Homework 07

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Exercise 1: Characteristics and Boundary Conditions

Consider the advection equation:

for $x \in (0,1),\ t > 0$ and $a \in \mathbb{R} \setminus \{0\}$ with initial condition $u(x,0) = u_0(x)$.

1. Sketch the characteristics in the $(x, t)$ plane.

2. Explain why a boundary condition is required at $x=0$ but not at $x=1$.

3. What happens if you impose a boundary condition at $x=1$?

Exercise 2: First-Order Hyperbolic System and Incoming Characteristics

Consider the 1D wave equation in first-order form:

for $x \in (0,1),\ t > 0$ with initial data $u(x,0), v(x,0)$.

1. Diagonalize the system using the change of variables $w^\pm = u \pm v$.

2. Determine the characteristic speeds of $w^+$ and $w^-$.

3. Based on the characteristic directions, determine:

   • how many boundary conditions are needed,

   • at which boundaries they must be imposed,

   • and for which variables.

Exercise 3: Energy Conservation in the Wave Equation

Let $u(x,t)$ satisfy the wave equation on $(0,L)$:

with smooth initial data $u(x,0) = u_0(x),\ u_t(x,0) = v_0(x)$ and smooth boundary conditions.

1. Derive the energy identity:

2. For which of the following boundary conditions is the energy conserved?

   • Dirichlet: $u(0,t) = u(L,t) = 0$

   • Neumann: $u_x(0,t) = u_x(L,t) = 0$

   • Mixed: $u(0,t) = 0, \ u_x(L,t) = 0$

3. Discuss how boundary conditions affect the stability and physical interpretation of the solution.