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## Multi-level and Runge-Kutta Time Differencing

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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

Table 4.2:

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.

Table: