A Parallel Algorithm for the Two-Dimensional Time Fractional Diffusion Equation with Implicit Difference Method

Algorithm 1

Parallel algorithm for 2D-TFDE.

(1) init parallel environment

(2) for  all MPI processes do in parallel

(3)    get the input parameters like 𝑀_𝑥, 𝑀_𝑦, 𝑁, 𝑃_𝑥, 𝑃_𝑦, 𝜖₀.

(4)    allocate local memory 𝑢, 𝑐₁, 𝑐₂, 𝑓, Part1, 𝑣 and so forth

(5)    init variables and arrays

(6)    get process id 𝑝_𝑥,𝑝_𝑦

(7)    compute the initial condition 𝑢⁰ with 𝜙(𝑥,𝑦) and boundary condition

(8)    record time 𝑇₁

(9)    for  𝑛=0  to  𝑁−1  do

(10)     compute 𝑐₁, 𝑐₂, 𝑓 et al.

(11)     𝑣_𝑖,𝑗←𝑢^𝑛_𝑖,𝑗 with 𝐼_𝑖,𝑗

(12)     𝑓_𝑖,𝑗←𝑓(𝑥_𝑖,𝑦_𝑗,(𝑛+1)𝜏) with 𝐼_𝑖,𝑗

(13)    Part1_𝑖,𝑗←𝑢^𝑛_𝑖,𝑗+𝜏^𝛼Γ(2−𝛼)𝑓^𝑛+1_𝑖,𝑗 with 𝐼_𝑖,𝑗

(14)     for  𝑠=1  to  𝑛  do

(15)      Part1_𝑖,𝑗←Part1_𝑖,𝑗−𝑏_𝑠𝑢^{𝑛+1−𝑠}_𝑖,𝑗+𝑏_𝑠𝑢^𝑛−𝑠_𝑖,𝑗 with 𝐼_𝑖,𝑗

(16)     while  𝜖≥𝜖₀  do

(17)      𝑢^𝑛+1_𝑖,𝑗←1/(1+2𝑐₁+2𝑐₂) (𝑐₁(𝑣_𝑖+1,𝑗+𝑣_{𝑖−1,𝑗})+𝑐₂(𝑣_𝑖,𝑗+1+𝑣_{𝑖,𝑗−1})) with 𝐼_𝑖,𝑗

(18)      if  𝑝_𝑥<𝑃_𝑥  then

(19)       send right boundary to its right neighbor

(20)       receive left boundary of its right neighbor

(21)      if   𝑝_𝑦<𝑃_𝑦  then

(22)       send top boundary to its top neighbor

(23)       receive bottom boundary of its top neighbor

(24)      if  𝑝_𝑥>0  then

(25)       send left boundary to its left neighbor

(26)       receive right boundary of its left neighbor

(27)      if  𝑝_𝑦<0  then

(28)       send bottom boundary to its bottom neighbor

(29)       receive top boundary of its bottom neighbor

(30)      𝜖←max|𝑣−𝑢^𝑛+1|  with  𝐼_𝑖,𝑗

(31)     get global maximum of ϵ of all processes

(32)      𝑣_𝑖,𝑗←𝑢^𝑛+1_𝑖,𝑗  with  𝐼_𝑖,𝑗

(33)  record time 𝑇₂

(34)  output 𝑇₂−𝑇₁

(35)  stop parallel environment