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Abstract In this paper the authors discuss a numerical simulation problem of three-dimensional contamination treatment from nuclear waste. The mathematical model is defined by a nonlinear system of an initial–boundary problem of four partial differential equations: an elliptic equation for the fluid pressure, three convection–diffusion equations for the brine, radionuclides and heat. The pressure appears within the concentration equations and heat conduction equation, and the Darcy velocity controls the concentrations and the temperature. The pressure is solved by the conservative mixed volume element method, and the order of accuracy is improved by the Darcy velocity. The concentrations and temperature are treated by a combination of a multistep method, the upwind approximation and a mixed volume element. The partial derivatives to time variable are approximated by a multistep method. A mixed volume element and an upwind scheme are used to discretize the diffusion and the convection, respectively. The composite method can solve the convection-dominated diffusion problems well because it eliminates numerical dispersion and nonphysical oscillation and has high order computational accuracy. The mixed volume element has the local conservation of mass and energy, and it can obtain the concentrations and temperature and their adjoint vector functions simultaneously. The conservation nature plays an important role in numerical simulation of underground fluid. By the technique of a priori estimates of differential equations, we derive an optimal second order result in L 2 norm. Numerical examples are given to show the effectiveness and practicability of our approach and the composite method is testified as a powerful tool for solving the challenging benchmark problem. [ABSTRACT FROM AUTHOR]

The paper is mainly concerned with the oscillations of numerical solutions by the linear θ − methods for a kind of linear delay differential equations with positive and negative coefficients. Some conditions under which the numerical solutions are oscillatory are obtained and it is proved that oscillations of the analytical solutions are preserved by the numerical solutions under mild conditions. Numerical experiments are given to demonstrate our results. [ABSTRACT FROM AUTHOR]

In this paper, we present a new multiscale model reduction technique for the Stokes flows in heterogeneous perforated domains. The challenge in the numerical simulations of this problem lies in the fact that the solution contains many multiscale features and requires a very fine mesh to resolve all details. In order to efficiently compute the solutions, some model reductions are necessary. To obtain a reduced model, we apply the generalized multiscale finite element approach, which is a framework allowing systematic construction of reduced models. Based on this general framework, we will first construct a local snapshot space, which contains many possible multiscale features of the solution. Using the snapshot space and a local spectral problem, we identify dominant modes in the snapshot space and use them as the multiscale basis functions. Our basis functions are constructed locally with non-overlapping supports, which enhances the sparsity of the resulting linear system. In order to enforce the mass conservation, we propose a hybridized technique, and uses a Lagrange multiplier to achieve mass conservation. We will mathematically analyze the stability and the convergence of the proposed method. In addition, we will present some numerical examples to show the performance of the scheme. We show that, with a few basis functions per coarse region, one can obtain a solution with excellent accuracy. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, the two-dimensional inverse Stokes problems, governed by bi-harmonic equations, are stably solved by the modified collocation Trefftz method (MCTM). In some practical applications of the Stokes problems, part of the boundary conditions cannot be measured in advance, so the mathematical descriptions of such problems are known as the inverse Stokes problems. When numerical simulation is adopted for solutions of the inverse Stokes problems, the solutions will become extremely unstable, which means that small perturbations in the boundary conditions will result in large errors of the final results. Hence, we adopted the MCTM for stably and efficiently analyzing the inverse Stokes problems. The MCTM is one kind of boundary-type meshless methods, so the mesh generation and the numerical quadrature can be avoided. Besides, the numerical solution is expressed as a linear combination of T-complete functions modified by a characteristic length. By enforcing the satisfactions of the boundary conditions at every boundary node, a system of linear algebraic equations will be yielded. The unknown coefficients in the solution expression can be acquired by directly inverting the coefficient matrix. The numerical solutions and their derivatives can be easily obtained by linear summation. Three numerical examples are provided to demonstrate the accuracy and the stability of the proposed meshless scheme for solving the two-dimensional inverse Stokes problems. [Copyright &y& Elsevier]

The main focus of this paper is to investigate the dynamics of the competition between rural and commercial bank with in the framework of fractal–fractional Atangana–Baleanu derivative sense. Initially, we take a model from literature and then utilize the real data since 2004–2014. We estimate effectively the parameters for the competition fractal–fractional model. Then, we provide a novel numerical technique for fractal–fractional model with many graphical results. We show graphical results for data fitting with the suggested operators and show that these two operators give the same results for the data fitting but varying both the values of the fractal and fractional order we have reasonable results. The results of the data fitting with these two operators are reliable and good are shown. Moreover, we present by comparing these two operators for many values of the fractal and fractional order parameters θ and α. [ABSTRACT FROM AUTHOR]