First Order Numerical Algorithms

These appendices are Professor Drummond's notes on the numerical algorithms discussed in the main part of this text. They are kept here as an alternative discussion to that given above.

How can we treat PDEs numerically?

First of all, it is clear that these PDE problems have many similarities to the coupled first-order differential equations (ODEs) treated earlier in the computational laboratory. Thus, many of the ideas from this field can be used here as well. This leads to the first obvious guess at how to treat this problem numerically, which is the first-order Euler method.

As a general rule, we can develop any method by considering a Taylor series expansion of the spatial and temporal derivatives around a given point, so as to give discrete approximations to the derivatives for numerical integration purposes. Thus, for example, if we divide time into a lattice of $t_{0},\ldots{},t_{N}$, with $t_{n+1} = t_{n} + \Delta t$, then:

$$\mbf{A}(t_{n+1},\mbf{x}) = \mbf{A}(t_{n},\mbf{x}) + \Delta t \frac{\partial }{\partial t} \mbf{A}(t_{n},\mbf{x}) + O(\Delta t^2).$$ (A.1)

In an explicit method, the propagation is calculated by stepping forward in time; and only using past or present values of the derivatives. Thus, in the Euler method, we simply evaluate the right-hand side of Eq.(A.1) at the initial time $t = t_0$ on a discretised lattice. This gives the derivative $\frac{\partial }{\partial t} \mbf{A}(t_0,\mbf{x})$ of the vector field $\mbf{A}$. Next, a step is taken in time, by adding $\Delta t \frac{\partial }{\partial t} \mbf{A}(t_{0},\mbf{x})$ to $\mbf{A}$, where we have yet to specify how this derivative is estimated numerically. The process is repeated as many times as necessary.