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AMATH 581 ( c○J. N. Kutz) 37n=N+1Δ xn=Nn=N-1Δ yyψ mnψm(n-1)ψ (m-1)nψ m(n+1)ψ (m+1)nn=3n=2n=1m=1 m=2 m=3m=M-1 m=M m=M+1xFigure 12: Discretization stencil for solving for the streamfunction with secondorderaccurate central difference schemes. Note that ψ mn = ψ(x m ,y n ).where⎡A=⎢⎣and−4 1 0 1 1 0 0 0 0 0 0 0 1 0 0 01 −4 1 0 0 1 0 0 0 0 0 0 0 1 0 00 1 −4 1 0 0 1 0 0 0 0 0 0 0 1 01 0 1 −4 0 0 0 1 0 0 0 0 0 0 0 11 0 0 0 −4 1 0 1 1 0 0 0 0 0 0 00 1 0 0 1 −4 1 0 0 1 0 0 0 0 0 00 0 1 0 0 1 −4 1 0 0 1 0 0 0 0 00 0 0 1 1 0 1 −4 0 0 0 1 0 0 0 00 0 0 0 1 0 0 0 −4 1 0 1 1 0 0 00 0 0 0 0 1 0 0 1 −4 1 0 0 1 0 00 0 0 0 0 0 1 0 0 1 −4 1 0 0 1 00 0 0 0 0 0 0 1 1 0 1 −4 0 0 0 11 0 0 0 0 0 0 0 1 0 0 0 −4 1 0 10 1 0 0 0 0 0 0 0 1 0 0 1 −4 1 00 0 1 0 0 0 0 0 0 0 1 0 0 1 −4 10 0 0 1 0 0 0 0 0 0 0 1 1 0 1 −4⎤⎥⎦(2.1.13)ψ =(ψ 11 ψ 12 ψ 13 ψ 14 ψ 21 ψ 22 ψ 23 ψ 24 ψ 31 ψ 32 ψ 33 ψ 34 ψ 41 ψ 42 ψ 43 ψ 44 ) T(2.1.14a)