Numeric Solutions of Thermal Problems Governed by Fractional Diffusion

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.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

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

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

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

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

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

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

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

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

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

Numeric Solutions of Thermal Problems Governed by Fractional Diffusion