240 research outputs found
An advanced meshless method for time fractional diffusion equation
Recently, because of the new developments in sustainable engineering and renewable energy, which are usually governed by a series of fractional partial differential equations (FPDEs), the numerical modelling and simulation for fractional calculus are attracting more and more attention from researchers. The current dominant numerical method for modeling FPDE is Finite Difference Method (FDM), which is based on a pre-defined grid leading to inherited issues or shortcomings including difficulty in simulation of problems with the complex problem domain and in using irregularly distributed nodes. Because of its distinguished advantages, the meshless method has good potential in simulation of FPDEs. This paper aims to develop an implicit meshless collocation technique for FPDE. The discrete system of FPDEs is obtained by using the meshless shape functions and the meshless collocation formulation. The stability and convergence of this meshless approach are investigated theoretically and numerically. The numerical examples with regular and irregular nodal distributions are used to validate and investigate accuracy and efficiency of the newly developed meshless formulation. It is concluded that the present meshless formulation is very effective for the modeling and simulation of fractional partial differential equations
Differential quadrature method for space-fractional diffusion equations on 2D irregular domains
In mathematical physics, the space-fractional diffusion equations are of
particular interest in the studies of physical phenomena modelled by L\'{e}vy
processes, which are sometimes called super-diffusion equations. In this
article, we develop the differential quadrature (DQ) methods for solving the 2D
space-fractional diffusion equations on irregular domains. The methods in
presence reduce the original equation into a set of ordinary differential
equations (ODEs) by introducing valid DQ formulations to fractional directional
derivatives based on the functional values at scattered nodal points on problem
domain. The required weighted coefficients are calculated by using radial basis
functions (RBFs) as trial functions, and the resultant ODEs are discretized by
the Crank-Nicolson scheme. The main advantages of our methods lie in their
flexibility and applicability to arbitrary domains. A series of illustrated
examples are finally provided to support these points.Comment: 25 pages, 25 figures, 7 table
An alternating direction implicit spectral method for solving two dimensional multi-term time fractional mixed diffusion and diffusion-wave equations
In this paper, we consider the initial boundary value problem of the two
dimensional multi-term time fractional mixed diffusion and diffusion-wave
equations. An alternating direction implicit (ADI) spectral method is developed
based on Legendre spectral approximation in space and finite difference
discretization in time. Numerical stability and convergence of the schemes are
proved, the optimal error is , where are the
polynomial degree, time step size and the regularity of the exact solution,
respectively. We also consider the non-smooth solution case by adding some
correction terms. Numerical experiments are presented to confirm our
theoretical analysis. These techniques can be used to model diffusion and
transport of viscoelastic non-Newtonian fluids
Fast Method of Particular Solutions for Solving Partial Differential Equations
Method of particular solutions (MPS) has been implemented in many science and engineering problems but obtaining the closed-form particular solutions, the selection of the good shape parameter for various radial basis functions (RBFs) and simulation of the large-scale problems are some of the challenges which need to overcome. In this dissertation, we have used several techniques to overcome such challenges.
The closed-form particular solutions for the Matérn and Gaussian RBFs were not known yet. With the help of the symbolic computational tools, we have derived the closed-form particular solutions of the Matérn and Gaussian RBFs for the Laplace and biharmonic operators in 2D and 3D. These derived particular solutions play an important role in solving inhomogeneous problems using MPS and boundary methods such as boundary element methods or boundary meshless methods.
In this dissertation, to select the good shape parameter, various existing variable shape parameter strategies and some well-known global optimization algorithms have also been applied. These good shape parameters provide high accurate solutions in many RBFs collocation methods.
Fast method of particular solutions (FMPS) has been developed for the simulation of the large-scale problems. FMPS is based on the global version of the MPS. In this method, partial differential equations are discretized by the usual MPS and the determination of the unknown coefficients is accelerated using a fast technique. Numerical results confirm the efficiency of the proposed technique for the PDEs with a large number of computational points in both two and three dimensions. We have also solved the time fractional diffusion equations by using MPS and FMPS
A RBF-based differential quadrature method for solving two-dimensional variable-order time fractional advection-diffusion equation
The file attached to this record is the author's final peer reviewed version. The Publisher's final version can be found by following the DOI link.Numerical simulation technique of two-dimensional variable-order time fractional advection-diffusion equation is developed in this paper using radial basis function-based differential quadrature method (RBF-DQ). To the best of the authors’ knowledge, this is the first application of this method to variable-order time fractional advection-diffusion equations. For the general case of irregular geometries, the meshless local form of RBF-DQ is used and the multiquadric type of radial basis functions is selected for the computations. This approach allows one to define a reconstruction of the local radial basis functions to treat accurately both the Dirichlet and Neumann boundary conditions on the irregular boundaries. The method is validated by the well documented test examples involving variable-order fractional modelling of air pollution. The numerical results demonstrate that the proposed method provides accurate solutions fortwo-dimensional variable-order time fractional advection-diffusion equations
Accurate, Meshless Methods for Magneto-Hydrodynamics
Recently, we developed a pair of meshless finite-volume Lagrangian methods
for hydrodynamics: the 'meshless finite mass' (MFM) and 'meshless finite
volume' (MFV) methods. These capture advantages of both smoothed-particle
hydrodynamics (SPH) and adaptive mesh-refinement (AMR) schemes. Here, we extend
these to include ideal magneto-hydrodynamics (MHD). The MHD equations are
second-order consistent and conservative. We augment these with a
divergence-cleaning scheme, which maintains div*B~0 to high accuracy. We
implement these in the code GIZMO, together with a state-of-the-art
implementation of SPH MHD. In every one of a large suite of test problems, the
new methods are competitive with moving-mesh and AMR schemes using constrained
transport (CT) to ensure div*B=0. They are able to correctly capture the growth
and structure of the magneto-rotational instability (MRI), MHD turbulence, and
the launching of magnetic jets, in some cases converging more rapidly than AMR
codes. Compared to SPH, the MFM/MFV methods exhibit proper convergence at fixed
neighbor number, sharper shock capturing, and dramatically reduced noise, div*B
errors, and diffusion. Still, 'modern' SPH is able to handle most of our tests,
at the cost of much larger kernels and 'by hand' adjustment of artificial
diffusion parameters. Compared to AMR, the new meshless methods exhibit
enhanced 'grid noise' but reduced advection errors and numerical diffusion,
velocity-independent errors, and superior angular momentum conservation and
coupling to N-body gravity solvers. As a result they converge more slowly on
some problems (involving smooth, slowly-moving flows) but more rapidly on
others (involving advection or rotation). In all cases, divergence-control
beyond the popular Powell 8-wave approach is necessary, or else all methods we
consider will systematically converge to unphysical solutions.Comment: 35 pages, 39 figures. MNRAS. Updated with published version. A public
version of the GIZMO MHD code, user's guide, test problem setups, and movies
are available at http://www.tapir.caltech.edu/~phopkins/Site/GIZMO.htm
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