Partial Differential Equations

We will now look at partial differential equations. There are 3 broad classes of PDEs that we will study: hyperbolic, elliptic, and parabolic PDEs. These will each require different solution methods.

See also

For more in-depth discussions of the methods used for PDEs in astrophysics, you can read my notes here: Introduction to Computational Astrophysical Hydrodynamics.

Hyperbolic PDEs

The canonical hyperbolic PDE is the wave equation:

\[\frac{\partial^2 \phi}{\partial t^2} = c^2 \frac{\partial^2 \phi}{\partial x^2}\]

The solution to this is traveling waves moving at speed \(c\) in both directions:

\[\phi(x, t) = A f_0(x - ct) + B g_0(x + ct)\]

A simpler example of this that we will look at is the linear advection equation:

\[\frac{\partial a}{\partial t} + u \frac{\partial a}{\partial x} = 0\]

The defining feature of hyperbolic PDEs is that there is a finite, real speed at which the solution changes.

In astrophysics, advection-like equations describe fluid dynamics.

Elliptic PDEs

The Poisson equation is the canonical elliptic PDE:

\[\nabla^2 \phi = f\]

Notice that there is no time variable here. Information about the solution moves at infinite speed, and the solution depends only on the boundary conditions and the source, \(f\).

In astrophysics, the Poisson equation is used to obtain the gravitational potential, via Poisson’s equation

\[\nabla^2 \Phi = 4 \pi G \rho\]

Parabolic PDEs

The heat equation is the canonical parabolic PDE:

\[\frac{\partial \phi}{\partial t} = k \frac{\partial^2 \phi}{\partial x^2}\]

As with elliptic PDEs, information about the solution is communicated instantaneously—there is no finite propagate speed like the hyperbolic case. But this is time-dependent.

Diffusion of photons or heat in stars is perhaps the main area where parabolic PDEs arise in astrophysics.