d
dt
190
4. Series-Expansion Methods
for which the forcing has the property that v F (v) = 0, where the overbar denotes
the integral over the spatial domain and v is any sufficiently smooth function that
satisfies the boundary conditions.
An example of this type of problem is the simulation of passive tracer transport by nondivergent flow in a periodic spatial domain , which is govemed by the
equation
at
a1/r
+ v . Vy, = 0.
In this case F (v) = v . Vv . One can verify that v F (v) = °if v is any periodic
function with continuous first derivatives by noting that
(
lD
v(v · Vu) dV =
(
21D
V · (v 2v) - v 2(V . v) dV = 0,
where the second equality follows from periodicity and the nondivergence of the
velocity field.
If ifJ is an approximate spectral solution to (4.1) in which the time dependence
is not discretized, then
aifJ + F(ifJ) = R(ifJ),
at
(4.33)
where R (ifJ) denotes the residual. Suppose that the partial differential equation
being approximated is a conservative system for which ifJF(ifJ) = 0, then multiplying (4.33) by i fJ and integrating over the spatial domain yields
1 aifJ2 - -
- - =ifJR(ifJ) ·
2 at
The right side of the preceding is zero because i fJ is a linear combination of the expansion functions and R(ifJ) is orthogonal to each individual expansion function.
As a consequence,
-lIifJII2=0,
(4.34)
implying that spectral approximations to conservative systems are not subject to
nonlinear instability because (4.34) holds independent of the linear or nonlinear
structure of F(y,). The only potential source of numerical instability is in the
discretization of the time derivative .
Neglecting time-differencing errors, spectral methods will also conserve (fJ provided that F(v) = 0, where once again v is any sufficiently smooth function that
satisfies the boundary conditions. The conservation of (fJ can be demonstrated by
integrating (4.33) over the domain to obtain
-
a(fJ
at = R(ifJ) -
where 'Po is the lowest-wave-number orthogonal expansion function , which is a
constant.