Consider the 1D linear advection diffusion PDE for the unknown u(t,x) for 0 0,0) = gL(t); ux(t>0,1) = gR(t,1) (a) What is the type of

Consider the 1D linear advection diffusion PDE for the unknown u(t,x) for 0 <= x <=1 and t>= 0 ut + aux + huxx = 0 where h and a are a positive constants, with the following initial (IC) and boundary conditions (BC) u(0,x) = g0(x) u(t>0,0) = gL(t); ux(t>0,1) = gR(t,1) (a)

What is the type of PDE? (2 marks) (b) Draw the computational domain and indicate for each boundary the type of BC adopted for this example (6 marks) (c) Formulate an implicit FD method first order in time and using the most appropriate space discretization for the convective term (aux) and for the diffusion term (huxx) relative to the nature of each term indicating the order of accuracy of each term and write the discrete form of the PDE with reference to nodes i, i+1, i-1, etc. and time levels, n, n+1, etc. including the initial and boundary conditions (15 marks) (d) Write the resulting linearised system of equations for a spatial mesh with x = 0.33333 and solving for the instants of time t0 and t1 with t =0.1 (13 marks) (e) Draw the stencil of the resulting method