Effects of Relative Permeability History Dependence on Two-Phase Flow in Porous Media

Hysteresis phenomena in multi-phase flow in porous media has been recognized by many researchers and widely believed to have significant effects on the flow. In an attempt to account for these effects, a theoretical model for history-dependent relative permeabilities is considered. This model is incorporated into 1-D two-phase nondiffusive flow system and the corresponding flow is predicted. Flow history is observed to have a notable impact on the saturation profile and fluids breakthrough.     

A discontinuous Galerkin method for time fractional diffusion equations with variable coefficients

We propose a piecewise-linear, time-stepping discontinuous Galerkin method to solve numerically a time fractional diffusion equation involving Caputo derivative of order μ ∈ (0, 1) with variable coefficients. For the spatial discretization, we apply the standard continuous Galerkin method of total degree ≤ 1 on each spatial mesh elements. Well-posedness of the fully discrete scheme and error analysis will be shown. For a time interval (0, T) and a spatial domain Ω, our analysis suggest that the error in \(L^{2}\left ((0,T),L^{2}({\Omega })\right )\)-norm is \(O(k^{2-\frac {\mu }{2}}+h^{2})\) (that is, short by order \(\frac {\mu }{2}\) from being optimal in time) where k denotes the maximum time step, and h is the maximum diameter of the elements of the (quasi-uniform) spatial mesh. However, our numerical experiments indicate optimal O(k² + h²) error bound in the stronger \(L^{\infty }\left ((0,T),L^{2}({\Omega })\right )\)-norm. Variable time steps are used to compensate the singularity of the continuous solution near t = 0.     

On the approximation of nonlinear singular self-adjoint second order boundary value problems

We investigate the approximation of the solutions of a class of nonlinear second order singular boundary value problems with a self-adjoint linear part. Our strategy involves two ingredients. First, we take advantage of certain boundary condition functions to obtain well behaved functions of the solutions. Second, we integrate the problem over an interval that avoids the singularity. We are able to prove a uniform convergence result for the approximate solutions. We describe how the approximation is constructed for the various values of the deficiency index associated with the differential equation. The solution of the nonlinear problem is obtained by a globally convergent iterative method.     

Long time behavior for a nonlinear fractional model

The asymptotic behavior for solutions of a weighted Cauchy-type nonlinear fractional problem is investigated. We find bounds for solutions on infinite time intervals and also provide sufficient conditions assuring decay to zero. This work improves earlier results by the same authors.     

Efficient high-order compact exponential time differencing method for space-fractional reaction-diffusion systems with nonhomogeneous boundary conditions

Numerical Algorithms - This paper introduces an efficient unconditionally stable fourth-order method for solving nonlinear space-fractional reaction-diffusion systems with nonhomogeneous Dirichlet...     

The locally extrapolated exponential splitting scheme for multi-dimensional nonlinear space-fractional Schrödinger equations

An efficient local extrapolation of the exponential operator splitting scheme is introduced to solve the multi-dimensional space-fractional nonlinear Schrödinger equations. Stability of the scheme is examined by investigating its amplification factor and by plotting the boundaries of the stability regions. Empirical convergence analysis and calculation of the local truncation error exhibit the second-order accuracy of the proposed scheme. The performance and reliability of the proposed scheme are tested by implementing it on two- and three-dimensional space-fractional nonlinear Schrödinger equations including the space-fractional Gross-Pitaevskii equation, which is used to model optical solitons in graded-index fibers.     

Asymptotic Behavior of Solutions for a Class of Fractional Integro-differential Equations

In this paper, we study the asymptotic behavior of solutions for a general class of fractional integro-differential equations. We consider the Caputo fractional derivative. Reasonable sufficient conditions under which the solutions behave like power functions at infinity are established. For this purpose, we use and generalize some well-known integral inequalities. It was found that the solutions behave like the solutions of the associated linear differential equation with zero right hand side. Our findings are supported by examples.     

The finite element-Galerkin method for singular self-adjoint differential equations

We investigate the finite element-Galerkin method for singular self-adjoint second-order differential expressions. The weak formulation of the problem involves integration by parts, which allows the use of the usual piecewise linear functions. Our analysis shows that the method produces the solution corresponding to a particular self-adjoint realization of the differential expression. We also propose two algorithms to approximate the solution of any self-adjoint realization. Numerical examples are given to illustrate the analysis as well as the proposed algorithms.