Problem 1. For $\phi(x)$, propose a third order finite difference (FD) discretization for $\frac{d^2 \phi}{dx^2}|_i$ using the values $\phi_{i-2}$, $\phi_{i-1}$, $\phi_i$, and $\phi_{i+1}$.
Problem 2. For a function $\phi(x, y)$, a consistent approximation for its second mixed derivative $\frac{\partial^2 \phi}{\partial x \partial y}|_{i,j}$ in terms of the four points $\phi_{i-1,j-1}$, $\phi_{i-1,j+1}$, $\phi_{i+1,j-1}$, and $\phi_{i+1,j+1}$ is given as:
$$\frac{\partial^2 \phi}{\partial x \partial y}|_{i,j} = \frac{\alpha \phi_{i-1,j-1} + \beta \phi_{i-1,j+1} + \gamma \phi_{i+1,j-1} + \delta \phi_{i+1,j+1} + O((\Delta x)^2, (\Delta y)^2)}{4 \Delta x \Delta y}.$$
If $\alpha = 1$, $\beta = -1$ and $\gamma = -1$, find $\delta$.
Problem 3. Check if the following scheme is consistent. If yes, find the order of accuracy.
$$\frac{d^2 \phi}{dx^2}|_i = \frac{-\phi_{i+3} + 4 \phi_{i+2} - 5 \phi_{i+1} + \phi_i}{(\Delta x)^2}$$
Problem 4.
(i) In the common form of the transport equation discussed in this course, the term $\nabla^2 \theta$ denotes the convection term. True or False.
(ii) What is meant by a well-posed problem? Discuss in 2-3 lines.
(iii) One-D wave equation is which type of equation? Elliptic, parabolic or hyperbolic?
Problem 5. For the unsteady 1-D heat conduction equation with left homogeneous Neumann and right Dirichlet boundary conditions (BCs):
(i) Discretise it using the forward time centre space method. Specify appropriate discretisations for the BCs too.
(ii) Specify the truncation error order. You do not need to derive it, just specify it.
(iii) Perform the Von Neumann stability analysis for the above FTCS discretisation.
(iv) Write an algorithm/pseudocode of how you would implement the method.
