School of Mathematics · TrustOpen
Build, implement, test, and defend computational models of real systems. Work through numerical methods for differential equations, Monte Carlo and discrete-event simulation, Markov chains, and linear programming with duality and sensitivity analysis — then integrate everything in a substantial simulation capstone.
Course description
MATH 4810 develops advanced methods for constructing, implementing, analyzing, and validating computational models.
Students study numerical methods for ordinary and partial differential equations, Monte Carlo simulation, stochastic state-transition models, discrete-event simulation, linear programming, duality, and sensitivity analysis. Models are implemented in spreadsheet software and, optionally, in a scientific programming environment.
Applications are drawn from the physical sciences, social sciences, finance, medicine, operations, and business. The course emphasizes reproducible model architecture, numerical stability, uncertainty analysis, verification, validation, and communication.
Each topic is divided into conceptual analysis and numerical simulation, with Microsoft Excel used to construct and implement models. Optional Python, computer algebra, discrete-event simulation, and optimization tools are supported for students who wish to extend their work.
Formal prerequisites: MATH 265, MATH 266, MATH 270, MATH 370, and MATH 376. There are no preclusions, and the course is not available for challenge credit.
Recommended preparation: single-variable calculus, linear algebra, real analysis, ordinary differential equations, probability, spreadsheet modelling, and basic optimization.
Access to the learning management system, Microsoft Excel 2010 or later (or an institutionally approved equivalent), a scientific calculator, and accessible equation rendering. A Python notebook environment, computer algebra system, discrete-event simulation software, or mathematical optimization software is optional.
Learning outcomes
Upon successful completion of MATH 4810, students will be able to:
Recurring process
Every module returns to this cycle. It is the backbone of the laboratories, the assignments, and the capstone.
Curriculum
Each unit pairs conceptual analysis with numerical simulation. Modules 1 through 6 carry a staged assignment; Module 7 supports the integrative capstone.
Model types, the modelling lifecycle, spreadsheet architecture, parameter estimation, and the distinction between verification and validation.
Discretization, Euler and modified Euler methods, fourth-order Runge–Kutta, local and global error, step-size selection, and stiffness awareness.
Spatial and temporal grids, finite-difference derivative approximations, boundary and initial conditions, explicit heat-equation schemes, and the time-step stability restriction.
Random experiments and estimators, sampling error and confidence intervals, pseudorandom generation, input transformation, replication, and variance reduction.
States and transition probabilities, steady-state behaviour, entities and queues, event calendars and clock advancement, and warm-up bias in output analysis.
Decision variables, objectives and constraints, spreadsheet solver use, the fundamental duality theorem, shadow prices, reduced costs, and sensitivity ranges.
Verification strategies, validation evidence, robustness analysis, scenario design, experimental planning, and technical and non-technical communication.
Assessment
Assignments build cumulatively across the modules. The capstone carries the greatest weight because it demonstrates integrated modelling competence end to end.
One applied modelling assignment per module across Modules 1–5, each worth 10 percent.
A substantial computational model, verified, validated, analyzed, and reported.
An accessible presentation of the model, its results, and its limitations.
| Assessment | Weight |
|---|---|
| Assignment 1: Model Architecture and Parameterization | 10% |
| Assignment 2: ODE Numerical Simulation | 10% |
| Assignment 3: PDE Finite-Difference Model | 10% |
| Assignment 4: Monte Carlo Simulation | 10% |
| Assignment 5: Markov, Discrete-Event, or Optimization Model | 10% |
| Simulation Capstone | 45% |
| Capstone Presentation | 5% |
| Total | 100% |
Students must achieve at least 50 percent overall, submit all five assignments, achieve at least 50 percent on the simulation capstone, submit functioning model files, submit the capstone presentation, and meet academic-integrity requirements.
Culminating work
Students construct, verify, validate, analyze, and communicate a substantial computational model of their choosing, subject to approval.
| Criterion | Weight |
|---|---|
| Problem definition and conceptual model | 8% |
| Mathematical formulation | 12% |
| Method selection | 10% |
| Implementation quality | 15% |
| Verification and numerical testing | 12% |
| Validation | 10% |
| Sensitivity and uncertainty analysis | 10% |
| Results and interpretation | 8% |
| Reproducibility and documentation | 8% |
| Communication and accessibility | 7% |
| Total | 100% |
Learning design
Every lesson follows the same structure so that theory, implementation, and judgement stay connected.
Why this method exists and where it is used.
What you should be able to do afterwards.
The mathematics you need before starting.
The model or method stated precisely.
How the formulation becomes an algorithm.
How the implementation is laid out.
A complete, runnable example.
Evidence that the implementation is correct.
What the output means in context.
Where the result could break down.
Mistakes to watch for.
Your turn to build and test.
Consolidate what changed in your thinking.
The key takeaways in brief.
Accessibility and responsible computing
Course production and student work both carry obligations around accessibility, transparency, and the ethical use of models.
Model files submitted for this course must be runnable by another person. That means documented inputs, labelled outputs, stated units, recorded seeds and step sizes, and no hidden manual steps.
Questions
No. The course is built around spreadsheet implementation, and Microsoft Excel 2010 or later (or an institutionally approved equivalent) is the required technology. A Python notebook environment, computer algebra system, discrete-event simulation software, or optimization package is optional for students who want to extend their work.
The formal prerequisites are MATH 265, MATH 266, MATH 270, MATH 370, and MATH 376. In practice you should be comfortable with single-variable calculus, linear algebra, real analysis, ordinary differential equations, and probability.
Individualized study online, asynchronously, over approximately 18 weeks. The estimated total workload is 135 hours, or roughly 7 to 9 hours per week.
No. Challenge credit is not available for MATH 4810, and there are no preclusions.
Required: the learning management system, Microsoft Excel 2010 or later (or an approved equivalent), a scientific calculator, and accessible equation rendering.
Optional: a Python notebook environment, a computer algebra system, discrete-event simulation software, and mathematical optimization software.
A substantial computational model of your own choosing, subject to approval. You define the problem, formulate the model, select and implement a method, verify and validate it, run sensitivity and uncertainty analyses, and communicate the results in a technical report, a plain-language executive summary, and an accessible presentation.
Achieve at least 50 percent overall, submit all five assignments, achieve at least 50 percent on the capstone, submit functioning model files, deliver the capstone presentation, and meet academic-integrity requirements.
Begin with the orientation to review course structure, technology setup, assessment expectations, and the modelling cycle that runs through every module.