Particle tracking methods for fractional ADEs

Transcription

1 Paricle racking mehods for fracional AEs Yong Zhang eser Research Insiue November 4 29 Yong.Zhang@dri.edu

2 ollaboraors Boris Baeumer avid Benson Eric LaBolle Mark Meerschaer Hans-Peer Scheffler

3 Why use paricle racking? Advanages: Grid free. Provide dynamics. oes no suffer he numerical dispersion. oes no modify such as runcaion he PE. The only viable ool for some cases: he vecor fae wih space-dependen parameers 2 he plume wih sharp concenraion fron. isadvanages: Resoluion paricle spliing ross verificaion by real soluions

4 Ouline Lagrangian solver for:. Space fae wih variable coefficiens; 2. Time fae wih scale inde < <; 3. Muliscaling vecor fae; 4. Time fae wih inde < <2; 5. oupled vs. decoupled TRW; 6. Super/sub-diffusion across discree inerface of medium.

5 . Space fae Langevin analysis in space [Zhang e al. JSP 26]: ompue adjoin o ge backward equaion 2 Build he generaor and he underlying Markov process 3 Track paricle dynamics and obain he paricle number densiy ffpe [ ] [ ] [ ] F-AE [ ] FF-AE FF-AE

6 Sep : Fracional Adjoin operaor Forward equaion ffpe Adjoin Backward equaion [ ] [ ] [ ] [ ] p B p B [ ] [ ] [ ]d g B p d dz z z g B p d dz z z p z B g d p B g Γ Γ P P P. Space fae

7 Backward equaion. Space fae Sep 2: Build he Markov process P P P Generaor Langevin equaion Lu u u y y u y 2 u π cos φ dy 2 / * [ X d] β μ dx X d σ Generae sandard Lévy -sable random variables:. The modified MS mehod [hambers Mallows and Suck JASA 976] 2. STABLE [Nolan SSM 997. hp://academic2.american.edu/~jpnolan] 3. The runcaed Pareo approach [Zhang e al. PRE 26] 4. The one-parameer Miag-Leffler funcion [Fulger e al. PRE 28]

8 . Space fae Sep 2: Build he Markov process on. densiy of X < < X densiy of X X densiy of X < < X densiy of X X

9 Posiion. Space fae Sep 3: Track paricle dynamics X X dx ne ne now Time ne now d /.2 * /.6.5 d X

10 Oher known faes wih space-dependen parameers. Space fae [ ] FF-AE Langevin equaion [ ] F-AE Langevin equaion [ ] * / * / σ μ β σ μ β d sign d X d X dx [ ] * / * / σ μ β σ μ β d sign d X d X dx

11 Oher known faes wih space-dependen parameers on.. Space fae FF-AE Langevin equaion F-AE n Langevin equaion [ ] [ ] / / * / d sign d X d X dx [ ] [ ] n n n / * / / ln ln d n n sign d sign d d dx

12 . Space fae Numerical eamples F-AE n [ n ] [ n ] n n.5/ T4 oncenraion Number of paricles RW np 4 RW np 5 RW np 6 F Number of ime seps RW n 4 RW n 4 RW n 4 F

13 Ouline Lagrangian solver for:. Space fae wih variable coefficiens; 2. Time fae wih scale inde < <; 3. Muliscaling vecor fae; 4. Time fae wih inde < <2; 5. oupled vs. decoupled TRW; 6. Super/sub-diffusion across discree inerface of medium.

14 2. The ime fae p p A p b β β Γ The densiy p can be calculaed by subordinaing he jump process agains he waiing ime process τ τ τ d h u p τ τ τ u A u τ β τ τ τ h h b h Moion process Hiing ime process 2 cos * / σ μ β τ π β τ d b d d Langevin analysis in ime < <: where -real ime τ-moion ime

15 2. The ime fae A simple hree-sep scheme: Sep - Transform he moion ime dτ o real ime d. Sep 2 - Simulae he paricle jump dx during operaional ime dτ. Sep 3 - Obain paricle rajecory: posiion dx and ime d. Moion ime τ Σdτ i dτ M dω M 2 Paricle posiion ΣdX Σdτ dτ dτ dω dτ 2 dω 2 dτ 3 dω 3 T end d d d2 d3... dm Real ime Σd

16 2. The ime fae Lagrangian soluion symbols vs. Semi-analyical soluion lines Snapsho BT p p isance from he source Time since injecion

17 Ouline Lagrangian solver for:. Space fae wih variable coefficiens; 2. Time fae wih scale inde < <; 3. Muliscaling vecor fae; 4. Time fae wih inde < <2; 5. oupled vs. decoupled TRW; 6. Super/sub-diffusion across discree inerface of medium.

18 Orhogonal eigenvecors in he scaling mari and he same orhogonal angelus in he miing measure.8 3. Muliscaling fae H I [ ] [ ] M Nolan s 2-d sable densiy surface y y F AI RW y Nolan's code RW -

19 ensiy The ime fae wih inde < <2 a p A p f δ.9 a. Operaional Time Operaional Time p.9 a Real Time Real ime

20 v o M o o A β Uncoupled oupled Γ v M A Jump Size: epending nonlinearly on he ime period. Waiing Time: Real ime vs. Operaional ime RW soluion f Waiing Time: F R k k k k k i k Δ Jump Size: RW soluion 5. oupled vs. decoupled TRW models

21 6. Super/sub-diffusion across discree inerface of medium iscree dispersion coefficien n n Time Sample pah for AE Time iscree porosiy n Paricle racking across he discree inerface Sample pah for fae

22 Summary Paricle racking approach is: fleible efficien he only viable ool more han we can use

23 Acknowledgemen Financial suppor from NSF and RI. References Zhang Y..A. Benson M.M. Meerschaer and H-P Scheffler. Journal of Saisical Physics Zhang Y..A. Benson M.M. Meerschaer and E.M. LaBolle. Physical Review E Zhang Y. and.a. Benson. Geophysical Research Leers 35 L Zhang Y. M.M. Meerschaer and B. Baeumer. Physical Review E Zhang Y. E.M. LaBolle and K. Pohlmann. Waer Resources Research 45 W Meerschaer M.M. Y. Zhang and B. Baeumer. ompuers and Mahemaics wih Applicaions 29 in press.