6.3 Application to the Generic Transport Equation
145
with n while, if all of them are smaller than 1, 6 will decay. Normally, the
eigenvalues of a matrix are difficult to estimate and, for a problem more
difficult than this, we would be in serious trouble. The saving feature of
this problem is that, because each diagonal of the matrix is constant, the
eigenvectors are easily found. They can be represented in terms of sines and
cosines but it is simpler to use the complex exponential form:
where i =
and a is a wavenumber that can be chosen arbitrarily; the
choice will be discussed below. If Eq. (6.29) is substituted into Eq. (6.25), the
complex exponential term eiaj is common to every term and can be removed
and we obtain an explicit expression for the eigenvalue u:
The magnitude of this quantity is what is important. Since the magnitude of
a complex quantity is the sum of the squares of the real and imaginary parts,
we have:
u2 = [l + ~ ~ ( C O S
a - 1)12 + 4c2 sin2 a .
(6.31)
We now investigate the conditions for u2 to be smaller than unity.
Since there are two independent parameters in the expression for a, it is
simplest to consider special cases first. When there is no diffusion (d = O),
u > 1 for any a and this method is unstable for any value of c, i.e. the method
is unconditionally unstable, rendering it useless. On the other hand, when
there is no convection (c = 0), we find that u is maximum when cosa = -1
so the method is stable provided d < a i.e. it is conditionally stable.
The requirement that the coefficients of all old nodal values be positive
leads to similar conclusions: d < 0.5 and c < 2d. The first condition leads to
the limit on At:
The second requirement imposes no limit on the time step, but gives a relation
between convection and diffusion coefficients:
i.e., the cell Peclet number should be smaller than two. This has already been
mentioned as a sufficient (but not necessary) condition for boundedness of
solutions obtained using CDS for convective fluxes.
Since the method is based on a combination of the explicit Euler method
for ordinary differential equations and the central difference approximation
for the spatial derivatives, it inherits the accuracy of each. The method is
therefore first order in time and second order in space. The requirement that
145
with n while, if all of them are smaller than 1, 6 will decay. Normally, the
eigenvalues of a matrix are difficult to estimate and, for a problem more
difficult than this, we would be in serious trouble. The saving feature of
this problem is that, because each diagonal of the matrix is constant, the
eigenvectors are easily found. They can be represented in terms of sines and
cosines but it is simpler to use the complex exponential form:
where i =
and a is a wavenumber that can be chosen arbitrarily; the
choice will be discussed below. If Eq. (6.29) is substituted into Eq. (6.25), the
complex exponential term eiaj is common to every term and can be removed
and we obtain an explicit expression for the eigenvalue u:
The magnitude of this quantity is what is important. Since the magnitude of
a complex quantity is the sum of the squares of the real and imaginary parts,
we have:
u2 = [l + ~ ~ ( C O S
a - 1)12 + 4c2 sin2 a .
(6.31)
We now investigate the conditions for u2 to be smaller than unity.
Since there are two independent parameters in the expression for a, it is
simplest to consider special cases first. When there is no diffusion (d = O),
u > 1 for any a and this method is unstable for any value of c, i.e. the method
is unconditionally unstable, rendering it useless. On the other hand, when
there is no convection (c = 0), we find that u is maximum when cosa = -1
so the method is stable provided d < a i.e. it is conditionally stable.
The requirement that the coefficients of all old nodal values be positive
leads to similar conclusions: d < 0.5 and c < 2d. The first condition leads to
the limit on At:
The second requirement imposes no limit on the time step, but gives a relation
between convection and diffusion coefficients:
i.e., the cell Peclet number should be smaller than two. This has already been
mentioned as a sufficient (but not necessary) condition for boundedness of
solutions obtained using CDS for convective fluxes.
Since the method is based on a combination of the explicit Euler method
for ordinary differential equations and the central difference approximation
for the spatial derivatives, it inherits the accuracy of each. The method is
therefore first order in time and second order in space. The requirement that