[16 MARKS TOTAL] QUESTION 3 Background-numbering schemes for 2D PDES The 2D heat equation for the temperature q in an rectangular domain is given by: 1829 = a at + ax y) (3) where, a is a constant thermal diffusivity. Suppose that this is defined over the domain 0 x1 and 0x2, shown in the left side of the following figure. The boundary conditions are aq/ax=0 at the left and right boundaries (insulating) q qe at the lower boundary q qr at the top boundary. The grey circles are boundary points where the values are either known, OR a gradient condition is specified. The white circles represent interior grid points. The grid spacing is assumed to be uniform in x and y and is equal to h for both. In this question, you will be asked to write finite difference equations for several different points in the grid. You might be asked to write the equation using grid indices OR node numbers - pay careful attention to what the question is asking. y 908 91.8 92.08 93.8 94.8 941 942 943 944 945 90,7 0 0 0 0 9.36 90.6 O 90.5 994 931 9.26 O O 921 O 0 0 O 903 916 0 0 40,2 91.2 92,2 93,2 942 911 912 913 914 915 99,1 911 921 931 0 94,1 96 97 98 99 919 90,0 91.0 920 93,0 94,0 x 92 94 95 Figure 1. Computational grid indicating grid indices (left) and node numbering (right) IMPORTANT: The boundary nodes ARE INCLUDED in the numbering scheme. The numbering scheme is "row major" where the rows here are radial lines (i.e., fixed y). To be clear Node 1 has grid indices (0,0), Node 2 has indices (1,0), node 3 indices (2,0), etc. as shown in the right figure. For the rest of this question, YOU MUST USE the Backwards in time, centered in space (BTCS) scheme to discretise the equation. Keep the grid spacing as h - don't substitute the numerica value. Finally, any gradient (Neumann) BCs must be formulated using centered differences. Q3a Write (by hand) an anonymous MATLAB function that returns the node number nn given grid indices (j,k) for this numbering scheme, i.e. nn = @(j,k) (your code goes here) Q3b Write the generic finite difference equation for this PDE at node (j,k) using the BTCS discretisation (use grid indices (j,k) NOT node numbers when writing the unknowns). Put all unknown values of on the LHS and all known values on the RHS. What order are the error terms in At and h?