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
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
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
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.