Solving Heat Equation with Interfaces Research
Explore the innovative research on solving the heat equation with interfaces. Discover applications in metallurgy, steel continuous casting, mathematical biology, cancer treatment, and ecological modeling. Dive into numerical treatments, temporal discretization, spatial discretization techniques, and numerical experiments with analytical solutions and error analysis.
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Presentation Transcript
WCUPA SOLVING THE HEAT EQUATION WITH INTERFACES RESEARCH AND CREATIVE ACTIVITY DAY APRIL 29, 2021 PRESENTER: MICHAEL BAUER WITH: REX LLEWELLYN, SHAUNA FRANK MENTORED BY: DR. CHUAN LI
Solving the Heat Equation with Interfaces ? diffusion coefficient u function of interest domain of interest interface zeroth jump condition first jump condition
Applications Metallurgy Steel Continuous Casting Mathematical Biology Cancer Treatment Ecological Modeling
Numerical Treatment
Temporal Discretization Euler Method 1st Order Accuracy
Spatial Discretizatio n At nodes adjacent to interface At regular nodes ~ ~
Spatial Discretizatio n Transformations Jump Conditions
Spatial Discretization ~ From we have ~ ~ From we have ~
Numerical Experiments ? ?= ( ?) + - in = + - + Jump Conditions ? diffusion coefficient u function of interest domain of interest interface function jump condition flux jump condition ? = ?+ ? = ?? = + ?+ ? ? = ?
Numerical Experiments We will look at the example where the analytical solution is given as With the following source terms
Numerical Experiment Result Error
Numerical Experiments We will look at the example where the analytical solution is given as With the following jump conditions
Numerical Experiment Result Error
Future Improvements Increase to 2D This will introduce additional complications at nodes near interfaces Increase complexity of Interfaces Utilize Peaceman-Rachford method Higher accuracy in temporal discretization Address corner cases with irregular interface geometries
Conclusion This exercise demonstrates the effectiveness of MIB in solving the Heat Equation with Interfaces Further improvements are required for applications to real- world tasks
