Fast finite volume methods for space-fractional diffusion equations

A generalized-Jacobi-function spectral method for space-time fractional reaction-diffusion equations with viscosity terms

“…In order to reduce the cost of computation, Wang et al [18] figured out the inhomogeneous Dirichlet boundaryvalue problems of space-fractional diffusion equations in finite element method. The space-fractional diffusion equations in fast finite volume methods and fast difference methods have been solved in [19,20], the authors dropped require storage and computational cost from O (N 2 ) and O (N 3 ) to O (N) and O (N log N). However, although researchers have got good results using local numerical methods, the approximations of the fractional parts largely waste their global properties.…”

A space‐time spectral method for one‐dimensional time fractional convection diffusion equations

“…At an early stage, a number of researchers attempted to achieve a better convergence via the finite difference method or (discontinuous) finite element method . Wang and Jia adopted fast finite difference methods for space‐fractional diffusion equations, and they have dropped the required storage and computational cost from O ( N 2 ) and O ( N 3 ) to O ( N log N ) and O ( N ). Yao, Sun, and Wu applied fractional alternating direction implicit method for solving a class of fractional subdiffusion equations, which led to a significant improvement in convergence.…”

“…Guo, Pu, and Huang [7] proved the existence and uniqueness of the fractional Ginzburg-Landau equation, the fractional QG equation, and the fractional Landau-Lifshitz equation with energy estimate, and Kaikina [8] proved the fractional evolution PDE via constructing Green's function.Another major challenge in solving TFCDEs is improving the convergence order. At an early stage, a number of researchers attempted to achieve a better convergence via the finite difference method or (discontinuous) finite element method [9][10][11][12][13][14]. Wang [9] and Jia [10] adopted fast finite difference methods for space-fractional diffusion equations, and they have dropped the required storage and computational cost from O N 2 and O N 3 to O .…”

“…At an early stage, a number of researchers attempted to achieve a better convergence via the finite difference method or (discontinuous) finite element method [9][10][11][12][13][14]. Wang [9] and Jia [10] adopted fast finite difference methods for space-fractional diffusion equations, and they have dropped the required storage and computational cost from O N 2 and O N 3 to O . NlogN/ and O.N/.…”

Finite difference approximation of space-fractional diffusion problems: The matrix transformation method

“…The finite difference methods for the conventional diffusion problems were extended in some sense including the development of higher-order methods for the spatial discretization [1], [2] and the time integration [3], generalization of ADI methods [4], [5], construction of appropriate iterative solvers [6] and computing on non-uniform meshes [7]. On the development of the computational efficiency we refer to [5] and [7].…”