Q4 Consider the following partial differential equation for T = T(y.t). aT +=a ат ат ot ду ay? where c > 0 and a>0 are both constants. This PDE is to be solved numerically over a spatial domain (0,H) and a time interval [0, tend) using finite differences. Constant spacing is used both in time (At = const) and in space (Ay = const). A cell-centred discretization is used with the grid points located at y) = ( - )y, j = 0,....N + 1, where Ay and N is the number of (interior) grid points in the spatial domain (N = 4). The computational grid is illustrated in figure Q4. The following boundary conditions are to be set • At the lower wall (y = 0) a Dirichlet boundary condition is applied T(H,t) = Twat (6) where Twar(t) is a known function. • At the upper boundary (y = H) a Robin boundary condition is applied ат T(H.t) +12 = F(t) "ун where is a constant and F(t) is a known function. In the following, use a consistent time-layer for the finite difference approximations of the PDE and its boundary conditions. (a) Find a finite difference representation of the PDE at an interior point y; (where j = 1,2,3 or 4) using second order accurate central differences in space and a first order accurate finite difference in time. Formulate a scheme that is implicit [6] (b) Draw the finite difference molecule for the scheme derived in step (a). [2] () Using the ghost point yo and interior points(s) discretise the boundary condition at the lower wall (y = 0). [2]
Yo Yi Y2 Y3 Y4 Y5 0 H Figure Q4: Spatial computational grid for question Q4. The filled circles indicate the locations of the interior grid points. The empty circles show the locations of the auxiliary ghost points.
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