Jacobi Basic Iterative Methods Intuitive Interpretation Instead of solving f we solve Wa+f, N1 t starting from an arbitrary u(a, 0 We expect u(m,t→∞)→u(a) SMA-HPC⊙2003MT Solution Methods: Iterative Techniques 5
Jacobi Basic Iterative Methods Intuitive Interpretation To solve at +f we use an inexpensive(explicit) method For instance 1- △22+1-20;+ t fi f=fir u+1=u+△t(f-Au)=(I-△tA)u+△tf SMA-HPC⊙2003MT Solution Methods: Iterative Techniques 6
Jacobi Basic Iterative Methods Intuitive Interpretation Stability dictates that 2 △t< 2 Thus, we take At as large as possible, i. e(At=h2/ 2) +1 2 A+。f +11 (a2+1+0-1+b2)for=1,m SMA-HPC⊙2003MT Solution Methods: Iterative Techniques 7
Jacobi Basic Iterative Methods Matrix Form Split A D: Diagonal A=D-L-U L: Lower triangular U: Upper triangular Au= f becomes (D-L-U)u= f Iterative method Dw+=(L+U)w”+f SMA-HPC⊙2003MT Solution Methods: Iterative Techniques 8
Jacobi Basic Iterative Methods Matrix Form uT+=D(L+U)u+D-If D(D-A)u+D- f (I-D-A)u+D-f +i=RJu+D-if RJ=(I-D- A): Jacobi iteration Matrix Da1=b2/2 +1 1 212+1+2-1+b2 f2)fori=1,…7 SMA-HPC⊙2003MT Solution Methods: Iterative Techniques 9