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We extend our forward Euler stability result for certain second- and third-order accurate multi-level and Runge-Kutta time-differencing.

To this end, we view our -approximate solution at time level , as an (*N*+1)-dimensional column vector which is uniquely realized
at the Gauss collocation nodes .

The forward Euler time-differencing (meth_cheb.7a)
with homogeneous boundary conditions
(meth_cheb.7b), reads

where *L* is an matrix which accounts for the
spatial spectral differencing together with the homogeneous boundary
conditions,

Theorem 4.1 tells us that if the CFL condition
(meth_cheb.12) holds, i.e., if

then is bounded in the -weighted induced
operator norm,

Let us consider an (*s* + 2)-level time differencing method of the form

In this case, is given by a
*convex* combination of stable forward Euler differencing, and we
conclude

__. Assume
that the following CFL condition holds,
Then the spectral approximation (cheb_RK.4)
is strongly stable, and the
following estimate holds
__

__
Second and third-order accurate multi-level time differencing methods
of the positive type (cheb_RK.4)
take the particularly
simple form
with positive coefficients, , given in Table
4.1
__

__
__

__
Similar arguments apply for Runge-Kutta time-differencing methods. In this
case the resulting positive type Runge-Kutta methods take the form
We arrive at
__

__
. Assume that the CFL
condition (meth_cheb.12) holds. Then the spectral approximation
(cheb_RK.8a)-(cheb_RK.8c)
with is strongly stable and the
stability estimate
(meth_cheb.13) holds.
__

__
Table 4.2
quotes second and third-order choices of positive-type
Runge-Kutta method.
__

Thu Jan 22 19:07:34 PST 1998