Numerical simulation
A calculated temperature has more digits than a thermometer. Which digits should you believe? This book follows that question from one decaying state to a spatial field, then compares two ways of approximating the same equation.
You can arrive directly from your favorite subject. A cooling body, an RC circuit, and a relaxing mechanical system all introduce time scales; heat conduction and elasticity both lead to spatial systems. The examples here are small enough that we can predict their behavior before computing it.
Choose a starting question
Section titled “Choose a starting question”| Question | Lesson | Mathematics used |
|---|---|---|
| My solver stopped. Is the answer accurate? | Errors and residuals | Vectors, matrix multiplication, norms |
| Why can a decaying system produce oscillations? | Time integration | Derivatives, exponentials, geometric sequences |
| How does a differential equation become a matrix? | Weak forms and finite elements | Integration by parts, piecewise linear functions |
| How does a mesh conserve a quantity? | Finite-volume balance | Flux, integration, outward normals |
| What does a refinement study actually show? | Refinement and reproducibility | Integrals, logarithms, error ratios |
Read in this order for a connected introduction. For physical context, start with heat transfer and return here when you want to understand the numerical choices. The short modeling foundations explain quantities, relations, and boundary conditions when you need them.
Two experiments, one habit
Section titled “Two experiments, one habit”The time experiment uses the same decay.eqi as Get started.
The spatial experiment uses a sine-shaped Poisson solution, first on an
interval and then conceptually on a square. Each lesson follows the same habit:
state a prediction, express the mathematics, compute, and try to disprove the
interpretation with a controlled change.
Follow Get started for the Python installation. Keep its eqiora-source
checkout: the Poisson comparison runs its maintained Cargo example from that
directory. The source revision attached to these pages supplies the models and
programs together.
The goal is a useful explanation of an answer: which equation it approximates, where it is sampled, what changes under refinement, and which physical assumptions still need examination.