Advisor Professor Anna Mazzucato Graduate Student Yajie Zhang Solving a Transmission Problem for the 1D Diffusion Equation Transmission Problem for the 1D Heat Equation Diffusion coefficient c jumps at x12 the interface Impose transmission conditions at interface Solve equation in ID: 799260
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Slide1
Creed Reilly, Sophomore, EngineeringAdvisor: Professor Anna MazzucatoGraduate Student: Yajie Zhang
Solving a Transmission Problem for the 1D Diffusion Equation
Slide2Transmission Problem for the 1D Heat EquationDiffusion 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
Slide3Our Project has Major Real World Applications
Model Composite Materials:
Slide4This 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
Slide5The General Equation was then DiscretizedDiscretization of the Exact Solution
:
Discretization of the Exact Solution:
Slide6The 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 BeneficialCL=1 CR=2 Δx=0.1CL=1 CR=2 Δx=0.025
Slide7Graphs and Tables
Δx
L2(1)
Linf(1)
L2(2)Linf(2)LogE0.13.03E-094.51E-094.05E-096.02E-09-0.422060.054.05E-096.02E-093.29E-094.89E-090.3028070.0253.29E-094.89E-092.09E-093.10E-09
0.656120.01252.09E-093.10E-091.17E-091.74E-090.829307
0.006251.17E-09
1.74E-09
6.23E-10
9.24E-10
0.915078
0.003125
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.