# Computational Astrophysics
## Partial Differential Equations. 01 Generalities

---
## Eduard Larrañaga

Observatorio Astronómico Nacional\
Facultad de Ciencias\
Universidad Nacional de Colombia

---

### About this notebook

In this notebook we present some of the generalities about systems of Partial Differential Equations.

`A. Garcia. Numerical Methods for Physics. (1999). Chapter 6 - 7 `

---

## Partial Differential Equations (PDEs)

A PDE is a relation between the partial derivatives of an unknown
function and the independent variables. The order of the highest derivative sets the *order* of the PDE.

A PDE is *linear* if it is of the first degree in the dependent 
variable (i.e. the unknown function) and in its partial derivatives.

If each term of a PDE contains either the dependent variable
or one of its partial derivatives, the PDE is called *homogeneous*.
Otherwise it is *non-homogeneous*.






---
### Types of Partial Differential Equations

There are three general types of PDEs

1. hyperbolic
2. parabolic
3. elliptic

Not all PDEs fall into one of these three types, but many
PDEs used in practice do.

These classes of PDEs model different sorts
of phenomena, display different behavior, and require different numerical
techniques for their solution.

It is not always straighforward to see (to show and proof) what type
of PDE is a given PDE!!. 


### Linear Second Order Differential Equation

Consider a function $u=u(x,y)$ satisfying the linear second order differential equation

\begin{equation}
a \partial^2_{xx} u + b \partial^2_{xy} u + c \partial^2_{yy} u + d \partial_x u + e \partial_y u + f u = g\,\,,
\end{equation}

This equation is straightforwardly categorized based on the discriminant,

\begin{equation}
b^2 - 4ac \left\{ \begin{array}{lcr}
< 0 & \rightarrow & \text{elliptic},\\
= 0 & \rightarrow & \text{parabolic},\\
> 0 & \rightarrow & \text{hyperbolic}.
\end{array}\right.
\end{equation}

The names come from analogy with conic sections in the theory of
ellipses.

---
## Hyperbolic PDEs

Hyperbolic equations in physics and astrophysics
describe **dynamical** processes and systems that generally start
at some initial time $t_0=0$ with some initial conditions. Hence, the equations are then integrated in time.


The prototypical linear second-order hyperbolic equation is the
homogeneous wave equation,

\begin{equation}
c^2 \partial^2_{xx} u - \partial^2_{tt} u = 0\,\,,
\end{equation}

where $c$ is the wave speed.

---
Another  class of hyperbolic equations are the **first-order
hyperbolic systems**. In one space dimension and assuming a linear
problem, this is

\begin{equation}
\partial_t u + A \partial_x u = 0\,\,,
\end{equation}

where $u(x,t)$ is a state vector with $s$ components and $A$
is a $s \times s$ matrix.

The problem is called *hyperbolic* if
$A$ has only real eigenvalues and is diagonizable, i.e., has a complet set
of linearly independent eigenvectors so that one can construct a 
matrix

\begin{equation}
\Lambda = Q^{-1} A Q\,\,,
\end{equation}

where $\Lambda$ is diagonal and has real numbers on its diagonal. 

**Example** 

An example of these equations is the linear **advection equation**, in which the function $u=u(t,x)$ satisfies

\begin{equation}
\partial_t u + v \partial_x u = 0\,\,,
\end{equation}

which is first order, linear, and homogeneous.

--- 

Other example is given by the non-linear first-order systems. Consider the equation

\begin{equation}
\partial_t u + \partial_x F(u) = 0\,\,,
\end{equation}

where $F(u)$ is the **flux** and may or may not be non-linear in $u(t,x)$.
We can re-write this PDE in **quasi-linear** form, by introducing
the Jacobian

\begin{equation}
\bar{A} = \frac{\partial F}{\partial u}\,\,,
\end{equation}

and writing

\begin{equation}
\partial_t u + \bar{A}\partial_x u = 0\,\,.
\label{eq:pde_quasilin1}
\end{equation}

This PDE is hyperbolic if $\bar{A}$ has real eigenvalues and is
diagonizable. 

The **equations of hydrodynamics** are a key example
of a non-linear, first-order hyperbolic PDE system.

