Classification of differential equations

Ordinary and partial differential equations

The first simple classification of differential equations relies on the number of independent variables:

• ordinary differential equation (ODE) is a differential equation containing one or more functions of one independent variable and the derivatives of those functions.
• partial differential equation (PDE) is a differential equation that contains beforehand unknown multivariable functions and their partial derivatives. 

We will see that the numerical methods used for simple ODE are adapted and used to solve PDE in the preferential propagation direction (usually this direction is the time arrow).

Linear Partial Differential equations

A differential equation is said to be linear if it is linear with respect to the unknown function and its  (partial) derivatives. Namely, in each term of the equation, the unknown function or its derivatives appear at its first power and they are not multiplied by each other. 

If an equation is non-linear, the standard way to solve it numerically is to linearize it. This means that where there is a product of the unknown function with one of its derivatives, one of the two terms is substituted by a known estimation term. This estimation term is generally computed throughout an iterative process. 

Example: Advection-Diffusion equation

The advection-diffusion equation

$$ \frac{\partial c}{\partial t}  = \nabla \cdot (D \nabla c) - \nabla \cdot (\mathbf{u} c) + R $$

is a linear PDE with respect to the function $c$, if $\mathbf{u}$ and $R$ known. Where: $c$ is the scalar function of interest, $D$ is the diffusivity, $\mathbf{u}$ is the velocity field and $R$ is the source term. 

Example: Navier-Stokes equation

 The incompressible convective Navier-Stokes equation

$$\rho \left( \frac{\partial \mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{u} \right) = - \nabla \bar{p} + \mu \, \nabla^2 \mathbf u + \tfrac13 \mu \, \nabla (\nabla\cdot\mathbf{u}) + \rho\mathbf{g} $$

is a non-linear PDE with respect to the velocity field $\mathbf{u}$. Where: $\rho$ is the density, $\bar p$ is the pressure (without the hydrostatic part), $\mathbf{u}$ is the velocity field, $\mu$ is the dynamical viscosity and $\mathbf{g}$ is the gravitational acceleration. 

To linearize the equation, the term $\mathbf{u} \cdot \nabla \mathbf{u}$ is replaced by $\mathbf{u}^k \cdot \nabla \mathbf{u}$, where $\mathbf{u}^k$ is the estimated value of the unknown computed ad the last $k-th$ iteration of the iterative linearization procedure. Now, since $\mathbf{u}^k$ is a vector of coefficients, the resulting equation is linear. By the way, the iterative process must be carried out until convergence ($\left\Vert\mathbf{u}^{k+1}-\mathbf{u}^k \right\Vert<\epsilon$ with $\epsilon$ small as desired).

Classification of linear PDE

A linear PDE can be classified referring to a couple of two of its independent variable $\xi$ and $\eta$. In this case is always possible to write the PDE in the following form

$$ A\frac{\partial^{2}\phi}{\partial\xi^{2}}+B\frac{\partial^{2}\phi}{\partial\xi\partial\eta}+C\frac{\partial^{2}\phi}{\partial\eta^{2}}+D\frac{\partial\phi}{\partial\xi}+E\frac{\partial\phi}{\partial\eta}+F\phi+G=0 $$

where $\phi$ is the dependent variable. We can classify these equations according to the discriminant $B^{2}-4AC$, in particular if:

• $B^{2}-4AC> 0$ the equation is hyperbolic,
• $B^{2}-4AC= 0$ the equation is parabolic,
• $B^{2}-4AC< 0$ the equation is elliptic.

 This classification is useful to determine the nature of an equation. An elliptic equation describes an isotropic diffusion process, on the contrary, a hyperbolic equation describes a phenomenon with a well-defined propagation direction. Finally, the parabolic equation describes an intermediate type of propagation. 

This mathematical characteristic of the PDEs is useful to define how the equation is going to be solved numerically:

• elliptic equation: the equation must be solved simultaneously for the pair of independent variables used for the classification. The typical example is the steady-state diffusion equation.
• parabolic or hyperbolic equation: can be solved with procedures that involve finite increments of the independent variable with respect to which the parabolic or hyperbolic character is expected (e.g. time in the problems in transitory regime). For example, it is possible to solve the equation of the two-dimensional conduction in a transient regime with three independent variables (one temporal and two spatial) as a succession of two-dimensional problems.