1: uk-1 Algorithm 1 Algorithm for computing u k+1 + previous solution 2: uk + current solution 3: 4: for each grid point [j,i] do 5: if is a boundary point then k+1 6: 0 "j, i 7: else hit Au,i 9: j, i -u + 2us, i +8tl, 'j, i 10: end if 11: end for u 8: k-1 Eqution (number) (2) (3) Ottu(t, x, y) Au(t, x, y) = 0, Ortuk (x, y) = uk-1(x, y) 2uk(x, y) + uk+1(x, y) St2 uk+1(x, y) = -4k-1(%, y) + 2uk (x, y) +st? A uk(r, y). Auia +uki u;i1 +u+1,i+ukin 8.x2 (4) 4u), i + u'91,1 j,i+1 (5) 3. Assuming dx = dy, derive the expression for computing the discrete Laplacian at the [j,i] grid point (Alg. 1, Line 8), given in Eq. 5. Hint: You will need to apply the centered difference from Eq. 3 to each spatial dimension. 4. How might you iterate over each grid point? (Alg 1, Line 4) 1: uk-1 Algorithm 1 Algorithm for computing u k+1 + previous solution 2: uk + current solution 3: 4: for each grid point [j,i] do 5: if is a boundary point then k+1 6: 0 "j, i 7: else hit Au,i 9: j, i -u + 2us, i +8tl, 'j, i 10: end if 11: end for u 8: k-1 Eqution (number) (2) (3) Ottu(t, x, y) Au(t, x, y) = 0, Ortuk (x, y) = uk-1(x, y) 2uk(x, y) + uk+1(x, y) St2 uk+1(x, y) = -4k-1(%, y) + 2uk (x, y) +st? A uk(r, y). Auia +uki u;i1 +u+1,i+ukin 8.x2 (4) 4u), i + u'91,1 j,i+1 (5) 3. Assuming dx = dy, derive the expression for computing the discrete Laplacian at the [j,i] grid point (Alg. 1, Line 8), given in Eq. 5. Hint: You will need to apply the centered difference from Eq. 3 to each spatial dimension. 4. How might you iterate over each grid point? (Alg 1, Line 4)