a numerical method to approximate solutions to the ......a numerical method to approximate solutions...

A numerical method to approximate solutions to thegeneralized diffusion equation with moving level sets

Ben Barr, Ryan Cinoman, and Gaby Martinez

Mentored by: Marianne Korten

SUMaR 2015 at Kansas State University

July 21st, 2015

Math REU funded by the NSF

Ben Barr, Ryan Cinoman, and Gaby Martinez Mentored by: Marianne Korten (SUMaR 2015 at Kansas State University)A numerical method to approximate solutions to the generalized diffusion equation with moving level setsJuly 21st, 2015 1 / 23

Introduction

Goal: To track the evolution of solutions to the generalizeddiffusion equation, ut = α(u)xx, through discretized level setanalysis

This equation is well-posed when the function α is non-decreasing,α(0) = 0, and u is made up of smooth, compactly supported initialdata.

Can be used to model temperature-dependent phase changes, energyflow, chemical concentration, population dynamics, etc.


Tracking Free Boundary Movement Analytically

Free Boundaries: Result from a conservative function α havingdiscontinuities or regions of zero-derivative within its domain

Velocity of free boundary movement in piecewise smooth solutionsover time can analytically be determined using the Rankine-Hugoniot,or jump condition:

x ′(t) =[−~∇α(u)]

[u]=

[−∂α(u)∂x ][u]


Single Jump Example

Figure: Left: Plot of α(u) = h(H(u − j)) ; Right: Plot of uI (x)

The solution has initial data as follows

uI (x) =

{0 x < jh x > j


Single Jump Example

Because α(u(x , t)) is discontinuous when u = j , it must be re-defined as apiecewise linear function U(x , t), such that Uxx = 0 and ut = 0, thatmodels the jump from α(u) = 0 to α(u) = h at u = j

U(x , t) =

0 x < s(t)

h( x−s(t)d(t)−s(t) ) s(t) ≤ x ≤ d(t)h x > d(t)

where s(t) and d(t) represent the left and right free boundariesrespectively. The speed of advance of these free boundaries is determinedby the jump condition:

s ′(t) =−h

d(t)− s(t)1

j − uI (s(t))

d ′(t) =h

d(t)− s(t)1

j − uI (d(t)).


Step Functions

What was “nice” about our Heaviside step function example?

Initial data composed of constants (easy to compute [u])

Piecewise linear α(u) function (explicit solutions to Uxx = 0)

s ′(t) and d ′(t) could be computed explicitly


What about examples with more complex initial data?

Figure: Plot of uI (x) = x3


Discretizing our initial data by level sets

What we have:

What we want:


Discretizing our initial data by level sets

Want: Discretized Initial Data, unI , which approximates uI (x)uniformly

A convergent series of step functions constructed from uI (x)

Allows for [u] to be computed between any two consecutive steps


Setting up our algorithm

Once our data is discretized, we can track the evolution of theboundaries of our steps (level sets) over time.

Successful Discretization = Piecewise Linear U(x , t)(re-parameterization of α(u))

Evolution of some free boundary, xi (t), can be modeled by:

x ′i (t) =− [∆U]

[u]=

−1ui − ui−1

(Ui+1 − Ui

xi+1(t)− xi (t)− Ui − Ui−1

xi (t)− xi−1(t)

)Use fourth-order Runge-Kutta algorithm to numerically track theevolution of all discretized free boundaries at a certain time, t.


Setting up our algorithm

If uI (x) is non-monotone, then the level sets of local extrema ofu(x , t) will shrink and reach zero in finite time.

Free boundary condition breaks down and becomes undefined when twofree boundaries meet

Must consider u(x , t0), at time t0 when two free boundaries meet, as anew set of initial data

Ignore the extreme level set and drop both free boundaries from data


Example: uI = sin2(x)

Figure: Diffusion of uI = sin2(x) with α(u) = u2


Convergence

Do our calculated solutions un(x, t) converge as n goes to ∞?

Not true for general α (i.e. α(u) = −u)Equation is not well-posed, non-unique solutions

Unique solutions have been shown to exist for several forwardequations

Standard heat equation (α(u) = u), porous media equation(α(u) = um, form > 1), Stefan Problem (α(u) = (u − 1)+), etc.

Discretized initial data converges in norm to uI, so the solutionsun(x, t) must converge to the unique solution of the equation ifit exists


Building an n-dimensional model

Can our level set discretization method be applied to higher spatialdimensions?

Yes it can! But to do so, we must re-parameterize α into a piecewisesmooth harmonic function where grad(α) is known at all boundaries.

One Dimension: Re-parametization of α (U(x , t)) is piecewise linear

Multiple Dimensions: Difficult to find explicit n-dimensional functionsthat satisfy these re-parameterization conditions


Building an n-dimensional model

Focus on radially symmetric data that is a function of one variable, r

Figure: Discretized radially symmetric initial data uI (x) = sin2(x)

Note: n-dimensional ”plane-like” data can also be modeled using our levelset discretization method


Radially Symmetric Solutions in n-dimensions

In spherical coordinates, the Laplacian operater acts on symmetricfunctions as

∆ f =1

rn−1∂

∂r

(rn−1

∂f

∂r

).

In between each free boundary, ut = 0, so we reparameterize U ∈ α(r , t)so that it is piecewise harmonic when uI is constant and in each region ofconstancy ,

∆U =1

rn−1∂

∂r

(rn−1

∂U

∂r

)= 0.


Radially Symmetric Evolution


sinrad.gifMedia File (image/gif)

Radially Symmetric Solutions in n-dimensions

Now we have reduced the problem to a first order ODE, which we canfurther reduce to

∂U

∂r= c0r

1−n

where c0 is a constant. In R2, this can be integrated to get

U(r , t) = c0 ln|r |+ c1

where c0 and c1 are constants determined by the initial position of the freeboundaries. For n > 3, the formula becomes

U(r , t) = c0r2−n + c1


Forward/Backward Phase

α(u) =

u + 1 u < −12−u −12 ≤ u ≤

12

u − 1 u > 12


Backward Phase Characteristics

Decreasing portion of α (when −12 ≤ u ≤12 )

Recall: diffusion equation is ill-posed for decreasing α functions

Models “reverse diffusion”

Agglutination can be expected within the backward phase of diffusion

Uniqueness?

Level set interaction?


Backward Phase Exploration

Figure: Backward Evolution of uI (x) = sin(x) with α(u) = −u

Decreasing α function, non-unique solutions to ut = uxx

The solution stops evolving after some finite time, T > 0.


Forward/Backward Phase

Figure: Forward/backward evolution of uI = sin2(x)

The backwards region stops evolving quickly.

Then, the solution evolves as if it were strictly a forward-equation.


References

J.E. Bouillet and J.I. Etcheverry. Numerical Experiments withut = α(u)xx . Rev. Un. Mat. Argent. 41, No. 1, 15-25 (1998).

J.E. Bouillet, M.K. Korten, and V. Marquez. Singular limits and themesa problem, Rev. Un. Mat. Argentina 41 (1998), 27-39.

J. Cheverie, L. Fosque, and M.K. Korten. Numerical Experiments witha Level-Set Tracking Algorithm for a Generalized Diffusion Equation.SUM@R 2014.


a numerical method to approximate solutions to the ......a numerical method to approximate solutions...

Documents