A Splitting Preconditioner for Toeplitz-Like Linear Systems Arising from Fractional Diffusion Equations

Abstract

In this paper, we study Toeplitz-like linear systems arising from time-dependent onedimensional and two-dimensional Riesz space-fractional diffusion equations with variable diffusion coefficients. The coefficient matrix is a sum of a scalar identity matrix and a diagonal-times-Toeplitz matrix which allows fast matrix-vector multiplication in iterative solvers. We propose and develop a splitting preconditioner for this kind of matrix and analyze the spectra of the preconditioned matrix. Under mild conditions on variable diffusion coefficients, we show that the singular values of the preconditioned matrix are bounded above and below by positive constants which are independent of temporal and spatial discretization step-sizes. When the preconditioned conjugate gradient squared method is employed to solve such preconditioned linear systems, the method converges linearly within an iteration number independent of the discretization step-sizes. Numerical examples are given to illustrate the theoretical results and demonstrate that the performance of the proposed preconditioner is better than other tested solvers.

On Minimization of Upper Bound for the Convergence Rate of the QHSS Iteration Method

“…and h x = h y = 2 M t +1 for different M t and/or N t ; see [31]. Tables 1, 2, 3 and 4 and h x = h y = 2 M t +1 , for different M t and/or N t .…”

Crank--Nicolson Alternative Direction Implicit Method for Space-Fractional Diffusion Equations with Nonseparable Coefficients

“…Therefore, instead of solving the large linear system (2.8) straightforwardly, we only need to solve at each time level 2M x M y -many one-dimensional linear systems of the following form: where M = M x or M y , DS = Q n j S x or K n i S y and y ∈ R M ×1 denotes some known vector. Actually, there has been a fast solver proposed in [16] analysis (see Theorems 3.11). Thus, we employ the fast solver proposed in [16] to solve (4.1).…”

“…Actually, there has been a fast solver proposed in [16] analysis (see Theorems 3.11). Thus, we employ the fast solver proposed in [16] to solve (4.1). And then, solving the N -many two-dimensional linear systems for N M Let M = M x = M y , h = h x = h y = 2/(M + 1) for some positive integer M .…”

“…Introduction. Consider a two-dimensional initial-boundary value problem of space-fractional diffusion equation (SFDE) [4,16,18]: where 0 <č ≤ d(x, y, t), w(x, y, t) ≤ĉ for some positive constantsč andĉ, Ω = (x L , x R ) × (y D , y U ), ∂Ω denotes boundary of Ω,Ω = Ω ∪ ∂Ω, the source term f (x, y, t) and the initial condition ψ(x, y) are known functions, ∂ t u denotes the first-order temporal derivative of u, and α, β ∈ (1, 2). Here, the Riesz fractional derivatives [30] are defined by (∂ α x u)(x, y, t) := σ(α) x L D α x + x D α x R u(x, y, t), (x, y, t) ∈ Ω × (0, T ], (∂ β y u)(x, y, t) := σ(β) y D D β y + y D β y U u(x, y, t), (x, y, t) ∈ Ω × (0, T ] with (1.4) σ(γ) := − 1 2 cos( πγ 2 ) > 0, γ ∈ (1, 2).…”

Finite Difference Method for Time-Space Fractional Advection–Diffusion Equations with Riesz Derivative

“…Finite difference methods [ 35 ], finite element methods [ 36 ], finite volume methods [ 37 ], homotopy perturbation methods [ 38 ] and spectral methods [ 39 , 40 ] are also employed to approximate the fractional ADE. Furthermore, recent advances in numerical linear algebra had a substantial impact on designing efficient methods for the solution of the resulting linear systems which are dense but whose computational cost can be essentially reduced to where N is the size of the underlying coefficient matrix (see [ 41 , 42 , 43 , 44 ] and references therein). In this article, we construct a numerical scheme for the time-space fractional ADE by transforming the fractional differential equations into equivalent Volterra integral equations.…”