Partial Differential Equation Symbols & Discretization

CS 493 Lecture, Dr. Lawlor

	Discrete	Continuous
Model:	Values change instantly, at clock ticks.	Values change smoothly.
Material:	Atoms	Metal
Light:	Photons	Electromagnetic waves
Written:	Computer program	System of equations
Unit:	Expression	Partial differential equation (PDE)
Domain:	Computer Science	Applied Mathematics
Benefits:	Easy to compute. Solutions are well-posed.	Easy to vary in space and time. Equations always stable.
Drawbacks:	Roundoff. Stability. Conservation. The code!	Ill-posed problems (singularities). The math!

One very common trick is "discretization": converting continuous equations to a discrete computer program. In theory this is always possible due to the limit definition of continuity, although it can be a lot of work in practice. Much less common is "continuization", recovering the continuous equations represented by a discrete program--this isn't always even possible. For example, the discrete logistic equation oscillates in a chaotic fashion that does not match any PDE.

The key trick in discretization or continuization is to use either the definition of the derivative df/dx as a limit:

lim

xf(x+

x)−f(x)

Or you can even use a finite difference, ∆f / ∆x. I usually use finite differences.

Here are some typical conversions. B and C are some arbitrary quantity.

Discrete	Finite	Continous
B = (right.z - left.z) / grid_size_x;	B=∆z/∆x	B = dz/dx (or ∂z/∂x)
C = (new.z - old.z) / dt;	C=∆z/∆t	C = dz/dt
new.z = old.z + C*dt;	∆z=C*∆t	C = dz/dt
P += V*dt;	∆P=V*∆t	dP/dt = V

So far, so good--one dimensional quantities have simple derivatives.

Discretizing the Shallow Water Wave Equations

Wikipedia lists the simplified velocity-based shallow water wave equations as:

$\begin{align} \frac{\partial u}{\partial t} - f v& = -g \frac{\partial h}{\partial x} - b u,\\[3pt] \frac{\partial v}{\partial t} + f u& = -g \frac{\partial h}{\partial y} - b v,\\[3pt] \frac{\partial h}{\partial t}& = - H \Bigl( \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} \Bigr) \end{align}$

Here:

(u,v) is the fluid's 2D velocity (meters/second). The key assumption in shallow-water work is the velocity is constant throughout the fluid column--that is, vertical velocities are small enough to ignore.
h is the current vertical deviation in fluid height from the mean (meters).
H is the mean fluid height above the bottom (meters).
b is the viscous drag coefficient (1/seconds). I prefer to stabilize the solution with artificial diffusion instead of using drag.
f is the Coriolis coefficient (1/seconds); this is zero for a non-rotating frame of reference.

Using finite differences, it's actually not terribly hard to derive these equations from first principles. Consider a 1D layer of thickness L meters, divided into cells of width ∆x meters.

Velocity to height: assume the cell to your right has a velocity difference, of ∆v meters/second.

The interface between cells has surface area H L square meters.
A flow rate of ∆v meters/second moves by ∆v ∆t meters in one timestep of size ∆t seconds.
The volume flowing across the interface is thus ∆v ∆t H L cubic meters.
This volume needs to go somewhere (conservation of volume), so your cell gets taller.
The top of your cell has area ∆x L square meters.
Adding a height of ∆h to your cell thus has a volume of ∆h ∆x L cubic meters.
Since the volume flowing in equals the volume caused height change, ∆h ∆x L = - ∆v ∆t H L.
L cancels out.
∆h / ∆t = - ∆v / ∆x H
This is the bottom equation above.

Height to velocity: assume the cell to your right has a height difference, of ∆h meters. You can similarly derive the official PDE by equating the pressure at the different cells.