respect to the Gibbs-Boltzmann distribution at equilibrium. We introduce two schemes for temporal integration of the overdamped Langevin equations of motion, one based on the Fixman midpoint method and the other based on a random finite difference approach, both of which ensure that the correct stochastic drift term is captured in a computationally efficient way. We study several examples of rigid colloidal particles diffusing near a no-slip boundary and demonstrate the importance of the choice of tracking point on the measured translational mean square displacement (MSD). We examine the average short-time as well as the long-time quasi-two-dimensional diffusion coefficient of a rigid particle sedimented near a bottom wall due to gravity. For several particle shapes, we find a choice of tracking point that makes the MSD essentially linear with time, allowing us to estimate the long-time diffusion coefficient efficiently using a Monte Carlo method. However, in general, such a special choice of tracking point does not exist, and numerical techniques for simulating long trajectories, such as the ones we introduce here, are necessary to study diffusion on long time scales.

Abstract: In this paper, a class of multi-term time fractional advection diffusion equations (MTFADEs) is considered. By finite difference method in temporal direction and finite element method in spatial direction, two fully discrete schemes of MTFADEs with different definitions on multi-term time fractional derivative are obtained. The stability and convergence of these numerical schemes are discussed. Next, a V-cycle multigrid method is proposed to solve the resulting linear systems. The convergence of the multigrid method is investigated. Finally, some numerical examples are given for verification of our theoretical analysis.

Abstract: The Kelvin–Helmholtz instability of multi-component (more than two) incompressible and immiscible fluids is studied numerically using a phase-field model. The instability is governed by the modified Navier–Stokes equations and the multi-component convective Cahn–Hilliard equations. A finite difference method is used to discretize the governing system. To solve the equations efficiently and accurately, we employ the Chorin’s projection method for the modified Navier–Stokes equations and the recently developed practically unconditionally stable method for the multi-component Cahn–Hilliard equations. Through our model and numerical solution, we investigate the effects of surface tension, density ratio, magnitude of velocity difference, and forcing on the Kelvin–Helmholtz instability of multi-component fluids. It is shown that increasing the surface tension or the density ratio reduces the growth of the Kelvin–Helmholtz instability. And it is also observed that as the initial horizontal velocity difference gets larger, the interface rolls up more. We also found that the billow height reaches its maximum more slowly as the initial wave amplitude gets smaller. And, for the linear growth rate for the Kelvin–Helmholtz instability of two-component fluids, the simulation results agree well with the analytical results. From comparison between the numerical growth rate of two- and three-component fluids, we observe that the inclusion of extra layers can alter the growth rate for the Kelvin–Helmholtz instability. Finally, we simulate the billowing cloud formation which is a classic example of the Kelvin–Helmholtz instability and cannot be seen in binary fluids. With our multi-component method, the details of the real flow (e.g., the asymmetry in the roll-up and the self-interaction of the shear layer) are well captured.

Abstract: A two-grid block-centered finite difference method is proposed for solving the two-dimensional Darcy--Forchheimer model describing non-Darcy flow in porous media. To construct the two-grid method we modify the original nonlinear elliptic operator of Darcy--Forchheimer flow to a twice continuously differentiable one by introducing a small and positive parameter $\varepsilon$. By using the two-grid method, solving a nonlinear equation on a fine grid is reduced to solving a nonlinear equation on a coarse grid together with solving a linear equation on a fine grid. Optimal order error estimates for pressure and velocity in discrete $L^2$ norms are obtained. Some numerical examples are given to show the accuracy and efficiency of the presented method.

Abstract: A two-point boundary value problem whose highest order term is a Caputo fractional derivative of order δ∈(1, 2) is considered. Al-Refai's comparison principle is improved and modified to fit our problem. Sharp a priori bounds on derivatives of the solution u of the boundary value problem are established, showing that u″(x) may be unbounded at the interval endpoint x=0. These bounds and a discrete comparison principle are used to prove pointwise convergence of a finite difference method for the problem, where the convective term is discretized using simple upwinding to yield stability on coarse meshes for all values of δ. Numerical results are presented to illustrate the performance of the method.

Abstract: This paper presents for the first time a full three-dimensional (3-D), multilayer, and multichip thermal component model, based on finite differences, with asymmetrical power distributions for dynamic electrothermal simulation. Finite difference methods (FDMs) are used to solve the heat conduction equation in three dimensions. The thermal component model is parameterized in terms of structural and material properties so it can be readily used to develop a library of component models for any available power module. The FDM model is validated with a full analytical Fourier series-based model in two dimensions. Finally, the FDM thermal model is compared against measured data acquired from a newly developed high-speed transient coupling measurement technique. By using the device threshold voltage as a time-dependent temperature-sensitive parameter (TSP), the thermal transient of a single device, along with the thermal coupling effect among nearby devices sharing common direct bond copper (DBC) substrates, can be studied under a variety of pulsed power conditions.

