QUESTION 3 Background-numbering schemes for 2D PDES [16 MARKS TOTAL] The 2D heat equation for the temperature q in an rectangular domain is given by: = 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 0 O 90,7 V 94.8 941 942 943 944 1945 O 0 0 936 0 o 99.6 90,5 99,4 0 90.3 C 931 926 921 0 O 0 916 90,2 91,2 92,2 93,2 942 0912 0913 0 914 915 90,1 911 92,1 931 0. 94.1 96 97 98 99 910 90,0 91.0 920 93,0 94,0 x 91 92 93 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 numerical 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 q on the LHS and all known values on the RHS. What order are the error terms in At and h? Q3c Write the difference equation at node (2,3) using the node-numbering scheme NOT (j,k) indices Q3d Write the difference equation at node (4,4) using the grid indices (j,k) - you will need to incorporate the Neumann BCS SHOW YOUR WORKING. Then write the same equation using the node- numbers. Q3e The system that needs to be solved to find the solution to this PDE can be written as a matrix problem. What is the size of the matrix? Write this system of finite difference equations as a matrix equation that depends on the identity matrix [I], the Laplacian matrix [L], a BC vector [BC] and vectors of the old and new unknowns y" and y+. Then write the solution for y+1 explicitly, (i.e. ensure your answer is written as 1+1 = something). SHOW ALL YOUR WORKING. Q3f Perform a von Neumann stability analysis of the BTCS scheme in Q3b and derive the stability restriction on the timestep for this method with diffusion coefficient x You will need to write the error as Ek AejAxelSkay where R, S are defined as R = 2tr Ly 2 s and S = Ly andr and s are the x and y wavenumbers of the Fourier mode you are considering. You should treat R, S the same way we treated k in the workshop for the 1D heat equation. Is the method unconditionally unstable, conditionally stable or unconditionally stable? Justify your answer and if it is conditionbally stable, write the timestep restriction.