1. NUMERIC SOLUTIONS OF THERMAL PROBLEMS GOVERNED BY FRACTIONAL DIFFUSION V.R. Voller, D.P Zielinski Department of Civil Engineering, University of Minnesota, Minneapolis, MN 55455 volle001@umn.eduvolle001@umn.edu, ziel0064@umn.edu Objective: Develop approximate solutions for the problem Where the flux is modeled as a fractional derivative e.g., Fraction –locality Skew An appropriate model when length-scales of heterogeneities are power-law distributed – e.g., fractal distribution of conductivity

2. n n-1 --- 3 2 w 1 e 0 local flux 1 st up-stream face gradient First start by defining the basic LOCAL FLUX via Finite Differrences Create Finite Difference Scheme from flux balance 0 0 0 x

3. n n-1 --- 3 2 w 1 e 0 non-local flux Weighted average of all up-stream face gradients Now define a NON-LOCAL FLUX Create Finite Difference Scheme from flux balance 0 The Control Volume Weighted Flux Scheme CVWFS

4. n n-1 --- 3 2 w 1 e 0 non-local flux Weighted average of all up-stream face gradients What's the Big Deal !! If we chose the power-law weights where locality In limit can be shown that The left-hand Caputo fractional derivative

5. n n-1 --- 3 2 w 1 e 0 So with appropriate choice of weights W We have a scheme for fractional derivative Can generalize for right-derivative And Multi-Dimensions

6. Alternative Monte-Carlo—domain shifting random walk Consider-domain with Dirichlet conditions (T_red and T_blue)—objective find value T_P Approach move (shift) centroid of domain by using steps picked from a suitable pdf PP Until domain crosses point P Then increment boundary counter (blue in case shown) And start over After n>>1 realizations—Value at point P can be approximated as Note this is the right-hand Levy distribution—fat tail on right associated with left hand Caputo fractional derivative

7. Results: First a simple 1-D problem CVWFS domain shift integer sol. x = 0 1

8. Testing of Alternative weighting schemes CVWFS—Voller, Paola, Zielinski, 2011 Classic Grünwald Weights (GW) L1/L2 Weights: e.g., Yang and Turner, 2011 Relative Error L1/L2 G.W CVWFS 0 -0.03 x = 0 1

9. And a 2-D problem 1 0,0 T=0 T=1 domain shift CVWFS

10. SO: 1. Fractional Diffusion -a non-local model appropriate in some heterogeneous media 2. Can be numerically modeled using a weighted non-local flux 3. Or with a domain shifting Random walk P 4. Gives accurate and consistent solutions 5. Approach Can and Has been extended to transient case 6. Work is on-going for a FEM implementations of the CVWFS