It is hard to find reliable numerical methods for the solution of partial differential equations (PDEs). Often they turn out to be either unstable or strongly diffusive, giving inaccurate solutions even to simple equations.
Our case study is one of the simplest PDEs, the advection equation
that describes the propagation of an unchanging shape at constant speed .
Trivially, this equation is solved by any function
To solve it numerically we approximate
to a discrete solution defined in a rectangular grid,
. For the spatial derivative we use a first-order, centered approximation
We can think now of many discrete approximations for the time derivative. The simplest one is the explicit Euler discretization:
Then we arrive at the method usually abbreviated FTCS (forward in time, centered in space). Unfortunately, this discretization turns out to be numerically unstable for any value of
A first improvement to the FTCS scheme would be to replace the term
in the time discretization by an average
. This approach, often called the Lax method, is equivalent to adding an artificial diffusion term to the advection equation. The resulting scheme is now stable if it satisfies the Courant–Friedrichs–Lewy
is called the Courant number. However, since we are adding an artificial diffusive term, the scheme becomes too diffusive and therefore inaccurate when
. The Lax method gives optimal results for
Fortunately, the differential equation solver of Mathematica, NDSolve
, comes with many numerical schemes that avoid the shortcomings of the FTCS and Lax methods. In this Demonstration you can choose some of these methods with a fixed-step time discretization.
R. Courant, K. Friedrichs, and H. Lewy, On the Partial Difference Equations of Mathematical Physics
, IBM J. Res. Develop., 11
, 1967 pp. 215–234.
W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing
, 3rd ed., Cambridge, UK: Cambridge University Press, 2007.