This problem explores how one has to be careful when choosing which finite difference rules to use whern discretizing differential equations. Consider an example of the classical convection-diffusion transport equation, which describes the transport of a fluid due to convection and diffusion: u(0) 0 u(1) = 0 Depending on the so-called Peclet number Pe-ad , where is the spacing between nodes in the finite difference approximation, we can get spurious behavior in the numerical solution if we use a central difference method for the convection (first-order derivative) term. In particular, if Pe > 1 (convection- dominated), then a different scheme should be used Assignment (B) Develop the exact solution to the differential equation in terms of a and v. (B2) By hand, discretize the differential equation, for an arbitrary number of points N, using a central difference approximation for both terms and write down the resulting equations in a form similar to equations (9.3) and (9.4) in the notes. When the Pe > 1, this scheme will be "unstable". (B3) An alternative scheme is based on "upwinding". By hand, discretize the differential equation, for an arbitrary number of points N, using a backward difference rule for the first-order derivative and a central difference rule for the second-order derivative. Write down the resulting equations in a form similar to equations (9.3) and (9.4) in the notes (BA) Based on the previous parts, write a MATLAB .m file that performs the following (a) Take a = v-1 and N = 21 . Compute the numerical solution using both discretizations developed previously, (part B2 and B3). You may use the "" command in MATLAB to solve the resulting linear systems (b) Plot both numerical approximations and the exact solution on the same plot (c) Take -500, v-1, and N 21, Recompute the numerical solutions and plot both approximations and the analytical solution on the same plot. This problem explores how one has to be careful when choosing which finite difference rules to use whern discretizing differential equations. Consider an example of the classical convection-diffusion transport equation, which describes the transport of a fluid due to convection and diffusion: u(0) 0 u(1) = 0 Depending on the so-called Peclet number Pe-ad , where is the spacing between nodes in the finite difference approximation, we can get spurious behavior in the numerical solution if we use a central difference method for the convection (first-order derivative) term. In particular, if Pe > 1 (convection- dominated), then a different scheme should be used Assignment (B) Develop the exact solution to the differential equation in terms of a and v. (B2) By hand, discretize the differential equation, for an arbitrary number of points N, using a central difference approximation for both terms and write down the resulting equations in a form similar to equations (9.3) and (9.4) in the notes. When the Pe > 1, this scheme will be "unstable". (B3) An alternative scheme is based on "upwinding". By hand, discretize the differential equation, for an arbitrary number of points N, using a backward difference rule for the first-order derivative and a central difference rule for the second-order derivative. Write down the resulting equations in a form similar to equations (9.3) and (9.4) in the notes (BA) Based on the previous parts, write a MATLAB .m file that performs the following (a) Take a = v-1 and N = 21 . Compute the numerical solution using both discretizations developed previously, (part B2 and B3). You may use the "" command in MATLAB to solve the resulting linear systems (b) Plot both numerical approximations and the exact solution on the same plot (c) Take -500, v-1, and N 21, Recompute the numerical solutions and plot both approximations and the analytical solution on the same plot