Creed Reilly, Sophomore, Engineering

Advisor: Professor Anna MazzucatoGraduate Student: Yajie Zhang

Solving a Transmission Problem for the 1D Diffusion Equation

Transmission Problem for the 1D Heat Equation

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

Our Project has Major Real World ApplicationsModel Composite Materials:

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 (Ut) The center-difference method was used for the second space derivative (Uxx) Because this is an explicit method, a convergence condition had to be observed:

The Method Used was Forward Euler’s

The General Equation was then Discretized

Discretization of the Exact Solution:

Discretization of the Exact Solution:

• The L2 Error Calculation is shown below:

• The program used is seen to converge as long as the L2 error decreases as the displacement step decreases.

• Since the error change is on a logarithmic scale, the equation should approach the value α of approximately 1.

The L2 Error Calculation Proved Most Beneficial

CL=1 CR=2 Δx=0.1 CL=1 CR=2 Δx=0.025

Graphs and TablesΔx L2(1) Linf(1) L2(2) Linf(2) LogE

0.1 3.03E-09 4.51E-09 4.05E-09 6.02E-09 -0.422060.05 4.05E-09 6.02E-09 3.29E-09 4.89E-09 0.302807

0.025 3.29E-09 4.89E-09 2.09E-09 3.10E-09 0.656120.0125 2.09E-09 3.10E-09 1.17E-09 1.74E-09 0.829307

0.00625 1.17E-09 1.74E-09 6.23E-10 9.24E-10 0.9150780.003125 6.23E-10 9.24E-10 No Mem No 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.