### Initial Conditions for Hyperbolic Problems

One must specify either von Neumann, Dirichlet, or Robin boundary
conditions:

1. **Dirichlet Boundary Conditions**

\begin{equation}
\begin{aligned}
u(x=0,t) &= \Phi_1(t)\,\,,\\
u(x=L,t) &= \Phi_2(t)\,\,.
\end{aligned}
\end{equation}

2. **von Neumann Boundary Conditions** 
\begin{equation}
\begin{aligned}
\partial_x u(x=0,t) &= \Psi_1(t)\,\,,\\
\partial_x u(x=L,t) &= \Psi_2(t)\,\,.
\end{aligned}
\end{equation}

Note that in a multi-dimensional problem $\partial_x$ turns into the derivative in the direction of the normal to the boundary.

3. **Robin Boundary Conditions**

$a_1, b_1, a_2, b_2$ be real numbers with $a_i \neq 0$ and $ b_i \neq 0$.

\begin{equation}
\begin{aligned}
a_1 u(x=0,t) + b_1 \partial_x u(x=0,t) &= \Psi_1(t)\,\,,\\
a_2 u(x=L,t) + b_2 \partial_x u(x=L,t) &= \Psi_2(t)\,\,.
\end{aligned}
\end{equation}

Dirichlet and von Neuman boundary conditions are recovered if either
both $a_i$ or both $b_i$ vanish.
Note that in a multi-D problem $\partial_x$ turns into
the derivative in the direction of the normal to the boundary.



---
## Parabolic PDEs

Parabolic PDEs describe processes that are slowly changing, such as
the slow diffusion of heat in a medium, sediments in ground water, and
radiation in an opaque medium. Parabolic PDEs are second order and
have generally the shape

\begin{equation}
\partial_t u - k \partial^2_{xx} u = f\,\,.
\end{equation}

### Initial Conditions for Parabolic Problems

One must specify $u(x,t=0)$ at all $x$.


### Boundary Conditions for Parabolic Problems

Dirichlet, von Neumann or Robin boundary conditions.
If the boundary conditions are independent of time, the system will
evolve towards a steady state ($\partial_t u = 0$). In this case, 
one may set $\partial_t u = 0$ for all times and treat the differential equation as an elliptic equation.

---
## Ellpitic PDEs

Elliptic equations describe systems that are static, in steady state
and/or in equilibrium. There is no time dependence. A typical elliptic
equation is the Poisson equation,

\begin{equation}
\nabla^2 u = f\,\,,
\end{equation}

which one encounters in Newtonian gravity and in electrodynamics.
$\nabla^2$ is the Laplace operator, and $f$ is a given scalar function
of position. Elliptic problems may be linear ($f$ does not depend on
$u$ or its derivatives) or non-linear ($f$ depends on $u$ or its
derivatives).


### Initial Conditions for Elliptic Problems

Do not apply, since there is no time dependence.

### Boundary Conditions for Elliptic Problems

Dirichlet, von Neumann or Robin boundary conditions.

---

---

## Numerical Methods for PDEs

There is no such thing as a general robust method for the solution of 
generic PDEs. Each type (and each sub-type) of PDE requires a different
approach. Real-life PDEs may be of mixed type or may have special 
properties that require knowledge about the underlying physics for their
successful solution.

There are three general classes of approaches to solving PDEs

### 1. Finite Difference Methods.
      
The differential operators are approximated using their finite-difference
      representation on a given grid. A sub-class of finite-difference methods,
      so-called finite-volume methods, can be used for PDEs arising from
      conservation laws (e.g., the hydrodynamics equations).

Finite difference/volume methods have polynomial convergence for
      smooth functions.
      
### 2. Finite Element Methods.

The domain is divided into cells (**elements**). The solution
      is represented as a single function (e.g., a polynomial) on each
      cell and the PDE is transformed to an algebraic problem for the matching
      conditions of the simple functions at cell interfaces.

Finite element methods can have polynomial or exponential convergence
      for smooth functions.

### 3. Spectral Methods.
The solution is represented by a linear combination of known
      functions (e.g. trigonometric functions or special
      polynomials). The PDE is transformed to a set of algebraic
      equations (or ODEs) for the amplitudes of the component
      functions. A sub-class of these methods are the collocation
      methods. In them, the solution is represented on a grid and the
      spectral decomposition of the solution in known functions is
      used to estimate to a high degree of accuracy the partial
      derivatives of the solution on the grid points.


```python

```

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Qwen/Qwen-72B
1. YES
2. YES