skade eaction ffusion ne-dimensional System - ZIB

More documents

Recommendations

Info

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)
Yet this frm not suited to be implemented straight-forward, becuse along with function evaluations there are still matrix*vector producs on the ighhand ide. The ransformation leads to the new ysem / "fi * — A 1 \ P-fu(u0) {u0 + J2 a J2— « = 1 r 7 i=1 j=1 uo ^ r (1. In KARDOS a special set of parameters, which guarantees L-stabiity of the chosen method, is used. Setting 711 = 722 = 733 := 7 only two function valuations and one matrix inversion are needed on the righthand side. The corresponding method was given by ROCHE [], who gives further sateent about convergence and consisteny. efficent 0 . 5 2 8 4 704526589 9 3 9 4 1 7309 0.940143465 09140443903 0. 1.99625219 0.874044410 1.99625219 4797 0. 223756213 ue to the drence of the pproximaed ues of er 3 and timeError := ||u (1- we have a good estimator of the main error term descibing the ocal error of the econd order method. The proposal for the new te ize is timeTol \ afeFa — — 1- timeError
Page 1 and 2: Tehnicl Repot TR 9-9 (July 193) Jen
Page 3: Contents Introducti efng a problem
Page 7 and 8: To cntrol the pace dicretiztion, we
Page 9 and 10: haper efining roblem At first the u
Page 11 and 12: haper C o n d ng All specal command
Page 13 and 14: haper r o g e r only This chapter g
Page 15 and 16: int nonNegativity ontrol of the non
Page 17 and 18: eal spaceTolFac paceTolglobTol, san
Page 19 and 20: espping
Page 21 and 22: haper le In this chapte how how to
Page 23 and 24: eal valO, vail; real *fVal, int equ
Page 25 and 26: fVals[l] = 0.0; /* */ else ZIBStdOu
Page 27 and 28: seldraw 1 timesteppin timestep 0.00

skade eaction ffusion ne-dimensional System - ZIB

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?