MATH 4810Computational Modelling

Module 03 · Capstone Project

A Verified PDE Simulation

Build a complete finite-difference simulation of a one-dimensional PDE. Verify against an analytical solution, demonstrate the stability restriction, and report the results with full documentation.

Weight Counts toward the module component of the course grade Duration ≈ 14 hours Prerequisite Units 3.1 and 3.2

Module 3 capstone

A Verified PDE Simulation

From grids and finite differences to a full simulation with stability analysis and verification.

§ Project brief

The Module 3 capstone is a complete finite-difference simulation of a one-dimensional partial differential equation. You will choose a problem, specify the domain and boundary conditions, implement the FTCS scheme, verify against an analytical solution on a tractable case, demonstrate the CFL stability restriction, and report the results with a claim-support statement.

The capstone draws on every skill of the module: grid construction, finite-difference stencils, boundary-condition handling, explicit scheme implementation, stability analysis, and convergence study.

Verification is the standard

The measure of success is not the visual appearance of the solution but the quantitative agreement with the analytical reference and the correct scaling of the error.

§ Learning outcomes addressed

This capstone assesses

  • Construct a spatial and temporal grid for a PDE problem.
  • Derive and implement finite-difference stencils.
  • Handle Dirichlet, Neumann, and Robin boundary conditions.
  • Implement the FTCS scheme for the heat equation.
  • Verify against an analytical solution and confirm the observed order.
  • Demonstrate the CFL stability restriction.
  • Report the results with a claim-support statement.

§ Suggested problems

Problem A

Heat equation with sine initial condition

u_t = α u_xx on [0, 1], u(0,t) = u(1,t) = 0, u(x,0) = sin(πx). Analytical solution available.

Problem B

Rod with one insulated end

Same heat equation, but with a homogeneous Neumann condition at x = 1. Requires handling a boundary condition numerically.

Problem C

Diffusion with Gaussian initial condition

u(x,0) = exp(−100(x − 0.5)²). Analytical solution available. Tests the grid's ability to resolve a narrow feature.

Problem D

Wave equation with reflecting ends

u_tt = c² u_xx. Requires two initial conditions. Different stability condition from the heat equation.

§ Deliverables

Deliverables and weights
Deliverable Weight What it should contain
Problem specification 10% PDE, domain, initial condition, boundary conditions, physical interpretation.
Grid and stencil design 15% Grid dimensions, spacing, diffusion number, chosen time step, and justification.
FTCS implementation 20% Working spreadsheet or code with the update rule and boundary handling documented.
Verification and convergence 25% Comparison against the analytical solution. Observed order at three grid refinements.
Stability demonstration 15% Numerical evidence of the CFL condition. Instability at r > 0.5 shown and interpreted.
Technical report 15% Three to five pages covering specification, implementation, verification, stability, and limitations.

§ Milestones

  1. Choose the problem and state it precisely PDE, domain, initial condition, boundary conditions. One hour.
  2. Design the grid Choose Δx, Δt, r, and document the stability margin. One hour.
  3. Implement the FTCS scheme Three-layer spreadsheet or Python code with documented boundary handling. Four hours.
  4. Verify against the analytical solution Error table at three grid resolutions. Confirm the observed order. Three hours.
  5. Demonstrate stability Run at r = 0.4 and r = 0.6; plot and interpret the difference. Two hours.
  6. Write the report Include the claim-support statement and the limitations. Three hours.

§ Rubric

Assessment criteria
Criterion Excellent Satisfactory Needs work
Specification Complete, precise, physically interpreted. Complete but terse. Missing one or more elements.
Grid design Stability margin computed; grid resolution justified. Grid dimensions stated; stability checked. Stability not checked or grid unjustified.
Implementation Clean, correct, boundary handling documented. Correct but boundaries handled informally. Boundary handling incorrect.
Verification Observed order within 10% of theory; error table complete. Observed order within 25%. Observed order not reported or does not match.
Stability Numerical demonstration at two diffusion numbers with interpretation. One demonstration without interpretation. Stability not demonstrated.
Report Complete, self-contained, with limitations. Complete but missing limitations. Incomplete or overclaims.

§ Common pitfalls

Watch for these

  • Off-by-one in grid dimensions. A domain divided into N intervals has N+1 grid points.
  • Applying the interior stencil at boundary points. Boundary values are imposed, not computed.
  • Assuming stability without checking r. Compute the diffusion number before every run.
  • Claiming convergence without matching the theoretical order. The observed order is the evidence; a visual match is not.
  • Confusing numerical error with model error. A verified scheme does not validate the PDE.

§ Submission

What to submit

  • A single ZIP archive named M3-capstone-yourid.zip.
  • The archive contains the spreadsheet or code, the analytical verification table, the convergence study, the stability demonstration, and the technical report.
  • Submit through the course LMS before the deadline listed on the course schedule.
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