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

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

Introduction & Background Cellular Automata (CA) 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.

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

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

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

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

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

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

Introduction & Background Celluar Automata Simulation

Introduction & Background Celluar Automata Simulation

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

Introduction & Background Vegetation Patterns

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

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

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

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

Converting Differential Equations to CA Discretization Techniques

Converting Differential Equations to CA Size of h x Large h x Small h x

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

Converting Differential Equations to CA Size of h t

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

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

Converting Differential Equations to CA Euler’s Methods Forward Euler’s Method: i=1 j j-1 j i=2 j j-1 j+1

CA Theoretical Constraints General Linear Form

CA Theoretical Constraints Convergence and Divergence

CA Theoretical Constraints Z-Transform 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

CA Theoretical Constraints Z-Transform

CA Theoretical Constraints Z-Transform 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

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

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

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

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

Convergence Maps & Guidelines CA Sim i=1 j j-1 j i=2 j j-1 j i=n j j-1 j i=n-1 j j-1 j+1... <

Convergence Maps & Guidelines CA Sim i=1 j j-1 j i=2 j j-1 j i=n j j-1 j i=n-1 j j-1 j+1... > 10 10

Convergence Maps & Guidelines CA Sim i=1 j j-1 j i=2 j j-1 j i=4000 j j-1 j i=3999 j j-1 j+1...

Convergence Maps & Guidelines Forward Convergence Map

Convergence Maps & Guidelines Backward Convergence Map

Convergence Maps & Guidelines a Parameters

Convergence Maps & Guidelines a Parameters a1a1

Convergence Maps & Guidelines a Parameters a2a2

Convergence Maps & Guidelines Forward Constraints Poles Constraint:

Convergence Maps & Guidelines Backward Constraints

Convergence Maps & Guidelines Simulation Speed

Convergence Maps & Guidelines a 3 Vertical Constraint

Convergence Maps & Guidelines a 3 Vertical Constraint Zeros Constraint:

Convergence Maps & Guidelines Substituting U j-1 and U j+1 Boundary Zero Values j j-1 j+1 00

Convergence Maps & Guidelines Zeros Boundary Constraint

Convergence Maps & Guidelines Zeros Boundary Constraint

Convergence Maps & Guidelines 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);

Convergence Maps & Guidelines Guidelines Example

Conclusion Partial Diff -> CA

Conclusion Theoretical Constraints Zeros Constraint: Poles Constraint:

Conclusion 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);

Conclusion Future Work Proofs of Observations Quadratic General Form: Efficient Parallelization Simulation Error

Conclusion References Paul Rochester. Euler's Numerical Method for Solving Differential Equations. November Region of Convergence. Wikipedia. November transform Keith Schubert. Cellular automaton for bioverms, October Jane Curnutt, Ernesto Gomez, and Keith Evan Schubert. Patterned growth in extreme environments Cell Image - Martin Gardner. The fantastic combinations of john conway’s new solitaire game ”life”. Scientific American, (223):120–123, T.A. Burton, editor. Modeling and Differential Equations in Biology. Pure and Applied Mathematics. Marcel Dekker Inc., J. von Hardenberg, E. Meron, M. Shachak, and Y. Zarmi1. Diversity of vegetation patterns and desertification. Physical Review Letters, 87(19), November 2001.

Conclusion Acknowledgements and Questions?