Simulating Spatial Partial Differential Equations with Cellular Automata

1. Simulating Spatial Partial Differential Equations with Cellular Automata By Brian Strader Adviser: Dr. Keith Schubert Committee: Dr. George Georgiou Dr. Ernesto Gomez

2. Introduction & Background • Partial Differential Equation, Cellular Automata (CA), & Biology • Converting Differential Equations to CA • CA Theoretical Constraints • Convergence Maps & Guidelines Topics Covered

3. Introduction & Background • CA Model uses simple rules about changes with time. • Rules are localized and involve the values of cell neighbors. • The set of rules are applied to the cells with the matrix after each time period. Cellular Automata (CA)

4. Introduction & Background Survival Rule: 2-3 Neighbors Conway’s Game of Life Death by Overpopulation: 4+ Neighbors

5. Introduction & Background Death by Isolation: 1 or Less Neighbors Conway’s Game of Life Birth: 3 Neighbors

6. Introduction & Background t = 0 Conway’s Game of Life

7. Introduction & Background t = 1 Conway’s Game of Life

8. Introduction & Background t = 2 Conway’s Game of Life

9. Introduction & Background t = 3 Conway’s Game of Life

10. Introduction & Background Celluar Automata Simulation

11. Introduction & Background Celluar Automata Simulation

12. Introduction & Background • Changes with respect to time. • Part of the equation depends on changes in space. Spatial Partial Diff. Equations

13. Introduction & Background Vegetation Patterns

14. Introduction & Background • Simple Rules - easy to understand • Discretized • Local Problem View • Highly Parallelizable CA Advantages

15. Converting Differential Equations to CA Conditions: for n(u) = up where p <= 1 for o(u) = up where p <= 1 Diff. Equation Form

16. Converting Differential Equations to CA Conditions: for n(u) = up where p <= 1 for o(u) = up where p <= 1 Diff. Equation Form

17. Converting Differential Equations to CA Conditions: for n(u) = up where p <= 1 for o(u) = up where p <= 1 Diff. Equation Form

18. Converting Differential Equations to CA Discretization Techniques

19. Converting Differential Equations to CA Large hx Size of hx Small hx

20. Converting Differential Equations to CA Forward Euler’s Method: Euler’s Methods

21. Converting Differential Equations to CA Size of ht

22. Converting Differential Equations to CA Backward Euler’s Method: Euler’s Methods

23. Converting Differential Equations to CA Forward Euler’s Method: Euler’s Methods Backward Euler’s Method:

24. Converting Differential Equations to CA Forward Euler’s Method: Euler’s Methods 1 2 3 4 5 i=1 j j-1 j+1 3.2 5.7 7.3 9.2 -7.5 i=2 j j-1 j+1

25. CA Theoretical Constraints General Linear Form

26. CA Theoretical Constraints Convergence and Divergence

27. CA Theoretical Constraints • Time Domain Frequency Domain • Discrete Form of Laplace Transform and related to the Fourier Transform • Transformation makes life easier • zeros when f(z)=0 poles when g(z)=0 Z-Transform

28. CA Theoretical Constraints Z-Transform

29. CA Theoretical Constraints 1. Perform z-transform 2. Solve for Uj 3. Find poles and zeros for Uj=f(z)/g(z) 4. Set poles and zeros values of z < 1 to converge Z-Transform

30. CA Theoretical Constraints Forward Euler’s Linear Form: Forward Euler’s Constraints Zeros Constraint:

31. CA Theoretical Constraints Forward Euler’s Linear Form: Forward Euler’s Constraints Poles Constraint:

32. CA Theoretical Constraints Backward Euler’s Linear Form: Backward Euler’s Constraints Zeros Constraint:

33. CA Theoretical Constraints Backward Euler’s Linear Form: Backward Euler’s Constraints Poles Constraint:

34. Convergence Maps & Guidelines 1 2 3 4 5 i=1 j j-1 j+1 1.1 1.9 2.8 2.6 5.4 i=2 ... j j-1 j+1 CA Sim 0.11 0.34 0.27 0.4 0.56 i=n-1 < 10-10 j j-1 j+1 0.1 0.35 0.27 0.4 0.57 i=n j j-1 j+1

35. Convergence Maps & Guidelines 1 2 3 4 5 i=1 j j-1 j+1 1.1 1.9 2.8 2.6 5.4 i=2 ... j j-1 j+1 CA Sim 1.2 872 927 -722 -256 i=n-1 > 1010 j j-1 j+1 541 -5623 -897 456 878 i=n j j-1 j+1

36. Convergence Maps & Guidelines 1 2 3 4 5 i=1 j j-1 j+1 1.1 1.9 2.8 2.6 5.4 i=2 ... j j-1 j+1 CA Sim 1 2.1 3.1 3.9 5 i=3999 j j-1 j+1 1.1 2.1 3 4 5.1 i=4000 j j-1 j+1

37. Convergence Maps & Guidelines Forward Convergence Map

38. Convergence Maps & Guidelines Backward Convergence Map

39. Convergence Maps & Guidelines a Parameters

40. Convergence Maps & Guidelines a1 a Parameters

41. Convergence Maps & Guidelines a2 a Parameters

42. Convergence Maps & Guidelines Poles Constraint: Forward Constraints

43. Convergence Maps & Guidelines Backward Constraints

44. Convergence Maps & Guidelines Simulation Speed

45. Convergence Maps & Guidelines a3 Vertical Constraint

46. Convergence Maps & Guidelines Zeros Constraint: a3 Vertical Constraint

47. Convergence Maps & Guidelines Substituting Uj-1 and Uj+1 • Boundary Zero Values 0 0.11 0.34 0.27 0.4 0.56 0 j j-1 j+1

48. Convergence Maps & Guidelines Zeros Boundary Constraint

49. Convergence Maps & Guidelines Zeros Boundary Constraint

50. Convergence Maps & Guidelines If ((upperZero and lowerPole intersects) and (intesection < initial point)) then htMax = intersection * safetyBuffer; Else htMax = initial point * safetyBuffer; End ht = userInput( < htMax); hx=lowerPole(ht); Guidelines