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AMATH 581 ( c○J. N. Kutz) 35and ∇ 2 = ∂x 2 +∂2 y is the two dimensional Laplacian. Note that this equation hasboth an advection component (hyperbolic) from [ψ, ω] andadiffusioncomponent(parabolic) from ν∇ 2 ω.Wewillassumethatwearegiventheinitialvalueof the vorticityω(x, y, t =0)=ω 0 (x, y) . (2.1.3)Additionally, we will proceed to solve this problem with periodic boundaryconditions. This gives the following set of boundary conditionsω(−L, y, t) =ω(L, y, t)ω(x, −L, t) =ω(x, L, t)ψ(−L, y, t) =ψ(L, y, t)ψ(x, −L, t) =ψ(x, L, t)(2.1.4a)(2.1.4b)(2.1.4c)(2.1.4d)where we are solving on the computational domain x ∈ [−L, L] andy ∈ [−L, L].Basic Algorithm StructureBefore discretizing the governing partial differential equation, it is important toclarify what the basic solution procedure will be. Two physical quantities needto be solved as functions of time:ψ(x, y, t) streamfunction (2.1.5a)ω(x, y, t) vorticity. (2.1.5b)We are given the initial vorticity ω 0 (x, y) andperiodicboundaryconditions.The solution procedure is as follows:1. Elliptic Solve: Solve the elliptic problem ∇ 2 ψ = ω 0 to find the streamfunctionat time zero ψ(x, y, t =0)=ψ 0 .2. Time-Stepping: Given initial ω 0 and ψ 0 ,solvetheadvection-diffusionproblem by time-stepping with a given method. The Euler method isillustrated belowω(x, y, t +∆t) =ω(x, y, t)+∆t ( ν∇ 2 ω(x, y, t) − [ψ(x, y, t),ω(x, y, t)] )This advances the solution ∆t into the future.3. Loop: With the updated value of ω(x, y, ∆t), we can repeat the processby again solving for ψ(x, y, ∆t) andupdatingthevorticityonceagain.This gives the basic algorithmic structure which must be implemented in orderto generate the solution for the vorticity and streamfunction as functions oftime. It only remains to discretize the problem and solve.