822 research outputs found
A fractional spline collocation-Galerkin method for the time-fractional diffusion equation
The aim of this paper is to numerically solve a diffusion differential
problem having time derivative of fractional order. To this end we propose a
collocation-Galerkin method that uses the fractional splines as approximating
functions. The main advantage is in that the derivatives of integer and
fractional order of the fractional splines can be expressed in a closed form
that involves just the generalized finite difference operator. This allows us
to construct an accurate and efficient numerical method. Several numerical
tests showing the effectiveness of the proposed method are presented.Comment: 15 page
A fractional spline collocation method for the fractional order logistic equation
We construct a collocation method based on the fractional B-splines to solve a nonlinear differential problem that involves fractional derivative, i.e. the fractional order logistic equation. The use of the fractional B-splines allows us to express the fractional derivative of the approximating function in an analytic form. Thus, the fractional collocation method is easy to implement, accurate and efficient. Several numerical tests illustrate the efficiency of the proposed collocation method.We construct a collocation method based on the fractional B-splines to solve a nonlinear differential problem that involves fractional derivative, i.e. the fractional order logistic equation. The use of the fractional B-splines allows us to express the fractional derivative of the approximating function in an analytic form. Thus, the fractional collocation method is easy to implement, accurate and efficient. Several numerical tests illustrate the efficiency of the proposed collocation method
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An exploration of the IGA method for efficient reservoir simulation
Novel numerical methods present exciting opportunities to improve the efficiency of reservoir simulators. Because potentially significant gains to computational speed and
accuracy may be obtained, it is worthwhile explore alternative computational algorithms
for both general and case-by-case application to the discretization of the equations of porous media flow, fluid-structure interaction, and/or production. In the present
work, the fairly new concept of isogeometric analysis (IGA) is evaluated for its suitability
to reservoir simulation via direct comparison with the industry standard finite difference (FD) method and 1st order standard finite element method (SFEM). To this end, two main studies are carried out to observe IGA’s performance with regards to geometrical modeling and ability to capture steep saturation fronts. The first study explores IGA’s ability to model complex reservoir geometries, observing L2 error convergence rates under a variety of refinement schemes. The numerical experimental setup includes an 'S' shaped line sink of varying curvature from which water is produced in a 2D homogenous domain. The accompanying study simplifies the domain to 1D, but adds in multiphase physics that traditionally introduce difficulties associated with modeling of a moving saturation front. Results overall demonstrate promise for the IGA method to be a particularly effective tool in handling geometrically difficult features while also managing typically challenging numerical phenomena.Petroleum and Geosystems Engineerin
A sparse-grid isogeometric solver
Isogeometric Analysis (IGA) typically adopts tensor-product splines and NURBS
as a basis for the approximation of the solution of PDEs. In this work, we
investigate to which extent IGA solvers can benefit from the so-called
sparse-grids construction in its combination technique form, which was first
introduced in the early 90s in the context of the approximation of
high-dimensional PDEs. The tests that we report show that, in accordance to the
literature, a sparse-grid construction can indeed be useful if the solution of
the PDE at hand is sufficiently smooth. Sparse grids can also be useful in the
case of non-smooth solutions when some a-priori knowledge on the location of
the singularities of the solution can be exploited to devise suitable
non-equispaced meshes. Finally, we remark that sparse grids can be seen as a
simple way to parallelize pre-existing serial IGA solvers in a straightforward
fashion, which can be beneficial in many practical situations.Comment: updated version after revie
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