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PLUTO
4.0
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PLUTO
4.0
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Runge-Kutta-Chebyshev driver for integration of diffusion terms. More...
#include "pluto.h"Functions | |
| void | RKC (const Data *d, Time_Step *Dts, Grid *grid) |
Take one step in the solution of the diffusion equation 
using Runge-Kutta-Chebyshev (RKC) method.
![\[ \begin{array}{lcl} Y_0 & = & Q^n \\ \noalign{\medskip} Y_1 & = & Y_0 + \tilde{\mu}_1\tau F_0 \\ \noalign{\medskip} Y_j & = & (1 - \mu_j - \nu_j) Y_0 + \mu_j Y_{j-1} + \nu_j Y_{j-2} + \tilde{\mu}_j \tau F_{j-1} + \tilde{\gamma}_j\tau F_0 \quad \rm{for} \quad j = 2,...,s \\ \noalign{\medskip} Q^{n+1} & = & Y_s \end{array} \]](form_176.png)
where
![\[\tau = \Delta t,\, \tilde{\mu}_1 = b_1 w_1,\, \mu_j = 2 b_j \frac{w_0}{b_{j-1}}, \, \nu_j = -\frac{b_j}{b_{j-2}},\, \tilde{\mu}_j = 2 b_j \frac{w_1}{b_{j-1}},\, \tilde{\gamma}_j = -\alpha_{j-1} \tilde{\mu}_j,\, \]](form_177.png)
![\[ \alpha_j = 1 - b_j T_j(w_0),\, w_0 = 1 + \epsilon/s^2,\, w_1 = T'_s(w_0)/T''_s(w_0),\, b_j = T''_j(w_0)/(T'_j(w_0))^2,\, b0=b1=b2,\, \epsilon = 2/13 \]](form_175.png)
Here the T's are the Chebyshev polynomial of the first kind:
![\[ T_j(x) = \cos(j {\rm arccos(x)}) ,\, T_0(x) = 1,\, T_1(x)=x,\, T_j(x)=2*x*T_jm1(x) - T_jm2(x), \quad \rm{for}\quad j = 2,...,s \]](form_178.png)
while sprad=Spectral_Radius-> 4,8,12 kappa/dx/dx (check)
Only the parabolic terms of the equations are advanced by a single step g_dt equal to the current integration time step and the number of RKC steps (Nrkc) is computed... The explicit parabolic time step is computed in the same manner as in STS.
References
1.8.2