Introduction 7
The approximation of derivatives by finite differences in an ODE or a
PDE results in the local replacement of the differential equation by an algebraic equation valid for the specific location. That algebraic equation is
formulated and is valid at a specific location of the solution domain of the
equation and contains as unknowns the values of the function in a number
of neighbouring points near f(x), such as f(x + Δx), f(x – Δx), f(x + 2Δx),
f(x – 2Δx) and so on. By discretizing the solution domain into a large number of fixed points, an equal number of algebraic equations can be derived.
Solution of the system would provide numerical values of the function f on
all those points. By definition, this is the numerical solution sought of the
differential equation, that is, the determination of the arithmetic values of
the unknown function f(x) on a sufficient number of prefixed locations in
the solution domain. This is in contrast to the analytic solution, which is
defined as the estimation of the functional form of the solution f(x) of the
differential equation, so that its value at any x point can be subsequently
calculated.
Depending on the procedure for the solution of the algebraic equations
corresponding to all the points in the solution domain, the finite difference
scheme used can be characterised as
• Explicit, when the deriving algebraic equations can be solved
independently
• Implicit, when those equations need to be solved simultaneously as a
system of algebraic equations
The locations where the numerical approximation of the differential
equation is being sought are ordered regularly in the solution domain,
usually at a constant in-between distance. This is done by discretizing
the solution domain using one-, two- or three-dimensional orthogonal
grids. The grid permits the decomposition of the solution domain into a
number of cells and nodes easily characterised by coordinate indices, with
positive integer values. For example, in a one-dimensional case at point
x i = (i – 1)Δx, from an origin at x 1 = 0, the function f(x) is denoted as f i ,
f(x = x i ) or f(x = (i – 1)Δx). Thus, if the solution domain for an ODE is 0 ≤
x ≤ 1 and the domain is discretized into 10 segments (Δx = 0.1), then the
unknown values of f(x) would be f 1 (x = 0), f 2 (x = 0.1), …, f 11 (x = 1.0). In
order to compute those 11 values of the function f(x), some of them need
to be provided (i.e. boundary conditions: f 1 = known) and the rest need to
be calculated by formulating and solving a number of algebraic equations
equal to the number of the 10 remaining unknown values of f(x i ).
The selection of the type of finite differences that will approximate the
derivatives in a differential equation and the solution procedure of the
group of the algebraic equations are not unique. In total they synthesise a
solution algorithm or a numerical solution scheme.
The approximation of derivatives by finite differences in an ODE or a
PDE results in the local replacement of the differential equation by an algebraic equation valid for the specific location. That algebraic equation is
formulated and is valid at a specific location of the solution domain of the
equation and contains as unknowns the values of the function in a number
of neighbouring points near f(x), such as f(x + Δx), f(x – Δx), f(x + 2Δx),
f(x – 2Δx) and so on. By discretizing the solution domain into a large number of fixed points, an equal number of algebraic equations can be derived.
Solution of the system would provide numerical values of the function f on
all those points. By definition, this is the numerical solution sought of the
differential equation, that is, the determination of the arithmetic values of
the unknown function f(x) on a sufficient number of prefixed locations in
the solution domain. This is in contrast to the analytic solution, which is
defined as the estimation of the functional form of the solution f(x) of the
differential equation, so that its value at any x point can be subsequently
calculated.
Depending on the procedure for the solution of the algebraic equations
corresponding to all the points in the solution domain, the finite difference
scheme used can be characterised as
• Explicit, when the deriving algebraic equations can be solved
independently
• Implicit, when those equations need to be solved simultaneously as a
system of algebraic equations
The locations where the numerical approximation of the differential
equation is being sought are ordered regularly in the solution domain,
usually at a constant in-between distance. This is done by discretizing
the solution domain using one-, two- or three-dimensional orthogonal
grids. The grid permits the decomposition of the solution domain into a
number of cells and nodes easily characterised by coordinate indices, with
positive integer values. For example, in a one-dimensional case at point
x i = (i – 1)Δx, from an origin at x 1 = 0, the function f(x) is denoted as f i ,
f(x = x i ) or f(x = (i – 1)Δx). Thus, if the solution domain for an ODE is 0 ≤
x ≤ 1 and the domain is discretized into 10 segments (Δx = 0.1), then the
unknown values of f(x) would be f 1 (x = 0), f 2 (x = 0.1), …, f 11 (x = 1.0). In
order to compute those 11 values of the function f(x), some of them need
to be provided (i.e. boundary conditions: f 1 = known) and the rest need to
be calculated by formulating and solving a number of algebraic equations
equal to the number of the 10 remaining unknown values of f(x i ).
The selection of the type of finite differences that will approximate the
derivatives in a differential equation and the solution procedure of the
group of the algebraic equations are not unique. In total they synthesise a
solution algorithm or a numerical solution scheme.