Question

In free surface flows the Laplace equation can be used to extend the vel- coity field out from the water into the air. For the r-component of velocity du au arauz = 0 (2) (a) Show the steps to arrive at a Jacobi approximation for solution which is **P = 2012) [+1 +-1; +B (43+1+4;-1)] (3) and, in so doing, define B. (b) What does the k index denote? (c) Consider the parabolic equation Use a FTCS explicit difference scheme to discretise (4) to provide ! (d) Assume Ar=Ay= A and At/A= 0.25. For what value of B from (a) are the Jacobi method in (a) and the FTCS scheme in (c) the same?

Answer 1

In this problem, we simply use the discretization technique to solve the whole problem.

simply, the central difference for space and forward difference for time.

part(c) that is the exact value of u(I,j) is provided at the last image as a derivation.

comment if any query.

བྱའབྱེད།༢ To reach to we discretize Jacobi opproscination your solution. above using centro diferente. Vatly-29gt Vols & Wiste-2 Uss+ Visi - 0 Ust - 2Usis+W; - 1 t Du? fuiste 22, U3, 3-1) = – རྗེ་ Oཆt , -ཏེ། ཡོད །) དཔར ད ( ནན་ 2 3 ནཱ མ་ ཆུ> ཆུག ། ། ར མ ཙེ ༈ ་ : ༢ ། , ད) ༤ གེ《 ། ༣ ) ༣. - » ལ ་ , ༡༢ ནས ( ༡ ） ༦ བ ས ཎ (༦) ར ལ ནས ༽

Index which ony denote the iteration k is the Count. ou - Bu & diu de dx dyt un - Est-2 Usists-is-sist 2 DX . tuss u DX = Dy=A - Un = at rustig + 8-1,3 + Ustit , 3-1 I – 4 Wars ] De Justus + Us 1 st US ist 1 + Usisht + (1 - 4 x Dz) Uds 0.25 stal = Dtsttu 1,37 Wagtit os For Both method to be some. 2(148) - D3 2(1+8) = 2 s (1+B²) = x², B²=1, B=1

ایک مرد 2- درمولا + دونا 2- سانا إلا - نا لا کدل2- امکان دادن به ندا 2- ردن ( الادلا احد دردنا - درد دل به مدت چی* (دادگا ؟ دونا ) = وتا- دو کیا مونگ ) او ( م . س ) دهد