2.2 Stability and Convergence
43
N
2.2.2 Von Neumann's Method
One drawback of the energy method is that each new problem requires fresh insight in order to define an appropriate energy and to show that the finite-difference
scheme preserves abound on that energy. Von Neumann 's method has the advantage that it can be applied by following a prescribed procedure ; however, it is applicable only to linear finite-difference equations with constant coefficients.? The
basic idea of the Von Neumann method is to represent the discretized solution at
some particular time step by a finite Fourier series of the form
cP' } = L a'kikj6x,
k= -N
and to examine the stability of the individual Fourier components. The total solution will be stable if and only if every Fourier component is stable. The use of
finite Fourier series is strictly appropriate only if the spatial domain is periodic .
When problems are posed with more general boundary conditions , a rigorous stability analysis is more difficult, but the Von Neumann method still provides a
useful way of weeding out obviously unsuitable schemes.
A key property of Fourier series is that individual Fourier modes are eigenfunctions of the derivative operator, i.e.,
d "k
ik
-e' x = ike ' x
dx
Finite Fourier series have an analogous propcrty in that individual modes eikj6x
are eigenfunctions of linear finite-difference operators. Thus, if the initial conditions for some linear, constant-coefficient finite-difference scheme are cP' } =
eikj6x. after one iteration the solution will have the form
",n +1 _ A eikj6x
't'j
-
k
,
where Ak is a complex constant, known as the amplificationfactor, that is determined by the form ofthe finite-difference formulae. Since the analysis is restricted
to linear constant-coefficient schemes, the amplification factor will not vary from
time step to time step, and if a'k denotes the amplitude of the kth finite Fourier
component at the nth time step, then
3In order to apply Von Neum ann's method to more general problems, the goveming finitedifference equa tions must be linearized and any variable coefficients must be froren at some constant
value . The Von Neumann stability of the family of linearized, frozen-coefficient systems may then be
examined. See Sect ion 3.5.
43
N
2.2.2 Von Neumann's Method
One drawback of the energy method is that each new problem requires fresh insight in order to define an appropriate energy and to show that the finite-difference
scheme preserves abound on that energy. Von Neumann 's method has the advantage that it can be applied by following a prescribed procedure ; however, it is applicable only to linear finite-difference equations with constant coefficients.? The
basic idea of the Von Neumann method is to represent the discretized solution at
some particular time step by a finite Fourier series of the form
cP' } = L a'kikj6x,
k= -N
and to examine the stability of the individual Fourier components. The total solution will be stable if and only if every Fourier component is stable. The use of
finite Fourier series is strictly appropriate only if the spatial domain is periodic .
When problems are posed with more general boundary conditions , a rigorous stability analysis is more difficult, but the Von Neumann method still provides a
useful way of weeding out obviously unsuitable schemes.
A key property of Fourier series is that individual Fourier modes are eigenfunctions of the derivative operator, i.e.,
d "k
ik
-e' x = ike ' x
dx
Finite Fourier series have an analogous propcrty in that individual modes eikj6x
are eigenfunctions of linear finite-difference operators. Thus, if the initial conditions for some linear, constant-coefficient finite-difference scheme are cP' } =
eikj6x. after one iteration the solution will have the form
",n +1 _ A eikj6x
't'j
-
k
,
where Ak is a complex constant, known as the amplificationfactor, that is determined by the form ofthe finite-difference formulae. Since the analysis is restricted
to linear constant-coefficient schemes, the amplification factor will not vary from
time step to time step, and if a'k denotes the amplitude of the kth finite Fourier
component at the nth time step, then
3In order to apply Von Neum ann's method to more general problems, the goveming finitedifference equa tions must be linearized and any variable coefficients must be froren at some constant
value . The Von Neumann stability of the family of linearized, frozen-coefficient systems may then be
examined. See Sect ion 3.5.