Implementing Finite Volume Methods 1

 Continue on Finite Volume Methods for Elliptic Equations  Finite Volumes in Two-Dimensions  Poisson’s Equation  Boundary Conditions ◦ Neumann ◦ Dirichlet  Using Excel 2

2-Dimensional grid development same as 1D. Note that the boundaries are on the control volume surfaces. [In the pictures, dotted lines represent control volumes, solid lines represent node adjacencies/data dependencies.] 3 Control Volumes Nodes placed at centre of control volumes

 Grid Generation - Similar as One-Dimension  Use grid spacing h (=Dx) in x direction and k (= Dy) in y  Consider Poisson's Equation 4 Control Volume P S W N E e w s n h k

 For an area A enclosed by the surface S, and vector field variable v, the DIVERGENCE theorem states:  Let  The Divergence theorem can be used on the Laplace equation for the scalar field u. Or 5 S A

 Using the divergence theorem it can be shown that:  The first order derivatives can be approximated by: 6 P S W N E e w s n

 Therefore at each node we have the following algebraic equations that represent the Poisson equation. 7 P S W N E e w s n h k

 The equation on the previous slide is in the standard form for finite volumes schemes  It is normal to write Ap in this way, rather than, say  This is because, if you use other volume shapes, such as triangles, rectangles or a mixture, the same sort of results (e.g. that Ap is the sum of the other coefficients) still hold 8

 The divergence theorem gives:  Example: at north boundary 9

 Therefore at each node on the north boundary we have 10

 Similarly at each node on the east boundary, if  Similar on other boundaries, but the signs are reversed (south) (west) 11

 The divergence theorem gives:  Example: at east boundary 12 Boundary P h k

 Therefore at each node on the east boundary we have 13

 Similarly at each node on the north boundary, if 14

 At each internal node 15 P S W N E e w s n h k

 Can use transformations of source term to calculate BCs  Remove the boundary term(s) A [B] =0 where [B]=N,S,E or W  Neumann (+east or –west) (+north or –south)  Dirichlet ( u = U B ) (east or west) (north or south) 16 Boundary P h k

 Finite Difference: ◦ Approximate derivatives at each node via a difference formula derived from Taylor series expansions on a regular grid. ◦ Advantages: Easy to implement. Easy to obtain higher order derivatives. ◦ Disadvantages: Need to use structured mesh. May not be conservative.  Finite Volume: ◦ Approximate first order derivatives on the faces of each control volume. ◦ Advantages: Can use unstructured meshes. Conservative. ◦ Disadvantage: Difficult to obtain higher order derivatives. 17

 Mesh nodes are at the centre of each cell (e.g. for at )  There are no known boundary cell values (even if a boundary value for u is known – different from Finite Differences)  Work out general formula, copy to all cells and apply boundary transformations 18

 Consider the Poisson equation on the following domain:  Using h = k = 0.5, g = 2, program the FV method in Excel [class demo] 19 (0,0) (0,5) (3,2) (3,3) (7,2) (7,3) (10,0) (10,5)

 At each internal node (see slide 15)  So in general  However, may be sensible to retain g, h, k in the formula so we can modify the Excel file easily (see example) 20

 At the West boundary we have Neumann BCs  Use general formula (slide 15) but apply boundary transformations (slide 16)  but (transformation)  (transformation)  So in general on West boundary 21

 At the East boundary we have Dirichlet BCs u=50  Use general formula (slide 15) but apply boundary transformations (slide 16)  but (transformation)  (transformation)  So in general on East boundary 22

 At the East boundary we have Dirichlet BCs u =50  At the South boundary we have Neumann BCs  but (S & E transformation)  (E transformation)  (S transformation)  So for bottom right hand corner 23

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