1. Creed Reilly, Sophomore, Engineering Advisor: Professor Anna Mazzucato Graduate Student: Yajie Zhang

2. Diffusion coefficient c jumps at x=1/2 (the interface). Impose transmission conditions at interface. Solve equation in [0,1]. Impose Dirichlet boundary conditions at x=0,1. Initial condition is sin(πx). General Heat Equation in 1 Dimension with Transmission Condition

3. Model Composite Materials:

4.  This is the simplest (explicit) first-order finite difference method to solve the heat equation.  First order Taylor expansion was used for the time derivative (U t )  The center-difference method was used for the second space derivative (U xx )  Because this is an explicit method, a convergence condition had to be observed:

6. C L =1C R =2Δx=0.1C L =1C R =2Δx=0.025

7. ΔxL2(1)Linf(1)L2(2)Linf(2)LogE 0.13.03E-094.51E-094.05E-096.02E-09-0.42206 0.054.05E-096.02E-093.29E-094.89E-090.302807 0.0253.29E-094.89E-092.09E-093.10E-090.65612 0.01252.09E-093.10E-091.17E-091.74E-090.829307 0.006251.17E-091.74E-096.23E-109.24E-100.915078 0.0031256.23E-109.24E-10No Mem N/A Table 1: L2 and L∞ error for various displacement steps Graph 1: Diffusion of energy when the left half has a C=1 and the right has a C=2. Graph 2: Diffusion of energy when the left half has a C=1 and the right has a C=100.