44
2. Basic Finite-DifferenceMethods
It follows that the stability of each Fourier component is detennined by the modulus of its amplification factor.
The Von Neumann stability condition, which is necessary and sufficient for
the stability of a linear constant-coefficient finite-difference equation," requires
the amplification factor of every Fourier component resolvable on the grid to be
bounded such that
(2.23)
where y is a constant independent of k, tlt, and Sx, This condition ensures that
a consistent finite-difference scheme satisfies the minimum stability criteria for
convergence in the limit tlx, tlt --* 0, (2.15). In applications where the true
solution is bounded by the nonn of the initial data, it is usually advantageous to
enforce the more stringent requirement that
(2.24)
which will guarantee satisfaction of the stability condition (2.16). When the Von
Neumann condition is satisfied , every finite Fourier component is stable, and the
full solution, being a linear combination of the individual Fourier components,
must also be stable.
As an illustration of the Von Neumann method, consider once again the finitedifference equation (2.17) . The solutions to the associated partial differential equation (2.10) do not grow with time, so we will require IAkI .:5 I. Substitution of an
arbitrary Fourier component, of the form e i kj tu, into (2.17) yields
Dividing out the common factor eik jtu gives
(2.25)
The magnitude of Ak is obtained by multiplying by its complex conjugate and
taking the square root. Thus ,
lAd = (l - J.L + J.Le-iktlX)(l - J.L + J.Le
i k6.x)
= 1 - 2J.L(l - J.L)(l - cosktlx).
(2.26)
The Von Neumann condition (2.24) will therefore be satisfied if
I - 2J.L(l - J.L)(l - cosktlx) .:5 1.
4The sufficiency of the Von Neumann condition holds only for single equations in one unknown.
The stability of systems of finite-difference equations in several unknown variables is discussed in
Seetion 3.1.
2. Basic Finite-DifferenceMethods
It follows that the stability of each Fourier component is detennined by the modulus of its amplification factor.
The Von Neumann stability condition, which is necessary and sufficient for
the stability of a linear constant-coefficient finite-difference equation," requires
the amplification factor of every Fourier component resolvable on the grid to be
bounded such that
(2.23)
where y is a constant independent of k, tlt, and Sx, This condition ensures that
a consistent finite-difference scheme satisfies the minimum stability criteria for
convergence in the limit tlx, tlt --* 0, (2.15). In applications where the true
solution is bounded by the nonn of the initial data, it is usually advantageous to
enforce the more stringent requirement that
(2.24)
which will guarantee satisfaction of the stability condition (2.16). When the Von
Neumann condition is satisfied , every finite Fourier component is stable, and the
full solution, being a linear combination of the individual Fourier components,
must also be stable.
As an illustration of the Von Neumann method, consider once again the finitedifference equation (2.17) . The solutions to the associated partial differential equation (2.10) do not grow with time, so we will require IAkI .:5 I. Substitution of an
arbitrary Fourier component, of the form e i kj tu, into (2.17) yields
Dividing out the common factor eik jtu gives
(2.25)
The magnitude of Ak is obtained by multiplying by its complex conjugate and
taking the square root. Thus ,
lAd = (l - J.L + J.Le-iktlX)(l - J.L + J.Le
i k6.x)
= 1 - 2J.L(l - J.L)(l - cosktlx).
(2.26)
The Von Neumann condition (2.24) will therefore be satisfied if
I - 2J.L(l - J.L)(l - cosktlx) .:5 1.
4The sufficiency of the Von Neumann condition holds only for single equations in one unknown.
The stability of systems of finite-difference equations in several unknown variables is discussed in
Seetion 3.1.