FleCSI Training Material: 2-D Heat Equation Solver

This repository contains hands-on training material used in the FleCSI training.
The exercises are based on a 2-D heat equation solver that is incrementally extended as new concepts are introduced.

The implementation uses a structured Cartesian grid, implicit Backward Euler time integration, and a Red–Black Gauss–Seidel (RBGS) iterative solver.
The code is written for clarity and instructional value.

---

Repository Layout

provided/
src/
solution_1/
solution_2/
solution_3/
solution_4/
solution_5/

• provided/
  Material supplied to reduce boilerplate and setup complexity.

• src/
  Primary working directory for the exercises.
  Contains source files, headers, CMakeLists.txt, and the build directory.
  Participants actively modify this directory during the training.

• solution_X/
  Reference solutions corresponding to the end of each module.
  solution_1 matches Module 1, solution_2 matches Module 2, etc.
  Each solution builds independently and reflects the expected code state for that module.

---

Problem Description

The application solves the 2-D heat equation on a uniform Cartesian mesh:

∂u/∂t = α ∇²u + s(x, y, t)

Domain:

(x, y) ∈ [0, Lx] × [0, Ly]

Features:

• Fixed Dirichlet boundary conditions on all sides
• Gaussian initial temperature distribution
• Optional source term s(x,y,t)
• Spatial discretization via five-point finite-difference Laplacian
• Time integration using implicit Backward Euler
• Linear system solved with Red–Black Gauss–Seidel (RBGS)
• Convergence monitored via update norms and residuals

An analytical solution for the Gaussian initial condition is used for verification and error analysis.

---

Numerical Method

Time Discretization

Backward Euler form:

(I − α Δt ∇²h) uⁿ⁺¹ = uⁿ + Δt sⁿ⁺¹

---

Spatial Discretization (Five-Point Stencil)

On a uniform grid:

(∇²u)ᵢⱼ ≈
  (uᵢ₊₁ⱼ − 2uᵢⱼ + uᵢ₋₁ⱼ) / Δx²
+ (uᵢⱼ₊₁ − 2uᵢⱼ + uᵢⱼ₋₁) / Δy²

The resulting linear system for each interior node:

aP uᵢⱼ − aE uᵢ₊₁ⱼ − aW uᵢ₋₁ⱼ − aN uᵢⱼ₊₁ − aS uᵢⱼ₋₁ = bᵢⱼ

Coefficients:

aE = aW = α Δt / Δx²
aN = aS = α Δt / Δy²
aP = 1 + 2(aE + aN)
bᵢⱼ = uᵢⱼⁿ + Δt sᵢⱼⁿ⁺¹

---

Solver: Red–Black Gauss–Seidel (RBGS)

• Grid colored as a checkerboard

  • Red: (i + j) even
  • Black: (i + j) odd

• Iteration procedure:

  1. Red sweep: update all red nodes using black neighbors
  2. Black sweep: update all black nodes using updated red values
  3. Re-apply Dirichlet boundary conditions after each sweep

Update formula:

uᵢⱼ(new) =
  (1 − ω) uᵢⱼ(old)
+ ω ( bᵢⱼ
      + aE uᵢ₊₁ⱼ
      + aW uᵢ₋₁ⱼ
      + aN uᵢⱼ₊₁
      + aS uᵢⱼ₋₁ ) / aP

ω is the relaxation factor, typically 0.7–1.0.

---

Source Organization (src/)

Initial layout inside src/:

CMakeLists.txt        Build configuration
main.cc               Program entry point
state.hh              Mesh, parameters, and state storage
physics.hh            Discretization, BCs, RBGS solver
advance.hh            Time integration loop
io.hh                 Output and diagnostics
visualization/viz.py  Python visualization script

Exact contents evolve across training modules.

---

Outputs

• CSV snapshots of the temperature field:

  u_000000.csv
  u_000100.csv
  …
  u_final.csv
  

• Python-based visualization (PNG, GIF, MP4)
• Relative L2 error against the analytical solution

---

Parameters

Parameter	Meaning	Typical Default
Nx, Ny	Grid resolution	101 × 101
Lx, Ly	Domain size	1.0 × 1.0
alpha	Thermal diffusivity	1e-3
dt	Time step	1e-4
t_final	Final simulation time	0.1
implicit_tol	Solver tolerance	1e-8
jacobi_omega	RBGS relaxation factor	0.8

---

Visualization

A Python script in visualization/ renders snapshots and animations.

• PNG frames
• GIF animations
• MP4 videos (if ffmpeg is available)

---

Notes

• The code compiles at every module boundary.
• Full execution is intentionally incomplete in Module 3.
• Reference solutions allow rapid recovery and comparison.
• Emphasis is on numerical methods, data layout, and FleCSI concepts rather than performance tuning.