Illustrated Guide to Writing a Finite Difference Code for Heat Equation
Explore the step-by-step process of writing a Finite Difference code for solving dynamic heat conduction problems, also known as the diffusion equation. Follow along as we find the temperature distribution for a small problem using a structured grid, initial and boundary conditions, and interior equations involving multiple points.
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Presentation Transcript
2023 EESC W3400 Lec 22: Writing a FD code illustrated with 1D+time heat flow equation Computational Earth Science Bill Menke, Instructor Emily Glazer, Teaching Assistant TR 2:40 3:55
Today Finite Difference Method for Solving Dynamic Heat Conduction Problem a.k.a. the Diffusion Equation
Today I will lead you through writing a FD code assuming small problem that doesn t need sparse matrices
Goal: Find temperature, ? ?,? for ???? ? = 0 0 ? source ? ?,? ? = 0 ???? ? = 0 ? ?? ?? ?2? satisfied in the interior ??2= ?
Finite Difference grid ? ?? 1 0 ? 0 ? ?? 1 ? ?? ?? ?2? satisfied in the interior ??2= ?
initial condition 0 ? 1 ? ?,? = 0 = 0 ?
boundary condition 1 ? ? = 0, ? ? ???? = 0 0 ? ? 1 ? ? = ????, ? ? ???? = 0
interior ?2? ?? ?? ??2 = ? ? ?,? ? ?,? ? ? ? ? ?,? ? 2? ?,? ? + ? ? + ?,? ? x2 ?? ?,? ? x2? ? ?,? ? + = ? ?,? 1 1 ?? ?,? 2 x2? ?,? ? x2? ? + ?,? + ? = ? ?,?
interior ?2? ?? ?? ??2 = ? ? ?,? ? ?,? ? ? ? ? ?,? ? 2? ?,? ? + ? ? + ?,? ? x2 ?? ?,? ? x2? ? ?,? ? + = ? ?,? 1 1 ?? ?,? 2 x2? ?,? ? x2? ? + ?,? + ? = ? ?,?
interior 1 1 2 x2? ?,? ? x2? ? ?,? ? ?? ?,? + ?+ x2? ? ?,? + ? = ? ?,?
interior ? ?? 1 0 ? 0 every interior equation involves 4 points ? ?? 1 1 1 2 x2? ?,? ? x2? ? ?,? ? ? ?? ?,? + ?+ x2? ? ?,? + ? = ? ?,?
Step 1: define x axis 0 ? 2 ? ? = ?? 1 ?
Step 2: define t axis 0 ? 2 ? ? = ?? 1 ?
Step 3: make tables to unwrap temperature matrix to vector ?0 ?? ?? 1 ?00 ?0,?? 1 ???,?? ? = ? = ??? 1,0 ??? 1,?? 1 ? = ?? ??
Step 3: make tables to unwrap temperature matrix to vector ?0 ?? ?? 1 ?00 ?0,?? 1 ???,?? ? = ? = ??? 1,0 ??? 1,?? 1
Step 3: make tables to unwrap temperature matrix to vector ?0 ?? ?? 1 ?00 ?0,?? 1 ???,?? ? = ? = ??? 1,0 ??? 1,?? 1
Steps 4, 5, 6 and 7: set up ? and ??? for ?? = ??? ? initial condition ? left boundary condition ? rt boundary condition ? interior equation ? ???
Step 4: initial conditions ? initial condition ? ???
Step 5: boundary conditions ? left boundary condition ? rt boundary condition ? ???
Steps 6: construct source ? Gaussian in ? impulsive in ? ? ? ???
Step 7: interior equation ? interior equation ? ???
Step 9: construct ? given ? ?0 ?? ?? 1 ?00 ?0,?? 1 ???,?? ? = ? = ??? 1,0 ??? 1,?? 1
trick is to build code incrementally step-by-step and test steps along the way