Abstract: In this paper, a meshless numerical scheme is adopted for solving two-dimensional inverse Cauchy problems which are governed by second-order linear partial differential equations. In Cauchy problems, over-specified boundary conditions are imposed on portions of the boundary while on parts of boundary no boundary conditions are imposed. The application of conventional numerical methods to Cauchy problems yields highly ill-conditioned matrices. Hence, small noise added in the boundary conditions will tremendously enlarge the computational errors. The generalized finite difference method (GFDM), which is a newly developed domain-type meshless method, is adopted to solve in a stable manner the two-dimensional Cauchy problems. The GFDM can overcome time-consuming mesh generation and numerical quadrature. Besides, Cauchy problems can be solved stably and accurately by the GFDM. We present three numerical examples to validate the accuracy and the simplicity of the meshless scheme. In addition, different levels o...

Abstract: We propose two stable and one conditionally stable finite difference schemes of second-order in both time and space for the time-fractional diffusion-wave equation. In the first scheme, we apply the fractional trapezoidal rule in time and the central difference in space. We use the generalized Newton---Gregory formula in time for the second scheme and its modification for the third scheme. While the second scheme is conditionally stable, the first and the third schemes are stable. We apply the methodology to the considered equation with also linear advection---reaction terms and also obtain second-order schemes both in time and space. Numerical examples with comparisons among the proposed schemes and the existing ones verify the theoretical analysis and show that the present schemes exhibit better performances than the known ones.

Abstract: Developments in gyrokinetic particle simulation enable the gyrokinetic toroidal code (GTC) to simulate turbulent transport in tokamaks with realistic equilibrium profiles and plasma geometry, which is a critical step in the code–experiment validation process. These new developments include numerical equilibrium representation using B-splines, a new Poisson solver based on finite difference using field-aligned mesh and magnetic flux coordinates, a new zonal flow solver for general geometry, and improvements on the conventional four-point gyroaverage with nonuniform background marker loading. The gyrokinetic Poisson equation is solved in the perpendicular plane instead of the poloidal plane. Exploiting these new features, GTC is able to simulate a typical DIII-D discharge with experimental magnetic geometry and profiles. The simulated turbulent heat diffusivity and its radial profile show good agreement with other gyrokinetic codes. The newly developed nonuniform loading method provides a modified radial transport profile to that of the conventional uniform loading method.

Abstract: The numerical simulation of wave propagation in media with solid and fluid layers is essential for marine seismic exploration data analysis. The numerical methods for wave propagation that are applicable to this physical settings can be broadly classified as partitioned or monolithic: The partitioned methods use separate simulations in the fluid and solid regions and explicitly satisfy the interface conditions, whereas the monolithic methods use the same method in all the domain without any special treatment of the fluid-solid interface. Despite the accuracy of the partitioned methods, the monolithic methods are more common in practice because of their convenience. In this paper, we analyse the accuracy of several monolithic methods for wave propagation in the presence of a fluid-solid interface. The analysis is based on grid-dispersion criteria and numerical examples. The methods studied here include: the classical finite-difference method (FDM) based on the second-order displacement formulation of the elastic wave equation (DFDM), the staggered-grid finite difference method (SGFDM), the velocity-stress FDM with a standard grid (VSFDM) and the spectral-element method (SEM). We observe that among these, DFDM and the first-order SEM have a large amount of grid dispersion in the fluid region which renders them impractical for this application. On the other hand, SGFDM, VSFDM and SEM of order greater or equal to 2 yield accurate results for the body waves in the fluid and solid regions if a sufficient number of nodes per wavelength is used. All of the considered methods yield limited accuracy for the surface waves because the proper boundary conditions are not incorporated into the numerical scheme. Overall, we demonstrate both by analytic treatment and numerical experiments, that a first-order velocity-stress formulation can, in general, be used in dealing with fluid-solid interfaces without using staggered grids necessarily.

Abstract: Purpose – The purpose of this paper is to conduct a detailed investigation of the two-dimensional natural convection flow of a dusty fluid. Therefore, the incompressible boundary layer flow of a two-phase particulate suspension is investigated numerically over a semi-infinite vertical flat plate. Comprehensive flow formations of the gas and particle phases are given in the boundary layer region. Primitive variable formulation is employed to convert the nondimensional governing equations into the non-conserved form. Three important two-phase mechanisms are discussed, namely, water-metal mixture, oil-metal mixture and air-metal mixture. Design/methodology/approach – The full coupled nonlinear system of equations is solved using implicit two point finite difference method along the whole length of the plate. Findings – The authors have presented numerical solution of the dusty boundary layer problem. Solutions obtained are depicted through the characteristic quantities, such as, wall shear stress coefficient...

Abstract: Modeling electromagnetic wave propagation in the upper atmosphere is important for space weather effects, satellite communications ionospheric modification experiments, and many other applications. We propose a new methodology for solving and incorporating the current equation into the finite-difference time-domain (FDTD) form of Maxwell's equations for modeling electromagnetic wave propagation in magnetized plasma. This approach employs a version of Boris's algorithm applied to particle-in-cell plasma computational models. There are four primary advantages of this new method over previously developed three-dimensional FDTD models of electromagnetic wave propagation in magnetized plasma. Specifically, it: 1) requires less memory; 2) is more than 50% faster; 3) is easier to implement; and 4) permits the use of two different time step increments when solving the current equation versus Maxwell's equations that is useful for modeling high collisional regimes. The new algorithm is faster because it solves all the equations explicitly and there is no need to solve complicated matrix equations. Modeling of higher altitude ranges and higher frequency electromagnetic waves is much more feasible using this new method. Results of the new FDTD magnetized plasma model are provided and validated.