3 Hyperbolic PDEs
Hyperbolic PDEs usually model wave phenomena and advective transport.
3.1 Linear Hyperbolic Equations
Let \(q(x,t) \in \mathbb{R}^n\) be some property.
Then assume the following equation, where \(A \in \mathbb{R}^{mxm}\): \[ q_t + Aq_x=0 \]
3.1.1 Simple cases: \(m=1\)
This could for instance model a pressure wave, i.e. sound that propagates with a speed of sound \(c\) in one direction only. \[ q_t + uq_x = 0 \] This equation is an advection equation, i.e. it models transport of some quantity (like energy, mass, …) in a flow with velocity u.
Let \(-\infty<x<\infty\) and a constant velocity \(u=\bar{u}\), \(q(x,t)=\tilde{q}\sim(x)\). Then: \[ q(x,t)=\tilde{q}(x-\bar{u}t) \]
We therefore know an analytic solution, because the initial condition simply shifts inside the domain with the constant in space velocity \(\tilde{u}\).
3.1.2 Simple cases: \(m=2\)
This could model acoustic waves that travel in two directions. Also for this we know an analytic solution:
Let us consider a system with a pressure \(p(x,t)\), a velocity $u(x,t), a density \(\rho\) and a constant \(K\).
The system then becomes: \[ \begin{align} p_t + Ku_x &= 0\\ u_t + \frac{1}{\rho}p_x &= 0\\ \end{align} \]
We can also rewrite this into a vector equation:
\[ \begin{bmatrix} p\\ u \end{bmatrix}_t+ \begin{bmatrix} 0 & K\\ \frac{1}{\rho} & 0\\ \end{bmatrix} \begin{bmatrix} p\\ u \end{bmatrix}_x = \begin{bmatrix} 0\\ 0\\ \end{bmatrix} \]
We now perform a change of variables with \(q \rightarrow w\): \[ \begin{align} w^1 &= p + \rho c u\\ w^2 &= p - \rho c u\\ \end{align} \]
This decouples the two equations: \[ \begin{bmatrix} w^1\\ w^2 \end{bmatrix}_t+ \begin{bmatrix} \rho c & 0\\ 0 & \rho c\\ \end{bmatrix} \begin{bmatrix} w^1\\ w^2 \end{bmatrix}_x = \begin{bmatrix} 0\\ 0\\ \end{bmatrix} \]
This leaves the question: When is \(q_t + Aq_x = 0\) hyperbolic?
For \(q_t + Aq_x = 0\) to be hyperbolic, we need:
- \(A\) has real eigenvalues
- \(A\) is diagonalizable (or equivalently: \(A\) is non-defective, or \(A\) is semi-simple)
3.1.3 Integral Form
When we look at hyperbolic equations, we often look at physical conservation laws. Physical conservation laws can often nicely be represented by an integral form.
Consider an equation in differential form, where \(f(q)\) is a nonlinear function representing flux. \[ q_t + [f(q)]_x=0 \] A differential form like above would be useful when \(q\) is smooth. With many applications of these equations, we however get discontinuities - think of shocks in flow problems. An integral form can be more useful in this situation. Consider an interval in a domain, say between two points \(x_1\) and \(x_2\) in the context of a mass conservation property.
The conservation of mass can then be represented in an integral form:
\[ \frac{d}{dt}\int_{x_1}^{x_2} q(x,t)\ dx = f(q(x_1,t)) - f(q(x_2,t)) + \text{source or sink terms} \tag{3.1}\]
If the quantity \(q\) is conserved on the interval, then it only changes by the fluxes \(F_1(t)=f(q(x_1,t))\) and \(F_2(t)=f(q(x_2,t))\). We can now also understand that, given a certain velocity \(u(x,t)\), the flux can be expressed as:
\[ \text{Flux}(x,t)=u(x,t)q(x,t) \]
We can insert this into Equation 3.1: \[ \int_{x_1}^{x_2} q_t(x,t)\ dx = -\int_{x_1}^{x_2} f_x(q,t)\ dx \] Rewriting this brings us to: \[ \int_{x_1}^{x_2} q(x,t)+ f_x(q,t)\ dx = 0 \] Which is equivalent to the original differential form.
The quasilinear form of this is: \[ q_t+f'(q)q_x=0 \]
- Jacobian \(f'(q)\)
- If this Jacobian is diagonalizable and has real eigenvalues, then we are looking at a hyperbolic equation
3.1.4 The Advection Equation
The advection equation mdels some “external” property in a fluid, for example some suspended substance in a fluid flw with a velocity \(\bar{u}\).
Mathematically, advection and convection is the same. However, the two terms are used in different contexts:
- Advection
- Used when talking about movement of some “external” property, like dye, pollutant, suspended particles that are carried by a fluid.
- Convection
- Used when talking about movement of an intrinsic property of the fluid, like mass.
The advection equation defines “characteristics”, where the solution has a special structure. For such characteristics, the solution of the advection equation, the solution stays constant: \[ x(t)=x_0+\bar{u}t \]