MATH 4810 Computational Modelling

Module 03 · Numerical PDEs

Numerical PDE Models

Extend finite differences from one dimension to grids in space and time. Build explicit schemes for the heat equation, understand the stability restriction that governs the time step, and interpret a full case study.

2 units · 6 lessons 2 laboratories Assignment 3 · 10% ≈ 20 hours

Module 03

Numerical PDE Models

Finite-difference grids, derivative approximations, boundary and initial conditions, and explicit schemes for diffusion and wave problems.

§ Overview

Where Module 2 stepped forward in time, this module steps forward in time and across space. You will construct finite-difference approximations, assemble them into an explicit update rule, and confront the stability restriction that limits the time step. The module closes with a full wave or diffusion case study.

Unit 3.1

Finite-difference foundations.

Unit 3.2

PDE simulation.

Laboratories 5–6

Finite-difference formulas and PDE simulation.

Assignment 3

Deterministic PDE model.

§ Learning outcomes

By the end of Module 3 you will be able to

  • Construct spatial and temporal grids and document the discretization they define.
  • Derive forward, backward, and central finite-difference approximations and state their error orders.
  • Implement an explicit finite-difference scheme for a parabolic or hyperbolic PDE.
  • State and apply the stability restriction on the time step.
  • Interpret a PDE simulation and identify where discretization error dominates.

3.1 Finite-Difference Foundations

A PDE is only well posed once its domain, boundary conditions, and initial conditions are fixed. This unit builds those foundations and the derivative approximations that sit on top of them.

Lesson 1

Spatial and temporal grids

Choose and document the discretization that defines the numerical problem.

 Lesson 2

Finite-difference derivative approximations

Forward, backward, and central differences, and their error orders.

 Lesson 3

Boundary and initial conditions

The conditions that determine whether a PDE problem is well posed.

3.2 PDE Simulation

Assemble the approximations into a working scheme. The explicit heat equation is the canonical example; the wave or diffusion case study shows the full workflow from formulation to interpretation.

Lesson 1

Explicit heat-equation schemes

Implement the FTCS scheme and read its output correctly.

 Lesson 2

Stability and the time-step restriction

The CFL-type condition and what happens when you violate it.

 Lesson 3

Wave or diffusion case study

A full worked simulation from formulation to interpretation.

§ Laboratories

Lab 5

Finite-difference formulas

Derive and verify forward, backward, and central difference formulas against known functions.

Lab 6

PDE simulation

Build a spreadsheet simulation of the heat equation and test it against the stability restriction.

§ Assignment 3

Deterministic PDE model · 10%

Formulate and implement a deterministic PDE model, justify the grid and time step, and present a numerical error and stability analysis of the simulation.

Full assignment brief →

§ Common pitfalls

Watch for these

  • A smaller time step is not automatically sufficient to guarantee a correct result. In PDE problems the spatial grid can dominate the error even when the time step is small.
  • Numerical convergence does not establish model validity. A stable, converged PDE solution may still be the solution to the wrong boundary-value problem.
  • A stable scheme is not automatically an accurate one. Stability is necessary for convergence, not sufficient for a physically meaningful result.
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