1. 1 IV European Conference of Computational Mechanics Hrvoje Gotovac, Veljko Srzić, Tonći Radelja, Vedrana Kozulić Hrvoje Gotovac, Veljko Srzić, Tonći Radelja, Vedrana Kozulić University of Split, Department of Civil and Architectural Engineering, Croatia University of Split, Department of Civil and Architectural Engineering, Croatia Explicit Adaptive Fup Collocation Method (EAFCM) for solving the parabolic problems Explicit Adaptive Fup Collocation Method (EAFCM) for solving the parabolic problemsPresentation ECCM, 21 May 2010, Paris, France.

2. 2 Presentation outline 1. General concept 2. Fup basis functions 3. Fup collocation transform (FCT) - space approximation 4. Explicit time integration for parabolic stiff problems 5. Numerical examples 6. Conclusions 7. Future directions

3. 3 1. General concept  Developing adaptive numerical method which can deal with parabolic flow and transport stiff problems having wide range of space and temporal scales  Ability to handle multiple heterogeneity scales  Application target: unsaturated and multiphase flow, reactive transport and density driven flow in porous media, as well as structural mechanics problems

4. 4 Saturated – unsaturated flow

5. 5 Interaction between surface and subsurface flow

6. 6 Geothermal convective processes in porous media

7. 7 Typical physical and numerical problems  Description of wide range of space and temporal scales  Sharp gradients, fronts and narrow transition zones (‘fingering‘ and ‘layering’)  Artificial oscillations and numerical dispersion – advection dominated problems  Description of heterogeneity structure  Strong nonlinear and coupled system of equations

8. 8 Motivation for EAFCM  Multi-resolution and meshless approach  Continuous representation of variables and all its derivatives (fluxes)  Adaptive strategy  Method of lines (MOL)  Explicit formulation (no system of equations!!!)  Perfectly suited for parallel processing

9. 9 Flow chart of EAFCM

10. 10

11. 11 2. Fup basis functions  Atomic or R bf class of functions  Function up(x)  Fourier transform of up(x) function  Function Fup n (x)

12. 12 Function Fup 2 (x)

13. 13 Function Fup 2 (x,y)

14. 14 Function Fup n (x)  Compact support  Linear combination of n+2 functions Fup n (x) exactly presents polynomial of order n  Good approximation properties  Universal vector space UP(x)  Vertexes of basis functions are suitable for collocation points  Fup n (x,y) is Cartesian product of Fup n (x) and Fup n (y)

15. 15  Any function u(x) is presented by linear combination of Fup basis functions: j - level (from zero to maximum level J) j - level (from zero to maximum level J) j min - resolution at the zero level j min - resolution at the zero level k - location index in the current level k - location index in the current level - Fup coefficients - Fup coefficients - Fup basis functions - Fup basis functions n - order of the Fup basis function n - order of the Fup basis function 3. Fup collocation transform (FCT)

16. 16

17. 17 Spatial derivatives

18. 18 4. Time numerical integration  Reduces to system of Ordinary Differential Equations (ODE) for adaptive grid and every time step (t – t+dt):  With appropriate initial conditions:

19. 19 Stabilized second-order Explicit Runge-Kutta method (SERK2)  Recently developed by Vaquero and Janssen (2009)  Extended stability domains along the negative real axis  Suitable for very large stiff parabolic ODE  Second – order method up to 320 stages  Public domain Fortran routine SERK2

20. 20 5. Numerical examples  1-D density driven flow problem  2-D Henry salwater intrusion problem

21. 21 Mathematical model  Pressure-concentration formulation  Fluid mass balance:  Salt mass balance:

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25. 25 Number of collocation points and compression coefficient

26. 26 6. Conclusions  Development of mesh-free adaptive collocation algorithm that enables efficient modeling of all space and time scales  Main feature of the method is the space adaptation strategy and explicit time integration  No discretization and solving of huge system of equations  Continuous approximation of fluxes

27. 27 7. Future directions  Multiresolution description of heterogeneity  Development of 3-D parallel EAFCM  Time subdomain integration  Description of complex domain with using other families of atomic basis functions  Further application to mentioned processes in porous media and other (multiphysics) problems