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Alternating-Direction Explicit (A.D.E.) finite-difference methods make use of two approximations that are implemented for computations proceeding in alternating directions, e.g., from left to right and from right to left, with each approximation being explicit in its respective direction of computation. Stable A.D.E. schemes for solving the linear parabolic partial differential equations that model heat diffusion are wellknown, as are stable A.D.E. schemes for solving the first-order equations of fluid advection. Several of these are combined here to derive A.D.E. schemes for solving time-dependent advection-diffusion equations, and their stability characteristics are discussed. In each case, it is found that it is the advection term that limits the stability of the scheme. The most stable of the combinations presented comprises an unconditionally stable approximation for computations carried out in the direction of advection of the system, from left to right in this case, and a conditionally stable approximation for computations proceeding in the opposite direction. To illustrate the application of the methods and verify the stability conditions, they are applied to some quasi-linear one-dimensional advection-diffusion problems.

In this article, a fast algorithm based on time two-mesh (TT-M) finite element (FE) scheme, which aims at solving nonlinear problems quickly, is considered to numerically solve the nonlinear space fractional Allen-Cahn equations with smooth and non-smooth solutions. The implicit second-order θ scheme containing both implicit Crank-Nicolson scheme and second-order backward difference method is applied to time direction, a fast TT-M method is used to increase the speed of calculation, and the FE method is developed to approximate the spacial direction. The TT-M FE algorithm includes the following main computing steps: firstly, a nonlinear implicit second-order θ FE scheme on the time coarse mesh τ c is solved by a nonlinear iterative method; secondly, based on the chosen initial iterative value, a linearized FE system on time fine mesh τ < τ c is solved, where some useful coarse numerical solutions are found by the Lagrange's interpolation formula. The analysis for both stability and a priori error estimates are made in detail. Finally, three numerical examples with smooth and non-smooth solutions are provided to illustrate the computational efficiency in solving nonlinear partial differential equations, from which it is easy to find that the computing time can be saved. RL D α c,y u = 1 Fast TT-M FE algorithm combined with second-order θ scheme is used to solve the nonlinear space 1 2 , (2.5) and norm u J β S (Ω) = ( u 2 + |u| 2 J β S (Ω) ) 1 2 , (2.6)and denote by J β S (Ω)(or J β S,0 (Ω)) the closure of C ∞ (Ω)(or C ∞ 0 (Ω)) with respect to · J β S (Ω) .

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In this article, we introduce two families of novel fractional θ-methods by constructing some new generating functions to discretize the Riemann-Liouville fractional calculus operator I α with a second order convergence rate. A new fractional BT-θ method connects the fractional BDF2 (when θ = 0) with fractional trapezoidal rule (when θ = 1/2), and another novel fractional BN-θ method joins the fractional BDF2 (when θ = 0) with the second order fractional Newton-Gregory formula (when θ = 1/2). To deal with the initial singularity, correction terms are added to achieve an optimal convergence order. In addition, stability regions of different θ-methods when applied to the Abel equations of the second kind are depicted, which demonstrate the fact that the fractional θ-methods are A(ϑ)-stable. Finally, numerical experiments are implemented to verify our theoretical result on the convergence analysis.

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