138
3. Beyond the One-Way Wave Equation
equation yields a set of algebraic equations that can be solved very efficiently,
when the same method is used to approximate the three-dimensional diffusion
equation, the matrix associated with the resulting algebraic system has a much
wider bandwidth, and a considerable increase in computational effort is required
to obtain its solution. Nevertheless , in comparison with explicit finite-difference
methods, the extra work per time step required by the trapezoidal scheme can
usually be more than offset by using a much larger time step.
3.4.2 Advection and Diffusion
Now consider the combined advection-diffusion problem , which in one dimension is govemed by the equation
a1fr
a1fr
a
21fr
- + c - = M - .
at
ax
ax 2
(3.72)
Before discussing how to approximate the time derivative in the preceding, we
will examine accuracy issues that arise solely from the approximation of the spatial derivatives. If the first spatial derivative is approximated by an upstream difference (with c > 0) and the second derivative is approximated by the standard
three-point stenciI, the resulting differential-difference approximation to (3.72) is
drP
-}' +c (rP}'-rP}'-I) = M }
(rP '+1-2rP} } .
'+rP'-I)
(3.73)
dt
L\x
(L\x)2
Evaluating the truncation error in the preceding shows that it is an 0 [(L\x)2]accurate approximation to the modified equation
(3.74)
where Pe = cSx / M is the numerical Peclet number. The Peclet number is a
nondimensional parameter cIassicaIly defined as the ratio of the strength of thermal advection to the strength of thermal diffusion.' Since the length scale in
the numerical Peclet number is the grid spacing, Pe is a measure of the relative strengths of advection and diffusion at the smallest spatial scales resolved on
the numerical mesh. A comparison ofthe modified equation (3.74) with the original advection-diffusion equation (3.72) shows that the differential-difference approximation (3.73) generates an inaccurate approximation to the diffusion term
unless Pe« I , i.e., unless diffusion dominates advective transport on the shortest resolvable scales. This difficulty arises because the total diffusion is dominated
3The Peclet number is completely analogous to the more familiar Reynolds number, which is
the ratio of momentum advection to momentum diffusion . The difference between the Peclet and
Reynolds numbers is due to the difference in the diffusivities of heat and momentum . In particular, the
ratio of the Peclet number to the Reynolds number is equal to the Prandtl number, which is the ratio
of the kinematic viscosity (or momentum diffusiv ity) to the thermal diffusivity.
3. Beyond the One-Way Wave Equation
equation yields a set of algebraic equations that can be solved very efficiently,
when the same method is used to approximate the three-dimensional diffusion
equation, the matrix associated with the resulting algebraic system has a much
wider bandwidth, and a considerable increase in computational effort is required
to obtain its solution. Nevertheless , in comparison with explicit finite-difference
methods, the extra work per time step required by the trapezoidal scheme can
usually be more than offset by using a much larger time step.
3.4.2 Advection and Diffusion
Now consider the combined advection-diffusion problem , which in one dimension is govemed by the equation
a1fr
a1fr
a
21fr
- + c - = M - .
at
ax
ax 2
(3.72)
Before discussing how to approximate the time derivative in the preceding, we
will examine accuracy issues that arise solely from the approximation of the spatial derivatives. If the first spatial derivative is approximated by an upstream difference (with c > 0) and the second derivative is approximated by the standard
three-point stenciI, the resulting differential-difference approximation to (3.72) is
drP
-}' +c (rP}'-rP}'-I) = M }
(rP '+1-2rP} } .
'+rP'-I)
(3.73)
dt
L\x
(L\x)2
Evaluating the truncation error in the preceding shows that it is an 0 [(L\x)2]accurate approximation to the modified equation
(3.74)
where Pe = cSx / M is the numerical Peclet number. The Peclet number is a
nondimensional parameter cIassicaIly defined as the ratio of the strength of thermal advection to the strength of thermal diffusion.' Since the length scale in
the numerical Peclet number is the grid spacing, Pe is a measure of the relative strengths of advection and diffusion at the smallest spatial scales resolved on
the numerical mesh. A comparison ofthe modified equation (3.74) with the original advection-diffusion equation (3.72) shows that the differential-difference approximation (3.73) generates an inaccurate approximation to the diffusion term
unless Pe« I , i.e., unless diffusion dominates advective transport on the shortest resolvable scales. This difficulty arises because the total diffusion is dominated
3The Peclet number is completely analogous to the more familiar Reynolds number, which is
the ratio of momentum advection to momentum diffusion . The difference between the Peclet and
Reynolds numbers is due to the difference in the diffusivities of heat and momentum . In particular, the
ratio of the Peclet number to the Reynolds number is equal to the Prandtl number, which is the ratio
of the kinematic viscosity (or momentum diffusiv ity) to the thermal diffusivity.