Stability and Convergence Analysis of Finite Difference Schemes for Time-Dependent Space-Fractional Diffusion Equations with Variable Diffusion Coefficients

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

“…Under the three conditions, we show the unconditional stability of the proposed CN-ADI method in discrete 2 norm and the consistency of cross perturbation terms arising from the CN-ADI method. We remark that our proof is different from that of CN-non-ADI schemes [18]. For example, we cannot use the norm of the local propagation matrix [18] to analyze the stability of the CN-ADI scheme.…”

“…We remark that our proof is different from that of CN-non-ADI schemes [18]. For example, we cannot use the norm of the local propagation matrix [18] to analyze the stability of the CN-ADI scheme. Moreover, the consistency of the cross terms in existing ADI schemes is rarely strictly discussed even if coefficients are separable functions (see, for instance, [4,5,15,19,33,34,37]).…”

“…Remark 2. Notice the regularity assumptions in Theorem 3.11. f ∈ B(D T Ω ) is commonly required in finite difference schemes for SFDEs (see, for example, [3,4,5,6,7,18,21,22,23,24,33,34]). ψ ∈ H β y (Ω) can be interpreted as the regularity of ∂ β y ψ.…”

“…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).…”

An Efficient Second-Order Convergent Scheme for One-Side Space Fractional Diffusion Equations with Variable Coefficients

“…In [16], the stability and convergence of the second-order numerical scheme for variable coefficient equations are established for α ∈ (α 0 , 2), where α 0 ≈ 1.5546 is a solution of the equation 3 3−γ − 4 × 2 3−γ + 6 = 0. In [10], a series of numerical schemes for Riesz space fractional diffusion equation have been proven to be convergent and stable. Nevertheless, the proof technique used in [10] heavily depends on the symmetry of discretization matrix of Riesz fractional derivative, which is not applicable to the OSFDE that involves the non-symmetric one-sided fractional derivatives weighted by variable coefficients.…”