129,464 research outputs found

    A semi-explicit multi-step method for solving incompressible navier-stokes equations

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    The fractional step method is a technique that results in a computationally-efficient implementation of Navier–Stokes solvers. In the finite element-based models, it is often applied in conjunction with implicit time integration schemes. On the other hand, in the framework of finite difference and finite volume methods, the fractional step method had been successfully applied to obtain predictor-corrector semi-explicit methods. In the present work, we derive a scheme based on using the fractional step technique in conjunction with explicit multi-step time integration within the framework of Galerkin-type stabilized finite element methods. We show that under certain assumptions, a Runge–Kutta scheme equipped with the fractional step leads to an efficient semi-explicit method, where the pressure Poisson equation is solved only once per time step. Thus, the computational cost of the implicit step of the scheme is minimized. The numerical example solved validates the resulting scheme and provides the insights regarding its accuracy and computational efficiency.Peer ReviewedPostprint (published version

    A space-time pseudospectral discretization method for solving diffusion optimal control problems with two-sided fractional derivatives

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    We propose a direct numerical method for the solution of an optimal control problem governed by a two-side space-fractional diffusion equation. The presented method contains two main steps. In the first step, the space variable is discretized by using the Jacobi-Gauss pseudospectral discretization and, in this way, the original problem is transformed into a classical integer-order optimal control problem. The main challenge, which we faced in this step, is to derive the left and right fractional differentiation matrices. In this respect, novel techniques for derivation of these matrices are presented. In the second step, the Legendre-Gauss-Radau pseudospectral method is employed. With these two steps, the original problem is converted into a convex quadratic optimization problem, which can be solved efficiently by available methods. Our approach can be easily implemented and extended to cover fractional optimal control problems with state constraints. Five test examples are provided to demonstrate the efficiency and validity of the presented method. The results show that our method reaches the solutions with good accuracy and a low CPU time.Comment: This is a preprint of a paper whose final and definite form is with 'Journal of Vibration and Control', available from [http://journals.sagepub.com/home/jvc]. Submitted 02-June-2018; Revised 03-Sept-2018; Accepted 12-Oct-201

    Modified fractional-step methods for the Navier-Stokes equations

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    Fractional step methods integrate the Navier-Stokes equations in a non-iterative fashion. The commonly used P2 pressure correction fractional step method introduces a second order in time error into the solution as a result of the non-iterative approach. In this paper two alternative fractional step methods are examined in which the additional error is third order in time. One of the methods extends the standard P2 method to increase the accuracy of the approximate pressure included in the momentum equations, and is denoted the P3 method. The other method solves a Poisson pressure equation prior to the solution of the momentum, and is denoted the pressure method. Both alternative methods are shown to reduce the overall error and increase the efficiency as compared to the standard method

    Computationally efficient methods for solving time-variable-order time-space fractional reaction-diffusion equation

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    Fractional differential equations are becoming more widely accepted as a powerful tool in modelling anomalous diffusion, which is exhibited by various materials and processes. Recently, researchers have suggested that rather than using constant order fractional operators, some processes are more accurately modelled using fractional orders that vary with time and/or space. In this paper we develop computationally efficient techniques for solving time-variable-order time-space fractional reaction-diffusion equations (tsfrde) using the finite difference scheme. We adopt the Coimbra variable order time fractional operator and variable order fractional Laplacian operator in space where both orders are functions of time. Because the fractional operator is nonlocal, it is challenging to efficiently deal with its long range dependence when using classical numerical techniques to solve such equations. The novelty of our method is that the numerical solution of the time-variable-order tsfrde is written in terms of a matrix function vector product at each time step. This product is approximated efficiently by the Lanczos method, which is a powerful iterative technique for approximating the action of a matrix function by projecting onto a Krylov subspace. Furthermore an adaptive preconditioner is constructed that dramatically reduces the size of the required Krylov subspaces and hence the overall computational cost. Numerical examples, including the variable-order fractional Fisher equation, are presented to demonstrate the accuracy and efficiency of the approach

    A semi-explicit multi-step method for solving incompressible navier-stokes equations

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    The fractional step method is a technique that results in a computationally-efficient implementation of Navier–Stokes solvers. In the finite element-based models, it is often applied in conjunction with implicit time integration schemes. On the other hand, in the framework of finite difference and finite volume methods, the fractional step method had been successfully applied to obtain predictor-corrector semi-explicit methods. In the present work, we derive a scheme based on using the fractional step technique in conjunction with explicit multi-step time integration within the framework of Galerkin-type stabilized finite element methods. We show that under certain assumptions, a Runge–Kutta scheme equipped with the fractional step leads to an efficient semi-explicit method, where the pressure Poisson equation is solved only once per time step. Thus, the computational cost of the implicit step of the scheme is minimized. The numerical example solved validates the resulting scheme and provides the insights regarding its accuracy and computational efficienc

    Fractional gradient methods via ψ-Hilfer derivative

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    Motivated by the increasing of practical applications in fractional calculus, we study the classical gradient method under the perspective of the ψ\psi-Hilfer derivative. This allows us to cover in our study several definitions of fractional derivatives that are found in the literature. The convergence of the ψ\psi-Hilfer continuous fractional gradient method is studied both for strongly and non-strongly convex cases. Using a series representation of the target function, we develop an algorithm for the ψ\psi-Hilfer fractional order gradient method. The numerical method obtained by truncating higher-order terms was tested and analyzed using benchmark functions. Considering variable order differentiation and optimizing the step size, the ψ\psi-Hilfer fractional gradient method shows better results in terms of speed and accuracy. Our results generalize previous works in the literature.publishe
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