Chapter 2
Motivation
Abstract This chapter introduces the reader to the “world” of numerical modelling
using the decay problem as a f rst benchmark. Discussed are finit differences,
explicit and implicit schemes, and conditions of consistency, accuracy, stability, and
efficien y. FORTRAN codes and SciLab scripts are used to create a firs numerical
prediction model and graphical display of results.
2.1 The Decay Problem
2.1.1 The Problem
The so-called decay problem is chosen as a f rst example to demonstrate what
numerical modelling is. In mathematical terms, this problem can be expressed as:
dC
dt
= −κ · C
(2.1)
where C is concentration of a substance, t is time, and κ (the Greek symbol “kappa”)
is a positive constant parameter. The d symbol refers to a change of a variable with
respect to some other parameter such as time as in the above equation.
2.1.2 Physical Interpretation
The term on the left-hand side of Eq. (2.1) refers to the temporal change of concentration per time unit. The right-hand side specifie this temporal change. For κ = 0,
there will no change and C remains unchanged at its initial value. With κ = 0, on the
other hand, the right-hand side is negative since concentration is always a positive
quantity. Accordingly, C will gradually decrease with time at a rate in proportion to
concentration itself at any time instance.
J. K¨ ampf, Ocean Modelling for Beginners,
DOI 10.1007/978-3-642-00820-7 2, C
Springer-Verlag Berlin Heidelberg 2009
5
Motivation
Abstract This chapter introduces the reader to the “world” of numerical modelling
using the decay problem as a f rst benchmark. Discussed are finit differences,
explicit and implicit schemes, and conditions of consistency, accuracy, stability, and
efficien y. FORTRAN codes and SciLab scripts are used to create a firs numerical
prediction model and graphical display of results.
2.1 The Decay Problem
2.1.1 The Problem
The so-called decay problem is chosen as a f rst example to demonstrate what
numerical modelling is. In mathematical terms, this problem can be expressed as:
dC
dt
= −κ · C
(2.1)
where C is concentration of a substance, t is time, and κ (the Greek symbol “kappa”)
is a positive constant parameter. The d symbol refers to a change of a variable with
respect to some other parameter such as time as in the above equation.
2.1.2 Physical Interpretation
The term on the left-hand side of Eq. (2.1) refers to the temporal change of concentration per time unit. The right-hand side specifie this temporal change. For κ = 0,
there will no change and C remains unchanged at its initial value. With κ = 0, on the
other hand, the right-hand side is negative since concentration is always a positive
quantity. Accordingly, C will gradually decrease with time at a rate in proportion to
concentration itself at any time instance.
J. K¨ ampf, Ocean Modelling for Beginners,
DOI 10.1007/978-3-642-00820-7 2, C
Springer-Verlag Berlin Heidelberg 2009
5