skade eaction ffusion ne-dimensional System - ZIB

haper
ntroduction
Usng the program KARDOS, sysems of semlinear parabolic initial boundary
value problems of the for
{x)ut ~ {D{X)U) F(u X G 0 C R\ t G [t t
Ci(, ^GTiCön
un + a(x)u £2(^ x x £ T2 C df
u(tx) = u0(x
are solvable. Here, u = (u u2,... ur) is a vector function The matrix
function P(x) may vanish on subset of ft. The discretizaton is done by
the Rothe method. In contrast to he widespread method of lines, time is
discretized first than space. The main advantage of this sequence is the
possibility o compute the pace discretizaton optimal during the time integration
b an adaptiv multileve finite element method. Therefore the
KASKADE toolbox is modified to handle one-dimensional problems. In
KARDOS a special embedded Runge-Kutta method of order 32) has bee
implemented. This method keeps its accuracy even in the case of ifferenial
algebraic equations (P(x) = 0 somewhere).
The above differetial equations can be reformulated into an abstract Cauchy
problem poibl of differenal-algebraic type in an appropriate Hilbrt
pace H
Pu f(u u G H ,t e[tt
u(0) = u0
Now, a three-stage embedded Rosenbroc ethod for his pure tim problem
looks as follows:
{P ~ 7" r /«(«o))A;8 = r/(u0 + Yl a
1.
) + T f{uo Y l i = 1 3
3=1 J=
Uo +
u0 + Y
3 =
1.3)